Malliavin calculus and densities for singular stochastic partial differential equations

Abstract

We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron–Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron–Martin norm. In particular, in many interesting situations we obtain a convergence and stability result for lifts of $L^2$-functions to models, which is of independent interest. The proof also involves two separate algebraic extensions of the regularity structure which are carried out in rather large generality.

1 Introduction

We establish Malliavin differentiability and subsequently study the existence of densities (with respect to Lebesgue measure) of finite dimensional projections of solution to singular stochastic partial differential equations (SPDEs). The equations we have in mind are formally given by systems of the form

(1)

where each component $u_i$ is in general a distribution on ${{\textbf {R}}}\times {{\textbf {T}}}^d$ for some $d \ge 1$, subject to some initial condition $u_i(0,\cdot )=u_{i,0}$. Here, is an elliptic differential operator involving only spatial derivatives, the functions $F_i$ and $F_i^j$ are smooth and allowed to depend on $u=(u_i)_{i\le m}$ and finitely many derivatives of u, and the random fields $\xi _j$, $j\le m$, are assumed to be jointly Gaussian.

Equations of type (1) have been subject to intensive study in recent years and lead to the development of novel technical approaches [15, 17, 31]. While these approaches differ in their scope and technical details, in situations where more then one of them can be applied, they lead to the same notion of solution. For the purpose of this paper we focus on the theory of regularity structures, originally developed in [17], and subsequently extended and generalized in a series of papers [4, 5, 10], see also [21]. Interesting examples that fall under this setting include the generalized KPZ equation [12, 19, 33]

$$\begin{aligned} \partial _t u_i = \partial _x^2 u_i + \sum _{k,l\le m} f_{k,l}^i(u) (\partial _x u_k) (\partial _x u_l) + \sum _{k \le n} g_k(u) \xi _k \end{aligned}$$

(2)

in $1+1$ dimensions, the $\Phi ^p_d$ [4, 17, 26, 34]equations

$$\begin{aligned} \partial _t u = \Delta u + \sum _{k < p} a_k u^k + \xi \end{aligned}$$

(3)

in $1+d$ dimensions for $d\le 3$ and the generalized PAM equation [15, 17]

$$\begin{aligned} \partial _t u = \Delta u + f(u) + \sum _{i,j \le d} f_{i,j}(u) (\partial _i u)(\partial _j u) + g(u)\xi . \end{aligned}$$

(4)

in $d=2$ or $d=3$ dimensions. Choosing $\xi$ as white noise, which is the natural choice in these examples, all of these equations have in common that there does not exist a solution in the classical sense. The robust solution theory of [4, 5, 10, 17] instead considers approximate, renormalized equations that take the form

(5)

subject to some initial condition $u^\varepsilon _{i}(0,\cdot )=u^\varepsilon _{i,0}$, where $\xi ^\varepsilon _j = \xi _j * \rho ^{(\varepsilon )}$ for some approximate $\delta$-distribution $\rho ^{(\varepsilon )}$. In [4, Thm. 2.21] it was shown that under some appropriate assumption on the equation there exists a choice of constants $c^\varepsilon _k$ with the property that the sequence of solutions $u^\varepsilon$ converges in probability to some limiting random distribution u as $\varepsilon \rightarrow 0$, and we call this limit u the (renormalized) solution to (1). The counter-terms $\Upsilon ^k_i$ and the renormalization constants $c^\varepsilon _k$ are given explicitly in [4, (2.12)]. We recall their definition in (16) below.

The first purpose of the present article is to establish the existence of continuous path-wise derivatives of the renormalized solution to (1) in the direction of Cameron–Martin functions (in the sense of [35, Def. 3.3.1]). This is in particular enough to obtain the existence of a localized version of Malliavin derivative ([28, Prop. 4.1.3], [7, Prop. 2.4]), which in turn is sufficient for the celebrated Bouleau–Hirsch criterion [3] to apply. The latter gives rather sharp conditions under which densities with respect to Lebesgue measure exist.

The second purpose of this article is to show that the conditions of the Bouleau–Hirsch criterion are indeed satisfied for an interesting class of equations. The equations for which we can show existence of densities include in particular the stochastic heat equation with multiplicative noise and the $\Phi ^p_d$-equations.

Malliavin calculus has been used in the context of SPDEs in different settings in the past. The strategy outlined above for singular SPDEs was already used in [7] to show existence of densities for the 2D-PAM equation, and a recent paper [13] treated the case of the $\Phi ^4_3$ equation. On the technical level, our approach for showing Malliavin differentiability uses extensions of the regularity structure and is strongly inspired by [7] (compare also [8, 9] for the rough path case), although the proofs given in the present paper differ in some key aspects, which in particular allows us to obtain statements that are more general. For the main technical step - the proof that the "shifted" model is well defined - we exploit a noise doubling strategy, which is similar to [22], where this has been done in the context of rough paths. In the second part of the paper we apply the Bouleau–Hirsch criterion by studying the "dual" to the tangent equation, an idea inspired from [30] where this has been used to study linear SPDEs driven by degenerate noise and the recent work [13] which studies the existence of densities for the $\Phi ^4_3$ equation.

Malliavin calculus has been used to study SPDEs interpreted in Ito’s theory, see for instance [32] and the references theirein. In [2] the authors consider the multiplicative stochastic heat equation with Neumann boundary conditions and prove the existence of smooth densities; this was later generalized to Dirichlet boundary conditions in [27]. Apart from the boundary conditions this setting is identical to (8) below; note that they obtain smoothness for densities of point evaluation, in the current paper we obtain densities (but not smoothness) after testing against test functions. In [24] stochastic heat equations with coloured noise in arbitrary dimensions are treated; note that Proposition 1 can deal in principle with coloured noises but doesn’t explore this topic in detail (see also the remark below this Proposition 1). In [29] existence and smoothness of law for a wide class of second order SPDEs has been shown, including the stochastic heat equation (generalizing previous results from [2, 24]). That paper also treats the stochastic wave equation in no more than 3 dimensions; wave equations do not fall under our setting. Relaxing the assumptions on the coefficients allowed [11] to treat (among other things) the parabolic Anderson model in one dimension.

Finally, a series of results concerning Hörmander theorems have also been obtained for SPDEs under "degenerate" forcing, see for instance [1, 25], or the recent generalization to rough forcing in [14]. Note that the main technical difficulty of these papers is quite different from ours. In these papers, showing the non-degeneracy of the Malliavin matrix is the main problem; in our setting the main problem is to show that a suitable path-wise generalization of the Malliavin derivative exists in the first place.

To make things more concrete, we put ourselves in the setting of the “black box” theorem [4, Thm. 2.21]. Given non-linearities $F_i$ and $F_i^j$, and a Gaussian noise $\xi$, this theorem establishes explicit formulae for the counterterms and renormalization constants appearing in (5), and works out concrete assumption on the equations under which the sequence of renormalized solutions converge.

Assumption 1

We assume throughout this paper that the assumptions of [4, Thm. 2.21] on the equation, the noises and the initial condition are satisfied. To be more precise, we assume [4, (2.5), Ass. 2.6, Ass. 2.8, Ass. 2.13, Ass. 2.15, Ass. 2.16], we assume that we are given jointly Gaussian random fields $(\xi _j)_{j \le n}$ in the sense of [4, Def. 2.17] and we assume that the initial condition can be decomposed as as in [4, (2.23)] with $\psi ^\varepsilon$ converging to some random initial condition $\psi$ in probability in as $\varepsilon \rightarrow 0$. We refer the reader to Sect. 2.3 for a summary of these assumptions and the definition of the space .

For the reader not familiar with these assumptions we recall briefly their purpose. In [4, (2.5)] the authors give a rigours meaning to the notion of sub-criticality. This is a key assumption which is seen in any of the theories developed in [15, 17, 31], and the equations are believed to behave quite differently when this assumption is violated. It also ensures that one can algebraically build a regularity structure adapted to the equation as in [5]. Assumption [4, Ass. 2.6] deals with compositions of the solution with smooth functions. It also limits the regularity blow-up at the initial time-slice to ensure that the solution is an actual distribution on the whole space (as opposed to just ${{\textbf {R}}}^+ \times {{\textbf {T}}}^d$). Throughout the solution theory developed in [17] the equations are treated in their mild formulation, and Assumption [4, Ass. 2.8] guarantees the existence of a Green’s function for , together with suitable analytic estimates. Assumption [4, Ass. 2.13] is a technical assumption that ensures that the solution to our equation can always be written as an explicit distribution-valued, stationary, random process, plus an implicit function-valued random perturbation (by explicit we mean that this process is given as a stationary solution to a linear equation and polynomial expressions in this solution). The explicit stationary process appearing for the regularized noise is denoted by and appears in the rather cumbersome way in which the initial conditions are phrased. This is needed, since in general the spaces in which converges as $\varepsilon \rightarrow 0$ are spaces of space-time distributions, and it follows that evaluating the limit process at a fixed time is in general not well defined. Finally, [4, Ass. 2.15 and 2.16] ensures that the analytic BPHZ theorem of [10, Thm. 2.33] can be applied, which in particular establishes the existence of a limit model. Most notably, this assumption rules out divergent variances in the “trees” used to build the regularity structure.

Remark 1

Assumption 1 is identical to the assumption made in [4, Thm. 2.21], which guarantees the existence of solutions, together with a precise definition of the renormalisation procedure, and is therefore a convenient starting point for our analysis. These assumptions are quite complicated. Fortunately, in the present paper we will never actually need them, other than for their black-box content: They imply that there is an "abstract fixed-point problem" in Hairer’s space of modelled distributions (we recall in Sect. 2.2.5 their definition) which (1) has a solution U and (2) its reconstruction coincides with the classical solution u to a renormalised SPDE whenever the noise is smooth.

We recall from [4, Thm. 2.21] that under Assumption 1 there exists a unique maximal random time $\tau =\tau (\omega )>0$ and a maximal solution $u = (u_i)_{i\le m}$ on $[0,\tau (\omega )) \times {{\textbf {T}}}^d$ to (1). To be more precise, there exists a choice of constants $c^\varepsilon _k$ for $\varepsilon >0$ and a sequence of random times $\tau ^\varepsilon =\tau ^\varepsilon (\omega )$ with $\tau ^\varepsilon \rightarrow \tau$ in probability as $\varepsilon \rightarrow 0$ and such that the classical solution $u^\varepsilon$ to (5) with $\Upsilon _i^k$ given as in (16) exits almost surely on $[0,\tau ^\varepsilon )$, and such that for $T>0$ the sequence $u^\varepsilon$ conditioned on the event $\{ \tau > T\}$ converges as $\varepsilon \rightarrow 0$ to u in probability in the space of space-time distributions . When restricted to positive times, this convergence also takes place in the Hölder-Besov space¹. Moreover, the random time $\tau$ can be chosen maximal, in the sense that the statement above does not hold for any random time ${\tilde{\tau }}$ such that ${\tilde{\tau }}>\tau$ with positive probability.

1.1 Main results

We want to study the finite-dimensional law of the random variables given by testing the solution u against a finite number of test-functions, that is, we study the law of

$$\begin{aligned} (u_i(\phi _l^i))_{l \le L, i\le m} \in {{\textbf {R}}}^{L \times m} \end{aligned}$$

(6)

for some $L \in {{{\textbf {N}}}}$, test functions . Note that our results are obtained for testing the solution against test functions, rather than considering point evaluations as in [7]. We will establish Malliavin differentiability [23, 28] of these random variables, and a fortiori study the existence of densities with respect to Lebesgue measure. As has already been observed in [7] and later in [13], the classical notion of Malliavin differentiability is to strong for our purposes, as it imposes moment bounds which are simply not true in general in our setting. Instead, we are lead to use a version of Malliavin differentiability more adapted to this setting, and we borrow the notion of local H-Fréchet differentiability from [35, Def. 3.3.1], which we recall in Definition 5 below. Denoting by $H^\xi$ the Cameron–Martin space for the jointly Gaussian random fields $\xi =(\xi _i)_{i \le n}$, our main result on Malliavin differentiability reads as follows.

Theorem 1

Under Assumption 1, let u be the solution to (1) given by [4, Thm. 2.21], let $\tau =\tau (\omega ) \in (0,\infty ]$ be the time of existence of u, let $\psi :=\lim _{\varepsilon \rightarrow 0}\psi ^\varepsilon$ and assume that $\psi ^\varepsilon$ and $\psi$ and are locally $H^\xi$-Fréchet differentiable for any $\varepsilon >0$. Then, for any $T>0$ and any $i \le m$ the solution $u_i$ restricted to $(0,T) \times {{\textbf {T}}}^d$ and conditioned on $\{ T < \tau \}$ is locally $H^\xi$-Fréchet differentiable with values in in the sense of Definition 5. The $H^\xi$-derivative $D_h u_i$ of $u_i$ in the direction of $h \in H^\xi$ is given by $v_{i,h}$, where $v_h$ is the renormalized solution to the equation

(7)

with initial condition $v_{i,h}(0) = D_{h} u_{i,0}$.

We refer the reader to (55) below for a precise formulation of what we mean by renormalized solution to (7).

Local H-Fréchet differentiability is a powerful tool to establish existence of densities due an argument by Bouleau and Hirsch [3], see also [28, Sec. 2.1.3] and the references therein. We show existence of densities under some simplifying assumptions which we introduce in Sect. 5 below. These assumptions are somewhat technical and we refrain from stating them precisely at this stage. Instead, we refer the reader to the paragraph below Theorem 2 for an informal discussion of these assumptions and to Proposition 1 for a class of interesting equations for which these assumptions are indeed satisfied. Taking Assumptions 4, 5 and 6 from Sect. 5 for granted at the moment, our main result concerning densities is the following.

Theorem 2

Assume that Assumptions 4, 5 and 6 below hold. Let furthermore $(\xi _i)_{i \le n}$ be a family of jointly Gaussian noises on some probability space $(\Omega ,{{\textbf {P}}})$ with Cameron–Martin space $H^{\xi }$. Let also $T>0$ and assume that ${{\textbf {P}}}(\{ T<\tau \})>0$. Finally, assume that $H^\xi$ is such that is dense in $L^2((0,T)\times {{\textbf {T}}}^d)$.

Then, for any $L\in {{{\textbf {N}}}}$ and any family $(\phi _l)_{l\le L}$ with of linearly independent, smooth, compactly supported ${{\textbf {R}}}^{{\mathfrak {L}}_-}$-valued test-functions, one has that the ${{\textbf {R}}}^{L}$-valued random variable

$$\begin{aligned} \left( \left\langle u,\phi _1 \right\rangle , \ldots , \left\langle u,\phi _L \right\rangle \right) \end{aligned}$$

conditioned on $\{ T < \tau \}$ has a density with respect to Lebesgue measure.

We briefly discuss the assumption of the previous theorem. Assumption 4 severly limits the explicit dependence of the right hand side of (5) on the derivatives of the solution. This is done mainly for convenience, as it simplifies many computations below. Assumption 5 ensures that the renormalization constants for the "dualized" tangent equation are identical to the constants appearing in the original tangent equation, which is needed in order to pass to the limit $\varepsilon \rightarrow 0$ in the dual equation. We believe that both of these assumptions are not really necessary and it will be the subject of future research to establish a density result that does not require them. Assumption 6 on the other hand is crucial, as it ensures that in case of multiplicative noise the term multiplying the noise does not make it degenerate.

Instead of giving the precise assumptions at this stage we will limit ourselves to the following examples in order to demonstrate the scope of Theorem 2. We also point out that Assumption 4 rules out that derivatives of the solution appear explicitly on the right hand side, which excludes in particular the (generalised) KPZ equation.

Proposition 1

Assumptions 4, 5 and 6 hold for the $\Phi ^p_2$ equations, the $\Phi ^4_3$ and the $\Phi ^{4}_{4-\varepsilon }$ equation.². It also holds for the multiplicative stochastic heat equation

$$\begin{aligned} \partial _t u = \Delta u + f(u) + g(u)\xi \end{aligned}$$

(8)

in $1+1$ dimension, as soon as the smooth function g does not vanish anywhere on ${{\textbf {R}}}$. In particular, the statement of Theorem 2 holds as soon as $\xi$ is a Gaussian noise with Cameron–Martin space dense in $L^2({{\textbf {R}}}\times {{\textbf {T}}}^d)$.

Remark 2

We remark that in [13] the authors obtained existence of densities for the $\Phi ^4_3$ equation under the same assumptions as above. Additionally, they obtained existence of densities for noises whose Cameron–Martin space is not dense in $L^2$, but are such that they are everywhere "rough enough" in a certain sense. We expect that it is possible to generalize these arguments to all of examples given above.

Remark 3

On the other hand, the result obtained for the 2D PAM equation in [7] falls out of our setting. Crucially, the purely spatial white noise driving the PAM equation violates our assumption on the density of the Cameron Martin space in $L^2((0,T)\times {{\textbf {T}}}^d)$. Another difference is that [7] considers point evaluations of the solutions, rather than testing against smooth test functions.

1.2 Main results on the level of the model

The main results of this work, Malliavin differentiability and existence of densities, are formulated on the level of the equation. However, the main novelty of this paper is actually contained in Theorem 7, which allows "lifting" Cameron–Martin functions and Gaussian noises jointly to an extended model. We refer to Sect. 1.4 for an outline of how to finish the proof of H-Frechet differentiability of solutions of SPDEs from there. For readers not familiar with the theory regularity structures [17], see also [4, 5, 10], we refer to Sect. 2.2 for a review of some notations and results from these papers. Recall that Models $Z=(\Pi ,\Gamma )$ are bounded in terms of expressions of the form

$$\begin{aligned} \sup _{\lambda \in (0,1)}|(\Pi _z \tau ) (\phi _z^\lambda )| \lambda ^{-|\tau |_{\mathfrak {s}}-\theta } \end{aligned}$$

(9)

for test functions $\phi$ and some $\theta >0$; see [17, Thm. 10.7] for a precise statement [we also make use of this argument in (33) below]. For this reason, the tricky part of our argument is to show that "renormalised extended" trees are well defined for Cameron–Martin functions. Once this is established, the necessary technical results (bound of moments of the extended model uniform in the norm of the Cameron–Martin function h, continuity of the extended lift as a function of the noise in some probabilistic sense, Lipschitz continuity in h, definition of the "shifted" model) follow from comparatively standard lines of argument.

To explain the main idea behind the "noise doubling strategy" that allows us to lift Cameron–Martin functions to extended renormalised models, we consider the cherry tree

from the $\Phi ^4_3$ equation. In the extended regularity structure, there are 4 trees corresponding to this cherry tree which we visualize as

Here we draw a black bullet for the abstract symbols $\Xi$, which is a place holder for white noise, and we draw a red square for the abstract symbol ${\hat{\Xi }}$, which is a place holder for a Cameron–Martin function. (Note that there are really only 3 trees, since the middle two trees are the same symbol due to symmetry.) Taking the second tree above as an example, we have for any $w \in {{\textbf {R}}}_+\times {{\textbf {T}}}^3$ the identity

where we write

Taking squares and expectations on both sides, we have

In the diagram in the middle we use a dashed line for a (regularised) $\delta _0$-distribution. The point of this argument is:

• 1. Both the "expectation" and the "$L^2(dy)$"-norm lead to the same type of contraction.

• 2. The final expression is identical (modulo the factor $\Vert h\Vert _{L^2}^2$) to the second moment of the original tree. The latter is bounded in large generality by [10, Thm. 2.31]; this allows us to show existence of extended models on the same level of generality. (For simplicity we evaluated the trees above at a point $z \in {{\textbf {R}}}_+ \times {{\textbf {T}}}^3$, for the actual bounds needed, one should integrate against a (rescaled) test-function instead.)

1.3 Application: multiplicative stochastic heat equation

We apply our results to the stochastic heat equation (8) driven by a space-time dependent Gaussian noise $\xi$ on ${{\textbf {R}}}\times {{\textbf {T}}}$ satisfying the assumptions of Sect. 2.2.4 and vector fields with $g>0$. We refrain from stating the precise assumptions on the noise at this point as they are somewhat convoluted, but we note that these assumptions allow in particular the case of space-time white-noise. The regularized and renormalized equation is given by

$$\begin{aligned} \partial _t u^\varepsilon= & {} \Delta u^\varepsilon + f(u^\varepsilon ) + g(u^\varepsilon )\xi ^\varepsilon + C_\varepsilon ^1 g'(u^\varepsilon )g(u^\varepsilon ) + C_\varepsilon ^2 g'(u^\varepsilon )^3 g(u^\varepsilon )^2\\{} & {} \quad + C_\varepsilon ^3 g''(u^\varepsilon ) g'(u^\varepsilon ) g(u^\varepsilon )^2 \end{aligned}$$

for some constants $C_\varepsilon ^i$ for $i=1,2,3$, subjection to (for simplicity deterministic) initial condition $u^\varepsilon (0)=u_0$. For space-time white noise $\xi$, this equation was first derived in [20], where it was also shown that in this case one can choose $C_\varepsilon ^2$ and $C_\varepsilon ^3$ independent of $\varepsilon$. For more general noises, it follows from [4] that given some initial condition there exists a choice of constants $C_\varepsilon ^i$, $i=1,2,3$, such that the regularized solution $u^\varepsilon$ conditioned on $\{ \tau >T \}$ converges to some limit u in probability the space .

By Theorem 1, the solution is locally $H^\xi$-Fréchet differentiable, and its derivative $v_h = D_h u$ satisfies the tangent equation

$$\begin{aligned} \partial _t v_h = \Delta v_h + f'(u_h)v_h + g'(u_h)v_h\xi + g(u_h) h. \end{aligned}$$

More precisely, one has $v_h = \lim _{\varepsilon \rightarrow 0}v_h^\varepsilon$, where $v_h^\varepsilon$ is the classical solution to

$$\begin{aligned} \partial _t v^\varepsilon _h= & {} \Delta v^\varepsilon _h + f'(u^\varepsilon )v^\varepsilon _h + g'(u^\varepsilon )v^\varepsilon _h \xi ^\varepsilon + g(u^\varepsilon ) h^\varepsilon \\{} & {} \quad \Big ( + C_\varepsilon ^1 \big ( g''(u^\varepsilon )g(u^\varepsilon ) + g'(u^\varepsilon )^2 \big ) + C_\varepsilon ^2 \big ( 3g''(u^\varepsilon )g'(u^\varepsilon )^2 g(u^\varepsilon )^2 + 2g'(u^\varepsilon )^4 g(u^\varepsilon ) \big ) \\{} & {} \quad + C_\varepsilon ^3 \big ( g'''(u^\varepsilon ) g'(u^\varepsilon ) g(u^\varepsilon )^2 + g''(u^\varepsilon )^2 g(u^\varepsilon )^2 + 2g''(u^\varepsilon ) g'(u^\varepsilon )^2 g(u^\varepsilon ) \big ) \Big ) v^\varepsilon _h, \end{aligned}$$

subject to the initial condition $v_h^\varepsilon (0) = 0$.

Furthermore, assuming that the Cameron–Martin space $H^\xi$ of $\xi$ is dense in $L^2({{\textbf {R}}}\times {{\textbf {T}}})$, then for any family of linearly independent test function $(\varphi _i)_{i \le L}$ with the ${{\textbf {R}}}^L$-valued random variable given by $( \left\langle u, \varphi _1 \right\rangle , \ldots , \left\langle u, \varphi _L \right\rangle )$ conditioned on the event $\{ \tau >T \}$ admits a density with respect to Lebesgue measure.

1.4 Outline of the paper

In the next section we review some of the results in the literature which we need later on. Most notation which we use in the paper will also be introduced in that section. In Sect. 2.1 we introduce general conventions on notation. We review the theory of regularity structures and singular SPDEs, see [4, 5, 10, 17], in Sects. 2.2 and 2.3, respectively. Finally, in Sect. 2.4 we review some classical results about Gaussian measure theory in infinite dimensional spaces.

The outline of the paper is as then follows. In Sects. 3 and 4 we show existence of H-Frechet derivatives in the following way.

• 1. As in [7], we first construct in a purely algebraic step an extended regularity structure in Sect. 3.1 by adding for any noise-type $\Xi$ a symbol ${\hat{\Xi }}$ that acts as an abstract place-holder for a fixed Cameron–Martin function. The extended set of trees is then given by allowing any appearance of any noise-type $\Xi$ in any tree to be replaced by ${\hat{\Xi }}$.

• 2. In Sect. 3.2 we perform the main analytic argument which shows that we can lift a fixed Cameron–Martin function $h_\Xi$ and Gaussian noise $\xi _\Xi$ to a renormalized model that in particular has the property that ${\varvec{\Pi }}\Xi = \xi _\Xi$ and ${\varvec{\Pi }} {\hat{\Xi }} = h_\Xi$. We will show that this lift is locally Lipschitz continuous in h.

• 3. An extended model can then be mapped in a locally Lipschitz continuous way onto a “shifted” model in Sect. 3.3, which in particular shows that the model behaves in a continuous way under shifting the noise by a Cameron–Martin function.

• 4. In Sects. 3.4 and 3.5 we show how to lift and shift abstract fixed point problems. This will in particular allow us to consider for fixed Cameron–Martin function h the equations driven by $\xi + rh$ for any $r \in {{\textbf {R}}}$ in an r-independent model, and is thus suited to study Gâteaux differentiability of the solution map in Cameron–Martin directions.

• 5. Gâteaux and Fréchet differentiability are then established in Sects. 4.1 and 4.2, respectively. Finally, in Sect. 4.3 this abstract theory is applied to singular SPDEs of the type (1) under Assumption 1, and we derive in particular the tangent equation (7).

In Sect. 5 we show the existence of densities. A rough outline of this section is as follows.

• 1. In order to establish the existence of densities we study the dual equation (57) of the tangent equation (7). We want to lift the dual equation again to an abstract fixed point problem, and since the dual equation is a stochastic PDE going backward in time, we are led to construct another extension of the regularity structure, this time extending the set of kernel-types by adding for any type ${\mathfrak {t}}$ a type ${ {\mathfrak {t}}'}$ representing the dualized kernel.

• 2. We then derive in Sect. 5.2 an abstract fixed point problem for the dual equation and we identify its reconstruction as the actual solution to the dual equation in Sect. 5.3. This step is not automatic, since it is a-priori not clear that the renormalization constants obtained in these two ways coincide (it is not even clear that they differ by something of order 1 in a suitable sense, which is the main reason that Sect. 5 is less general then the rest of the paper).

• 3. This identification relies on Assumption 5 which basically enforces the identity that we need, and we show in Sect. C that Assumption 5 is satisfied when considering single equations (as opposed to systems of equations). Finally, we derive the existence of densities in a spirit similar to [13] by showing that the solution to the dual equation does not vanish identically in Sect. 5.4.

2 Setting and notation

2.1 General conventions on notation

We introduce some notation that is used throughout this article. Given $M\in {{{\textbf {N}}}}$ we write $[M]:=\{1, \ldots , M \}$. We fix a spatial dimension $d\ge 1$ and write ${{{\textbf {D}}}}:= {{\textbf {R}}}\times {{\textbf {T}}}^d$. Given $z \in {{{\textbf {D}}}}$ we often write $z=(z_0,z_1,\ldots ,z_d)$ with $z_0 \in {{\textbf {R}}}$ and $(z_1,\ldots z_d) \in {{\textbf {T}}}^d$. Given a finite set A, a subset $B\subseteq A$, and a variable $z \in {{{\textbf {D}}}}^A$, we write $z_B:= (z_a)_{a \in B}$. We also fix a space-time scaling ${\mathfrak {s}}:\{0, \ldots , d\} \rightarrow {{{\textbf {N}}}}$, and we write $|{\mathfrak {s}}|:=\sum _{i=0}^d {\mathfrak {s}}(i)$ for the effective space-time dimension. For a multi-index $k \in {{{\textbf {N}}}}^{\{0, \ldots d \}}$ we write $|k|_{\mathfrak {s}}:= \sum _{i =0}^d {\mathfrak {s}}(i)k_i$, and for $z \in {{{\textbf {D}}}}$ we write $|z|_{\mathfrak {s}}:= \sum _{i=0}^d |z_i|^{\frac{1}{ {\mathfrak {s}}(i) } }$. We use the convention that sums of the form

$$\begin{aligned} \sum _{|k|_{\mathfrak {s}}\le r} \cdots \end{aligned}$$

always run over all multi-indices $k \in {{{\textbf {N}}}}^{\{0,\ldots ,d\} }$ with $|k|_{\mathfrak {s}}\le r$.

For any open subset ${\mathcal {O}}\subseteq {{\textbf {R}}}^n$ we write for the space of compactly supported, smooth functions $\phi : {\mathcal {O}} \rightarrow {{\textbf {R}}}$, we endow this space with the topology given by the system of semi-norms

$$\begin{aligned} \Vert \phi \Vert _{K,r}:= \sup _{z\in K} \sup _{|k|_{\mathfrak {s}}\le r} |\partial ^k \phi (z)| \end{aligned}$$

for $K\subseteq {\mathcal {O}}$ compact and $r \in {{{\textbf {N}}}}$, and we write for the dual space of . We call a mollifier if $\int \rho (x) dx=1$, and in this case we define

$$\begin{aligned} \rho ^{(\varepsilon )}(z):= \varepsilon ^{-|{\mathfrak {s}}|} \rho \left( \frac{z_0}{\varepsilon ^{{\mathfrak {s}}(0)} }, \ldots , \frac{z_d}{\varepsilon ^{{\mathfrak {s}}(d)} }\right) . \end{aligned}$$

Finally, the following terminology of multi-sets will be useful. A multiset $\texttt {m}$ with values in A is an element of ${{{\textbf {N}}}}^A$. Given two multisets $\texttt {m}, \texttt {n}\in {{{\textbf {N}}}}^A$ we write $\texttt {m}\sqcup \texttt {n}\in {{{\textbf {N}}}}^A$ for the multiset given by $(\texttt {m}\sqcup \texttt {n})(a):= \texttt {m}(a) + \texttt {n}(a)$, and we write ${\mathfrak {m}}\sqsubset {\mathfrak {n}}$ if ${\mathfrak {m}}\le {\mathfrak {n}}$. We also naturally identify a subset $B\subseteq A$ with the multiset ${{{\textbf {I}}}}_B: A \rightarrow {{{\textbf {N}}}}$. Given any finite set I and a map $\varphi : I \rightarrow A$ we write $[I,\varphi ]$ for the multiset with values in A given by

$$\begin{aligned}{}[I,\varphi ]_a := \#\{ i \in I : \varphi (i) = a \} \end{aligned}$$

(10)

for any $a \in A$.

We sometimes discuss concepts in detail for concrete examples in order to clarify notation. In these cases we use notations of the form $[a,b,c,\ldots ]$, to denote multisets. For instance, we write $[a,a,b]:=2{{{\textbf {I}}}}_a + {{{\textbf {I}}}}_b$.

Given a multiset $\texttt {m}$ as above and a function f on A we also freely use the notation $\sum _{a \in \texttt {m}} f(a)$ and $\prod _{a \in \texttt {m}} f(a)$. These expression should be interpreted as

$$\begin{aligned} \sum _{a \in \texttt {m}} f(a):= \sum _{a \in A} \texttt {m}(a) f(a) \qquad \text { and }\qquad \prod _{a \in \texttt {m}} f(a):= \prod _{a \in A} {f(a)}^{\texttt {m}(a)}. \end{aligned}$$

Sometimes it will be useful to consider functions f whose domain is formally given by ${{{\textbf {D}}}}^\texttt {m}$ for some multiset $\texttt {m}$. Setting , when we write $f: {{{\textbf {D}}}}^\texttt {m}\rightarrow {{\textbf {R}}}$ we really mean that is a function which is symmetric under any permutation $\sigma$ of with the property that for any there exits $l \le \texttt {m}(a)$ such that $\sigma (a,k) = (a,l)$.

2.2 Regularity structures

In this section we recall the main notations and results about regularity structures that we will use in the sequel. Throughout this paper we assume we are given a finite set of types ${\mathfrak {L}}={\mathfrak {L}}_-\sqcup {\mathfrak {L}}_+$. The finite set ${\mathfrak {L}}_+$ will index the components of the equation, while the finite set ${\mathfrak {L}}_-$ will index the Gaussian noises appearing on the right hand side of the equation. We assume that ${\mathfrak {L}}$ is equipped with a homogeneity assignment $|\cdot |_{\mathfrak {s}}:{\mathfrak {L}}_\star \rightarrow {{\textbf {R}}}^\star$ for $\star \in \{+,-\}$. Recall from [5, Def. 5.14] that a rule R is a collection $(R({\mathfrak {t}}))_{{\mathfrak {t}}\in {\mathfrak {L}}}$ that assigns to any type ${\mathfrak {t}}\in {\mathfrak {L}}$ a set of multisets $R({\mathfrak {t}})$ with values in ${\mathfrak {L}}\times {{{\textbf {N}}}}^{d+1}$. We recall the notions of normal, sub-critical and complete from [5, Def. 5.7, Def. 5.14, Def. 5.22]. Let us especially recall that a rule is subcritical if there exists a map ${\text {reg}}: {\mathfrak {L}}\rightarrow {{\textbf {R}}}$ with the property that

$$\begin{aligned} {\text {reg}}({\mathfrak {t}}) < |{\mathfrak {t}}|_{\mathfrak {s}}+ \inf _{N \in R({\mathfrak {t}})} {\text {reg}}(N), \end{aligned}$$

where we set ${\text {reg}}(N):= \sum _{ ({\mathfrak {l}},k) \in N} ({\text {reg}}({\mathfrak {l}}) - |k|_{\mathfrak {s}})$ for any multiset $N \in {{{\textbf {N}}}}^{{\mathfrak {L}}\times {{{\textbf {N}}}}^{d+1}}$. A rule R is normal if $R({\mathfrak {t}})$ is stable under taking multi-subsets of any $N\in R({\mathfrak {t}})$, and additionally if $R({\mathfrak {t}}):=\{\emptyset \}$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_-$. Completeness ensures that the set of tree conforming to the rule R (c.f. [5, Def. 5.8]) is stable under the action of renormalization.

Example 1

In case of stochastic heat equation (8) one has a unique kernel type ${\mathfrak {t}}$ and a unique noise type $\Xi$ and the rule R is given by $R(\Xi ):= \{ \emptyset \}$ and $R({\mathfrak {t}})$ contains all multisets of the form

$$\begin{aligned}{}[({\mathfrak {t}},0),\ldots ]\qquad or \qquad [(\Xi ,0), ({\mathfrak {t}},0), \ldots ] \end{aligned}$$

where we write $({\mathfrak {t}},0),\ldots$ denote an arbitrary (possible vanishing) number of occurrences of $({\mathfrak {t}},0)$.

We assume we are given a normal, subcritical, complete rule R and we denote by the regularity structure constructed as in [5, Def. 5.26]. We will actually work with a slightly simplified structure as far as the extended decoration is concerned, compare Sect. 2.2.1 below. We extend the homogeneity assignment $|\cdot |_{\mathfrak {s}}$ to in the usual way, taking into account the extended decoration³, and we write for the reduced regularity structure obtained as in [5, Sec. 6.4]. We will very rarely need the homogeneity assignment that neglects the extended decoration, but in these situations we will denote this by $|\cdot |_-$ as in [5, Def. 5.3]. We write ${\mathscr {T}}^{\text {ex}}$ and ${\mathscr {T}}$ for the set of trees in and , respectively, so that and are freely generated by ${\mathscr {T}}^{\text {ex}}$ and ${\mathscr {T}}$ as linear spaces. We write ${\mathscr {T}}^{\text {ex}}_\alpha$ for the set of trees $\tau \in {\mathscr {T}}^{\text {ex}}$ with the property that $|\tau |_{\mathfrak {s}}= \alpha$, we write , and for $\gamma \in {{\textbf {R}}}$ we write for the projection of onto .

Finally, we make the following Assumption on the regularity structure, which is needed to apply the results of [10, Thm. 2.31,Thm. 2.33].

Assumption 2

For any tree $\tau \in {\mathscr {T}}$ one has

$$\begin{aligned} |\tau |_{\mathfrak {s}}> \bigg ( -\frac{|{\mathfrak {s}}|}{2}\bigg ) \vee \max _{u \in L(\tau )} |{\mathfrak {t}}(u)|_{\mathfrak {s}}\vee \bigg ( - |{\mathfrak {s}}| - \min _{\Xi \in {\mathfrak {L}}_-} |\Xi |_{\mathfrak {s}}\bigg ). \end{aligned}$$

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Remark 4

Under Assumption 1 we are indeed in this setting, compare [5, Sec. 5.5] and [4, Sec. 3.1]. In particular, Assumption 2 follows from [4, Ass. 2.15 and 2.16].

Singular SPDEs are the application that we have in mind in the present publication, and the formulation of Theorem 1 is therefore on the level of the equation. However, the main technical contribution of this paper is the noise doubling strategy which shows that Cameron–Martin functions can be "lifted" to models in Theorem 7. This theorem is completely carried out on the level of the regularity structure and does not require that we are in the setting of a singular SPDE.

Finally, we make the simplifying assumption on the rule that we do not allow products or derivatives of noises to appear on the right hand side of the equation. (Such an assumption does not seem to be crucial for the statement, but simplifies certain arguments.)

Assumption 3

We assume that for any ${\mathfrak {t}}\in {\mathfrak {L}}$ and any $N \in R({\mathfrak {t}})$ there exists at most one pair $(\Xi ,k) \in {\mathfrak {L}}_- \times {{{\textbf {N}}}}^{d+1}$ such that $N_{(\Xi ,k)} \ne 0$, and this case $k=0$ and $N_{(\Xi ,0)}=1$.

2.2.1 Trees

Trees $\tau \in {\mathscr {T}}^{\text {ex}}$ can be written as typed, decorated trees $\tau =(T^{{\mathfrak {n}},{\mathfrak {o}}}_{\mathfrak {e}}, {\mathfrak {t}})$, where T is a rooted tree with vertex set V(T), edge set E(T) and root $\rho _T$, the map ${\mathfrak {t}}$ assigns types to edges and is formally a map ${\mathfrak {t}}:E(T)\rightarrow {\mathfrak {L}}$, and the decorations ${\mathfrak {n}},{\mathfrak {e}},{\mathfrak {o}}$ are maps ${\mathfrak {n}}:N(T) \rightarrow {{{\textbf {N}}}}^d$, ${\mathfrak {e}}: E(T) \rightarrow {{{\textbf {N}}}}^d$ and ${\mathfrak {o}}: N(T) \rightarrow (-\infty ,0]$. We call ${\mathfrak {o}}$ the extended decoration. Here we define the decomposition of the set of edges into $E(T) = L(T) \sqcup K(T)$ with $e \in L(T)$ (resp. $e \in K(T)$) if and only if ${\mathfrak {t}}(e) \in {\mathfrak {L}}_-$ (resp. ${\mathfrak {t}}(e) \in {\mathfrak {L}}_+$), and we write $N(T) \subseteq V(T)$ for the set of $u \in V(T)$ such that there does not exist $e \in L(\tau )$ such that $u=e^\uparrow$. We will often abuse notation slightly and leave the type map ${\mathfrak {t}}$ and the root $\rho _\tau$ implicit. We recall that the relation between the homogeneity assignments $|\cdot |_{\mathfrak {s}}$ and $|\cdot |_-$ is given by $|\cdot |_{\mathfrak {s}}= |\cdot |_- + \sum _{u \in N(T)}{\mathfrak {o}}(u),$ so that in particular one has $|\tau |_{\mathfrak {s}}\le |\tau |_-$ for any tree $\tau \in {\mathscr {T}}^{\text {ex}}$.

On a rooted tree T we define a total order $\le$ on V(T) by setting $u \le v$ if and only if u lies on the unique shortest path from v to the root $\rho _T$ and we write edges $e \in E(T)$ as order pairs $e=(e^\uparrow , e^\downarrow )$ with $e^\uparrow \ge e^\downarrow$. If $u \in V(T) \backslash \{\rho _T \}$, then there exists a unique edge $e \in E(T)$ such that $u = e^\uparrow$, and in this case we write $u^\downarrow := e$. Recall that it follows from the fact that R is normal (c.f. [5, Def. 5.7]) that elements $u \in V(T)\backslash N(T)$ are leaves of the tree T.

Given a typed, decorated tree $\tau$ as above, $k \in {{{\textbf {N}}}}^{d+1}$ and ${\mathfrak {t}}\in {\mathfrak {L}}_+$ we write for the planted, decorated, typed tree obtained from $\tau$ by attaching an edge e to the root with type ${\mathfrak {t}}$ and ${\mathfrak {e}}(e)=k$. We write ${\mathscr {T}}_{{\mathfrak {t}}} \subseteq {\mathscr {T}}$ for the set of trees $\tau \in {\mathscr {T}}$ such that .

Example 2

Throughout the paper we will consider examples from stochastic heat equation (8) whenever we need to clarify notations. In particular, we often consider the tree , where we introduce the following graphical conventions:

2.2.2 Algebraic notation

We use the notation , , , , and from [5] for the respective spaces defined in [5, Def. 5.26, (5.23), Def. 5.36], and we write for the reduced renormalization group defined above [5, Thm. 6.29]. We recall that and form Hopf algebras and and are defined as their respective character groups. We use the notation $\Delta _-^{\text {ex}}$ and $\Delta _+^{\text {ex}}$ for the co-products for negative and positive renormalization respectively, as in [5, Cor. 5.32], and we write and for the twisted antipodes defined in [5, Prop. 6.3, Prop.6.6].

2.2.3 Models

We assume that for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ we are given a decomposition of the Green’s function into $G_{\mathfrak {t}}=K_{\mathfrak {t}}+ R_{\mathfrak {t}}$ with and such that satisfies [17, Ass. 5.1, Ass. 5.4], and given the kernel assignment $(K_{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ we recall the definition of admissible models [17, Def. 2.7, Def. 8.29]. We call a model $Z=(\Pi ,\Gamma )$ smooth if for any $\tau \in {\mathscr {T}}^{\text {ex}}$ and any $x \in {{{\textbf {D}}}}$, and we call Z reduced if $\Pi _x \tau$ does not depend on the extended decoration of $\tau$. Given an admissible tuple for we write for the model constructed as in [5, (6.11),(6.12)], whenever this is well defined, and we write for the set of smooth, reduced, admissible models for of the form . We write for the closure of in the space of models, and given a probability space $(\Omega ,{{\textbf {P}}})$ we write and for the spaces of and valued random variables on $(\Omega ,{{\textbf {P}}})$, respectively, endowed with the topology induced by convergence in probability. We write and given $f \in \Omega _\infty$ we write for the canonical lift of f to a model , c.f. [5, Rem. 6.13]. We finally write .

Remark 5

Again, we remark that under Assumption 1 we are in this setting, compare [4, Ass. 2.6].

Remark 6

We use the notation $\Omega _\infty$ and $\Omega _0$ to be consistent with previous papers on this topic, e.g. [10, Def. 2.13]. This should not be confused with the (abstract) probability space $(\Omega ,{{\textbf {P}}})$ used throughout the paper.

2.2.4 Gaussian driving noises

We now discuss in some detail the assumptions we need on the driving noises; these assumptions are identical to those made in [10, (2.7)]. First, given a probability space $(\Omega ,{{\textbf {P}}})$ we write for the space of $\Omega _\infty$-valued centred, stationary, jointly Gaussian random fields $(\eta _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_-}$ on $(\Omega ,{{\textbf {P}}})$. Next, we want to introduce a set of $\Omega _0$-valued Gaussian random fields $(\eta _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_-}$ that we will consider as driving noises for our SPDE.

Given a $\Omega _0$-valued jointly Gaussian, stationary, centred random noise $\eta$ on $(\Omega ,{{\textbf {P}}})$, we denote by the distributional covariance of $\eta _{\mathfrak {t}}$ and $\eta _{{\mathfrak {t}}'}$ defined via the identity

$$\begin{aligned} {{{\textbf {E}}}}[\eta _{\mathfrak {t}}(\varphi ) \eta _{{\mathfrak {t}}'}(\psi )] = {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}^\eta \bigg (\int \varphi (x- \cdot )\psi (x)dx\bigg ) \end{aligned}$$

for any . We note that this is well defined by stationarity. The next definition is as in [10, (2.7)]. We fix for any $k \in {{{\textbf {N}}}}^{d+1}$ a function such that $P_k(x) = x^k$ in a neighbourhood of the origin.

Definition 1

We write ${\mathfrak {C}}({\mathfrak {L}}_-)$ for the space of families of kernels $({\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'})_{{\mathfrak {t}},{\mathfrak {t}}' \in {\mathfrak {L}}_-}$ such that ${\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}^* = {\mathscr {C}}_{{\mathfrak {t}}',{\mathfrak {t}}}$ in the sense that one has

$$\begin{aligned} \left\langle {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'} * \phi , \psi \right\rangle _{L^2} = \left\langle \phi , {\mathscr {C}}_{{\mathfrak {t}}',{\mathfrak {t}}} * \psi \right\rangle _{L^2} \end{aligned}$$

for any ${\mathfrak {t}},{\mathfrak {t}}' \in {\mathfrak {L}}_-$ and any choice of test functions , such that ${\mathscr {C}}$ is non-negative definite in the sense that one has

$$\begin{aligned} \sum _{{\mathfrak {t}}} \left\langle \sum _{{\mathfrak {t}}'} {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'} * \phi _{{\mathfrak {t}}'}, \phi _{\mathfrak {t}} \right\rangle _{L^2} \ge 0 \end{aligned}$$

for any family , such that the singular support of the distribution is contained in $\{0\}$, and denoting by the smooth function representing away from the origin, we require that

• for any test function such that $D^k \varphi (0)=0$ for any $|k|_{\mathfrak {s}}< -|{\mathfrak {t}}|_{\mathfrak {s}}- |{\mathfrak {t}}'|_{\mathfrak {s}}-|{\mathfrak {s}}|$, one has ; and

• there exists $\theta >0$ such that one has $\Vert {\mathscr {C}}\Vert _{|\cdot |_{\mathfrak {s}}} <\infty$.

Here, we define the quantity

$$\begin{aligned} \Vert {\mathscr {C}}\Vert _{|\cdot |_{\mathfrak {s}}} := \texttt {d}({\mathscr {C}})+ \max _{{\mathfrak {t}},{\mathfrak {t}}' \in {\mathfrak {L}}_-} \sup _{x \in {{{\textbf {D}}}}\backslash \{0\}} { \sup _{\begin{array}{c} k \in {{{\textbf {N}}}}^{d+1} \\ |k|_{\mathfrak {s}}\le 6|{\mathfrak {s}}| \end{array}} } |D^k {\hat{{\mathscr {C}}}}_{{\mathfrak {t}}, {\mathfrak {t}}'}(x) | |x|^{-|{\mathfrak {t}}|_{\mathfrak {s}}-|{\mathfrak {t}}'|_{\mathfrak {s}}+|k|_{\mathfrak {s}}-\theta } \end{aligned}$$

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with

$$\begin{aligned} \texttt {d}({\mathscr {C}}):= \max _{{\mathfrak {t}},{\mathfrak {t}}' \in {\mathfrak {L}}_-} { \sup _{\begin{array}{c} k \in {{{\textbf {N}}}}^{d+1} \\ |k|_{\mathfrak {s}}< -|{\mathfrak {t}}|_{\mathfrak {s}}-|{\mathfrak {t}}'|_{\mathfrak {s}}-|{\mathfrak {s}}| \end{array}} } |{\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'} (P_k)|. \end{aligned}$$

We write for the set of $\Omega _0$-valued, jointly Gaussian, centred, stationary random fields $\eta$ with the property that ${\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}^\eta \in {\mathfrak {C}}({\mathfrak {L}}_-)$, and we write $\Vert \eta \Vert _{|\cdot |_{\mathfrak {s}}}:=\Vert {\mathscr {C}}^\eta \Vert _{|\cdot |_{\mathfrak {s}}}$.

Remark 7

One can show that for any Gaussian field there exists a version such that $\eta _{\mathfrak {t}}$ is $|{\mathfrak {t}}|_{\mathfrak {s}}$-Hölder continuous almost surely. Compare the remark at the end of [4, Def. 2.17]. A possible way to see this fact is to combine the convergence theorem [10, Thm. 2.34] with the reconstruction theorem [17, Thm. 3.10].

We finally make the following precise to type of approximations used below:

Definition 2

Given an element and a mollifier with $\int \rho (x) dx = 1$, we call the sequence an approximation by mollification of $\eta$. We say that a map X from into a topological space can be extended continuously by mollification to if there exists a map that extends X and is such that whenever is an approximation by mollification of in the sense above, then $X(\eta ^\varepsilon ) \rightarrow X(\eta )$.

2.2.5 Modelled distributions

We recall the terminology and notation from [17, Sec. 3, Sec. 6] of modelled distributions. Given a model and $\gamma >0$, $\eta \in {{\textbf {R}}}$ we write for the space of singular modelled distributions defined in [17, Def. 6.2] allowing a singularity at the hyperplane $t=0$. More precisely consists of all maps with the property that, setting $P=\{z \in {{{\textbf {D}}}}: z_0 = 0\}$, one has that

$$\begin{aligned} \Vert f\Vert _{\gamma ,\eta ,K} := \sup _{ z, {\bar{z}} } \sup _{\beta <\gamma } \bigg ( \frac{\Vert f(z)\Vert _\beta }{ ( |z_0|^{\frac{1}{{\mathfrak {s}}_0}} \wedge 1 )^{(\eta -\beta )\wedge 0} } \!+\! \frac{\Vert f(z) - \Gamma _{z, {\bar{z}}} f({\bar{z}})\Vert _\beta }{\Vert z-{\bar{z}}\Vert _{\mathfrak {s}}^{\gamma -\beta } (|z_0|^{\frac{1}{{\mathfrak {s}}_0}}\wedge |{\bar{z}}_0|^{\frac{1}{{\mathfrak {s}}_0}} \wedge 1)^{\eta -\gamma } } \!\bigg ) \end{aligned}$$

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is finite for any compact $K \subseteq {{{\textbf {D}}}}$. Here, the first supremum runs over all $z,{\bar{z}} \in K\backslash P$ with the property that $\Vert z - {\bar{z}} \Vert _{{\mathfrak {s}}} \le |z_0|^{\frac{1}{{\mathfrak {s}}_0}}\wedge |\bar{z}_0|^{\frac{1}{{\mathfrak {s}}_0}}$. Given a sector V of we write for the space of such that $f(x) \in V$ for any $x \in {{{\textbf {D}}}}$, and we write if we want to emphasise the underlying model. Often we want to consider localized version of these spaces that contain functions f that only live on a bounded time interval [0, T) for some $T>0$. We write for the space of all functions satisfying the bound (13) for any compact $K \subseteq [0,T) \times {{\textbf {T}}}^d$. The notation and then have meanings analogue to above.

On the spaces for $\gamma >0$ we denote by the reconstruction operator defined in [17, Prop. 6.9], provided that $\alpha \wedge \eta > -|{\mathfrak {s}}|+{\mathfrak {s}}(0)$, where $\alpha \le 0$ denotes the regularity of the sector V.

Finally, we denote for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ by the operator constructed in [17, (5.15)] acting between and for any sector V of regularity $\alpha$ such that one has for any . We also define the operator for any . Of course, the operators and depend slightly on $\gamma$. Since $\gamma$ will always be clear from the context, we leave it implicit in this notation.

In [17, Sec. 6] basic properties of certain maps (multiplication, differentiation, integration, composition) between space of modelled distributions were derived and we summarize them in Proposition 8 below.

2.2.6 BPHZ theorem

Given a smooth Gaussian field we define as in [5, (6.23)] a character $g^\eta _-$ on by setting $g^\eta _-(\tau ):= {{{\textbf {E}}}}({\varvec{\Pi }}^\eta \tau )(0)$ for any tree , and extending this linearly and multiplicatively, where ${\varvec{\Pi }}^\eta$ is such that is the canonical lift of $\eta$. We then define the BPHZ-character as in [5, (6.24)] by setting

for any $\tau \in {\mathscr {T}}_-^{\text {ex}}:=\{\tau \in {\mathscr {T}}^{\text {ex}}: |\tau |_-<0\}$, and extending this linearly and multiplicatively. For any character we use the notation for the linear operator given by

and for a smooth Gaussian field we set

for any $\tau \in {\mathscr {T}}^{\text {ex}}$, and we defined the BPHZ-renormalized model , compare [5, Thm. 6.18].

The following is then a direct consequence of Hairer’s and Chandra’s analytic BPHZ theorem.

Theorem 3

The map $\eta \mapsto {\hat{Z}}^\eta _{\textrm{BPHZ}}$ can be extended by mollification to a map from into .

Proof

This is a consequence of [10, Thm. 2.15]. Note that approximations by mollification in the sense 2 imply that $\eta ^\varepsilon$ is uniformly compatible in the sense introduced in the paragraph immediately above [10, Thm. 2.15]. The three assumptions made in [10, Thm. 2.15] follow from [4, Ass. 2.15 and 2.16], which we always we assume in this paper. $\square$

2.3 Singular SPDEs

In [4] the authors established a black box theorem for solving a large class of singular SPDEs of the form (1). We briefly recall the notations introduced in this paper, as far as we are going to need it later on. In order to unify the notation, we assume that $\# {\mathfrak {L}}_+ = n$ and $\# {\mathfrak {L}}_- = m$ and we write $(u_{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ and $(\xi _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_-}$ rather than $(u_i)_{i \le n}$ and $(\xi _j)_{j \le m}$. We recall that we assume that for ${\mathfrak {t}}\in {\mathfrak {L}}_+$ we are given a differential operator involving only spatial derivatives and such that admits a Green’s function $G_{\mathfrak {t}}$ satisfying Assumption [4, Ass. 2.6]. Furthermore, we recall from [4, Sec. 2.5] that we assume we are given two functions ${\text {reg}}: {\mathfrak {L}}\rightarrow {{\textbf {R}}}$ and ${\text {ireg}}: {\mathfrak {L}}_+ \rightarrow {{\textbf {R}}}$ satisfying [4, Def. 2.3] and [4, Ass. 2.6]. We define

and we write for the map given by [4, (A.10)], and we assume that the initial condition $u_0^\varepsilon$ is of the form

for a sequence of -valued random fields $\psi ^\varepsilon$ such that $\psi ^\varepsilon \rightarrow \psi$ in probability as $\varepsilon \rightarrow 0$.

We now recall the definition of the counter-terms appearing on the right hand side of (5). For this we borrow some more notation from [4].

For ${\mathfrak {t}}\in {\mathfrak {L}}_+$ we often write $F_{\mathfrak {t}}^\bullet := F_{\mathfrak {t}}$ in order to avoid case distinctions. The smooth functions $F^\Xi _{\mathfrak {t}}$ are allowed to depend on $D^k u_{\mathfrak {l}}$ where ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and where $k \in {{{\textbf {N}}}}^{d+1}$ ranges over a finite set of multi-indices, say $|k|_{\mathfrak {s}}\le r$. Consequently, it makes sense to define for any ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and $k\in N^{d+1}$ the derivative $D_{({\mathfrak {l}},k)}F_{\mathfrak {t}}^\Xi$ of $F_{\mathfrak {t}}^\Xi$ in the direction of $(D^k u_{\mathfrak {l}})$. We will reserve the symbol $\partial$ for derivatives in direction of space-time variables.

For any tree and node $\mu \in N(\tau )$ we write $\Xi [\mu ]:={\mathfrak {t}}(e)$ if there exists a (necessarily unique) edge $e \in L(\tau )$ with $e^\downarrow = \mu$. We write $\Xi [\mu ]:= \bullet$ otherwise. Moreover, we write $n[\mu ]:=\# \{e \in K(\tau ): e^\downarrow = \mu \}$, and we write $e_j[\mu ]$, $j\le n[\mu ]$ for the $n[\mu ]$ distinct edges $e \in K(\tau )$ such that $e^\downarrow = \mu$. Note that this is uniquely defined up to order of $e_j[\mu ]$. Finally, we write ${\mathfrak {t}}_j[\mu ]:= {\mathfrak {t}}(e_j[\mu ])$ and $k_j[\mu ]:={\mathfrak {e}}(e_j[\mu ])$ for the type and derivative decoration of $e_j[\mu ]$, respectively. Note that any tree can now be written in the form

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for some decorated, typed trees .

Definition 3

For ${\mathfrak {l}}\in {\mathfrak {L}}_+$ we say that a tree $\tau \in {\mathscr {T}}$ is ${\mathfrak {l}}$-non-vanishing if for any $\mu \in N(\tau )$ one has that

$$\begin{aligned} \Big (\partial ^{{\mathfrak {n}}(\mu )} \prod _{j=1} ^{n[\mu ]} D_{({\mathfrak {t}}_j[\mu ], k_j [\mu ])} \Big ) F_{{\mathfrak {t}}(\mu )}^{\Xi [\mu ]} (u, \nabla u, \ldots ) \end{aligned}$$

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does not vanish identically for any smooth function $u:{{{\textbf {D}}}}\rightarrow {{\textbf {R}}}^{{\mathfrak {L}}_+}$, where we set ${\mathfrak {t}}(\rho _\tau ):={\mathfrak {l}}$, and ${\mathfrak {t}}(\mu ):= {\mathfrak {t}}(\mu ^\downarrow )$ if $\mu \ne \rho _\tau$. We write ${\mathscr {T}}_{\mathfrak {l}}^F$ for the set of tree $\tau \in {\mathscr {T}}$ that are ${\mathfrak {l}}$-non-vanishing and are such that , and we write ${\mathscr {T}}_{{\mathfrak {l}},-}^F$ for the set of $\tau \in {\mathscr {T}}_{\mathfrak {l}}^F$ such that $|\tau |_{\mathfrak {s}}<0$, and we write ${\tilde{{\mathscr {T}}}}_{\mathfrak {l}}^F:= {\mathscr {T}}_{{\mathfrak {l}},-}^F \sqcup \{ \bullet \}$.

It follows from a straight forward inductive argument that the definition of ${\mathfrak {l}}$-non-vanshing given above coincides with [4, Def. 2.12]. We now define the counter-terms appearing in the renormalized equation as in [4, (2.12)].

Definition 4

For ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and $\tau \in {\tilde{{\mathscr {T}}}}_{\mathfrak {l}}^F$ we define the function

$$\begin{aligned} \Upsilon _{\mathfrak {l}}^F[\tau ] := \prod _{\mu \in N(\tau )} \Bigg ( \partial ^{{\mathfrak {n}}(\mu )} \prod _{j=1} ^{n[\mu ]} D_{({\mathfrak {t}}_j[\mu ], k_j [\mu ])} F_{{\mathfrak {t}}(\mu )}^{\Xi [\mu ]} (u, \nabla u, \ldots ) \Bigg ) \end{aligned}$$

(16)

where we use the notation from Definition 3 for the type ${\mathfrak {t}}(\mu ) \in {\mathfrak {L}}_+$.

Example 3

In case of stochastic heat equation (8) one has $F_{\mathfrak {t}}^\bullet = f$ and $F_{\mathfrak {t}}^\Xi = g$. Moreover, for the tree from Example 2 one has

Given a character and a smooth Gaussian field we define the g-renormalization of (1) by

(17)

Here $S(\tau ) \in {{{\textbf {N}}}}$ is a symmetry factor explicitly given in [4, (2.22)]. We are thus in the setting of (5) with $K = \# {\mathscr {T}}_{{\mathfrak {t}},-}^F$. Given initial conditions as above and letting $u^\varepsilon$ be the solution to (17) with $\eta$ replaced by $\xi ^\varepsilon$ and g replaced by $g_{\textrm{BPHZ}}^\varepsilon$, then the statement of [4, Thm. 2.21] precisely says that $u^\varepsilon$ converges to some limit u in probability as $\varepsilon \rightarrow 0$.

2.4 Gaussian measure theory

In this section we review basic facts about Gaussian measures on infinite dimensional spaces as far as it is needed for the purpose of this paper. We follow in this section mostly the lecture notes [16] for basic properties of the Cameron–Martin space. For more details we refer the reader to standard literature [28]. Given a separable Fréchet space we call a centered probability measure $\mu$ on equipped with the Borel sigma field Gaussian if all finite dimensional projections of $\mu$ are Gaussian. A Gaussian measure $\mu$ is uniquely determined by its covariance operator defined via the identity $l^*({\mathscr {C}}_\mu (k^*))= {{{\textbf {E}}}}^\mu [k^*l^*]$ for any . We denote the image of the covariance operator by and we equip this space with a scalar product given by $\langle h,k \rangle _\mu := {{{\textbf {E}}}}^\mu [ k^* l^*]$ where are such that and . The closure $H_\mu$ of under the norm induced from this scalar product is a Hilbert space know as Cameron–Martin space. It is well known that continuously, the space $H_\mu$ determines the Gaussian measure $\mu$ uniquely, and one has the following classical result due to Cameron and Martin [6].

Theorem 4

(Cameron–Martin) Let $\mu$ be a Gaussian measure on some Fréchet space and define for the operator by $T_h (x):= x + h$. Then one has $(T_h)_* \mu \ll \mu$ if and only if $h \in H_\mu$ and in this case one has $(T_h)_* \mu \simeq \mu$.

We want to use non-degeneracy of the pathwise derivative in Cameron–Martin directions to establish existence of densities. The classical Malliavin derivative imposes moment bounds on this derivative which are not available in our setting. An approach more adapted to our situation is the notion of local Cameron–Martin Fréchet differentiability, in the form introduced in [35, Def. 3.3.1].

Definition 5

Let be a Banach space. We call a -valued random variable X on is called locally $H_\mu$-Fréchet differentiable if there exists a $\mu$-null set N such that for any $\omega \in N^c$ the map , $h \mapsto X( \omega + h )$, is Fréchet differentiable in a $H_\mu$-neighbourhood of the origin.

We note that this version of local H-Fréchet differentiability was also used in [7, Def. 2.2] and [13, Def. 4.1]. If X is locally $H_\mu$-Fréchet differentiable we denote its $H_\mu$-Fréchet derivative by DX. The main motivation for this definition is the criterion by Bouleau–Hirsch [3] for the existence of densities. In order to deal with situations in which we the solution does not exist globally, we use a slightly generalized version, and for this we make the following construction. Let be a measurable subset of . We say that U is $H_\mu$-open if for any $x \in U$ there exists $\varepsilon >0$ such that for any $h \in H_\mu$ with $\Vert h\Vert _{H_\mu }<\varepsilon$ one has $x+h \in U$. We fix an $H_\mu$-open set U and we define for $\varepsilon >0$ the set $U_\varepsilon$ as the $\varepsilon$-involution of U in $H_\mu$, i.e.

$$\begin{aligned} U_\varepsilon := \{ x \in U: \forall h \in H_\mu \text { with } \Vert h\Vert _{H_\mu } \le \varepsilon \text { one has } x+h \in U \}. \end{aligned}$$

We then assume that there exists a sequence of locally $H_\mu$-Fréchet differentiable random variables for $\varepsilon >0$ that approximates the indicator function ${{{\textbf {I}}}}_U$ from the inside in the sense that one has $0 \le \varphi _\varepsilon \le 1$, and $\varphi _\varepsilon (x)=1$ for any $x \in U_\varepsilon$ and $\varphi _\varepsilon (x)=0$ for $x \in U^c$. If such a sequence exists, we say that U can be approximated from the inside.

Theorem 5

(Bouleau–Hirsch) Let X be an ${{\textbf {R}}}^n$-valued random variable on a Gaussian probability space with separable Cameron–Martin space $H_\mu$, and let be an $H_\mu$-open measurable subset of such that U can be approximated from the inside. Let moreover be the event that DX has full rank, and assume that $\mu (U \cap \Omega _X)>0$. Then X conditioned on the event $U \cap \Omega _X$ admits a density with respect to Lebesgue measure.

Proof

See [7, Prop. 2.4]. $\square$

We will see that the time of existence $\tau$ is lower semi-continuous with respect to $H_\mu$-shifts, see Remark 12 below, which implies in particular that the event $\{ \tau > T \}$ is $H_\mu$-open. The fact that $\{ \tau >T \}$ can be approximated from the inside can be shown exactly as in [7, Lem. 5.3], and this leads to the following.

Corollary 1

Under Assumption 1, let $T>0$, let be the solution to a singular SPDE of the form (1) on a Gaussian probability space $(\Omega ,{{\textbf {P}}})$, let be a $C^1$ map, and assume that ${{\textbf {P}}}( \Omega _{X(u)} \cap \{ \tau>T \})>0$. Then X(u) restricted on $\Omega _{X(u)} \cap \{ \tau >T\}$ admits a density with respect to Lebesgue measure.

We now implement the general constructions from this section in the situation that the underlying Fréchet space is given by , and the Gaussian probability measure ${{\textbf {P}}}$ on $\Omega$ is a stationary, centred Gaussian measure such that its covariance ${\mathscr {C}}_{{\textbf {P}}}$ is an element of ${\mathfrak {C}}({\mathfrak {L}}_-)$. Note in particular that in this case the random field $\xi$ which ${{\textbf {P}}}$-almost surely agrees with the identity on is an element of . Given such a Gaussian measure ${{\textbf {P}}}$, we will henceforth use the convention that $\xi$ denotes this particular random Gaussian field, while $\eta$ will still be used to denote more general random fields on $\Omega$ whose laws under ${{\textbf {P}}}$ are Gaussian. We will usually leave the measure ${{\textbf {P}}}$ implicit in the notation and one should always think of ${{\textbf {P}}}$ as arbitrary but fixed. It is straightforward to see that the space is then given by

It follows in particular that one has .

We finish this section by introducing the following terminology.

Definition 6

Given a kernel ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-)$ we write for the space given by

endowed with the scalar product $\left\langle \cdot ,\cdot \right\rangle _{H[{\mathfrak {L}}_-,{\mathscr {C}}]}$ given by

$$\begin{aligned} \left\langle h,k \right\rangle _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} := \sum _{{\mathfrak {t}}, {\mathfrak {t}}' \in {\mathfrak {L}}_-} \int _{{{{\textbf {D}}}}\times {{{\textbf {D}}}}} dx dy \, {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}(x-y) \varphi _{{\mathfrak {t}}'}(x)\psi _{\mathfrak {t}}(y) \end{aligned}$$

(18)

for any , where are such that $h=(\sum _{{\mathfrak {t}}' \in {\mathfrak {L}}_-} {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}*\varphi _{{\mathfrak {t}}'})_{ {\mathfrak {t}}\in {\mathfrak {L}}_-}$ and $k=(\sum _{{\mathfrak {t}}' \in {\mathfrak {L}}_-} {\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}*\psi _{{\mathfrak {t}}'})_{ {\mathfrak {t}}\in {\mathfrak {L}}_-}$. Finally, we write $H[{\mathfrak {L}}_-,{\mathscr {C}}]$ for the Hilbert space given as the closure of under the induced norm.

It is then not hard to see that for any $\xi$ as above one has that and $H[{\mathfrak {L}}_-,{\mathscr {C}}^\xi ]$ agree with and $H^\xi$, respectively.

3 Extension and translation of models

In this section we introduce the main technical tools and show key estimates needed to prove Theorem 1. On the technical level, pathwise differentiability (in Cameron Martin directions) of solutions to singular SPDEs can be effectively studied by introducing an extended regularity structure. The basic idea, which was already used in [7], is to extend the regularity structure by adding for any noise type $\Xi$ a new noise type ${\hat{\Xi }}$, which plays the role of an abstract place holder for a fixed Cameron–Martin function. We perform this extension in two separate steps. We first construct in a purely algebraic step, using the formalism developed in [5], an extended regularity structure. Afterwards we show in an analytic step, building up on the result of [10, Thm. 2.31], that any fixed Cameron–Martin function h can be indeed be "lifted" to a renormalized extended model, and, crucially, this lift is locally Lipschitz continuous in h.

For any $r \in {{\textbf {R}}}$ the original regularity structure then maps into the extended structure via a map ${\mathscr {S}}_r$, which is essentially the multiplicative extension of the map $\Xi \mapsto \Xi + r{\hat{\Xi }}$. Conversely, any extended model maps onto a model for the original structure via the "dual" map ${\mathscr {S}}_r^*$, which can be viewed as implementing this shift on an analytic level.

At the end of this section we are going to lift abstract fixed point problems to the extended regularity structure, and, using the shift operator, we also make sense of shifted fixed point problems.

3.1 Extension of the regularity structure

In the sequel it will be useful to consider general extensions of the set of noise types, and we are led to make the following general construction. Given a finite set I we define a new set of noise types by ${ {\hat{{\mathfrak {L}}}}_-^{I} } :={\mathfrak {L}}_- \times I$, and we write . We call the extended set of noise types. We also define the extended set of types by . There exists a natural map , which acts as the identity on ${\mathfrak {L}}$ and removes the second variable on ${\hat{{\mathfrak {L}}}}_-^I$, i.e. one has for any $\Xi \in {\mathfrak {L}}_-$ and any $i \in I$. We extend the homogeneity assignment $|\cdot |_{\mathfrak {s}}$ to by setting $|(\Xi ,i)|_{\mathfrak {s}}:=|\Xi |_{\mathfrak {s}}$ for any $\Xi \in {\mathfrak {L}}_-$ and any $i \in I$. In order to avoid case distinctions, we will sometimes add a distinct element $\star$ to the index set by setting $I^\star := I \sqcup \{\star \}$, and we identify with ${\mathfrak {L}}_- \times I^\star$.

Starting from the set of noise-types and the (unchanged) set of kernel-types ${\mathfrak {L}}_+$, we can consider an extension of the rule R to a rule , which is defined by allowing any appearance of any noise types $\Xi \in {\mathfrak {L}}_-$ being replaced by any extended noise type of the form $(\Xi ,i)$ for $i \in I$.

To be more precise, with the notation , we define a rule by setting

(19)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, and for . Here, we define by setting

(20)

for any ${\mathfrak {t}}\in {\mathfrak {L}}$ and any $k \in {{{\textbf {N}}}}^{d+1}$, where the sum runs over all with . The following Lemma shows that one can construct a regularity structure starting from the extended rule as in [5].

Lemma 1

For any finite set I the rule is a complete and subcritical rule. In particular, we can define the extended regularity structure as in [5, Sec. 5.5]. Then coincides with the span of all decorated trees $\tau = (T^{{\mathfrak {n}},{\mathfrak {o}}}_{\mathfrak {e}},{\mathfrak {t}})$ with and with the property that .

Proof

In order to see that is subcritical, recall from [5, Def. 5.14] and the fact that R is subcritical that there exists a function ${\text {reg}}: {\mathfrak {L}}\rightarrow {{\textbf {R}}}$ with the property that

$$\begin{aligned} {\text {reg}}({\mathfrak {t}}) < |{\mathfrak {t}}|_{\mathfrak {s}}+ \inf _{N \in R({\mathfrak {t}})} {\text {reg}}(N) \end{aligned}$$

(21)

for any ${\mathfrak {t}}\in {\mathfrak {L}}$. We extend ${\text {reg}}$ to a function by setting for any . Then one has , where is as in (20), and thus the fact that (21) holds for any is a trivial consequence from the respective bound for and the fact that . Completeness (c.f. [5, Def. 5.22]) is a little tedious to verify, but completely straight forward. $\square$

Note that we could always consider the completion of as in [5, Prop. 5.21], so that showing completeness is not really crucial. The construction in [5, Sec. 5.5,Sec.6] results in a number of spaces which are all completely determined by the rule . We adopt the convention that we use the notation to denote a space X constructed from , and we sometimes drop [I] from the notation, whenever I is clear from the context. In particular, we write and for algebras constructed in [5, Def. 5.26], we write and for the Hopf algebras constructed in [5, (5.23)], and we write and for the character group of and , respectively, compare [5, Def. 5.36].

For we write ${\hat{L}}(\tau ):= \{u \in L(\tau ): {\mathfrak {t}}(u) \in {\hat{{\mathfrak {L}}}}_-^I\}$. One has the obvious embedding , and multiplicatively extended we obtain a Hopf algebra monomorphism . This embedding between the Hopf algebras induces a natural group monomorphism between their character groups , which is defined by extending any character in such a way that $g(\tau )$ vanished for any tree $\tau$ outside of . We will use all of these embeddings implicitly, so that in particular we view as a sub Hopf algebra of , and we view as a subgroup of .

Given an admissible family for any we write for the admissible model constructed as in [5, (6.12),(6.13)] and we write similar to before and for the set of smooth reduced admissible models of the form and for their closure in the space of models, respectively. For any finite set I we consider the space , and we write elements of this set as tuples $(\eta ,h)$ or $(\eta ,(h_i)_{i\in I})$, with $h \in \Omega _\infty ({\hat{{\mathfrak {L}}}}_-^I)$ and , depending on the situation. Note that one has almost surely, and thus the canonical lift of $(\eta ,h)$ to a random admissible model is well defined. Finally, we denote by

the BPHZ renormalization. Here, we denote by the BPHZ character for the smooth stationary noise $\eta$, and we use the convention introduced above to view any character also as a character of .

The particular case that $I=\{1\}$ will play the most important role in the sequel. In this case we use the shorter notation $\hat{\Xi }:=(\Xi ,1)$ for any $\Xi \in {\mathfrak {L}}_-$, and we write and , and similar for the other spaces defined above. We call the onefold extension of . More generally, if $I=\{1, \ldots , m\}$, then we call the m-fold extension of .

Extensions of the set of noise-types can be used to conveniently encode shifts and differences between canonical lifts of smooth functions to models. This construction will allow us later in particular to almost automatically obtain Lipschitz bounds from uniform bounds applied to extended regularity structures. We make two constructions that we will use throughout the paper. For this, let be a regularity structure obtained from some noise-type set ${\mathfrak {L}}_-$, and let as above be its onefold extension.

The first constructions concerns "shifts" of models. For this, we introduce the operator by setting

$$\begin{aligned} {\mathscr {S}}(\tau ,{\mathfrak {t}}) := \sum _{{\tilde{{\mathfrak {t}}}}} (\tau ,{\tilde{{\mathfrak {t}}}}) \end{aligned}$$

(22)

for any typed, decorated tree , where the sum over ${\tilde{{\mathfrak {t}}}}$ runs over all maps such that . The shift operator algebraically encodes a binomial expansion of a tree $\tau$ when it is interpreted for the "shifted" noise $f+h$. The following Lemma shows in particular that this binomial expansion interacts nicely with the action of renormalization.

Lemma 2

For any $h,k \in \Omega _\infty ({\mathfrak {L}}_-)$ write $Z^{h+k}=(\Pi ^{h+k},\Gamma ^{h+k})$ and . Then, for any one has the identity

(23)

on for any $z \in {{{\textbf {D}}}}$.

Example 4

As an example, consider the tree form Example 2. We graphically represent the shifted noise type ${\hat{\Xi }}$ by , so that one has . The left and the right hand side of Eq. (23) for $g = {{\textbf {1}}}^\star$ then read respectively and .

Proof

For the identity element , this follows directly from the definition of the canonical lift and the definition of the shift operator (22). Indeed, since the canonical lift is multiplicative, it suffices to show (23) on planted trees $\tau$. If $\tau = \Xi$ for some $\Xi \in {\mathfrak {L}}_-$, then ${\mathscr {S}}\tau = \Xi + {\hat{\Xi }}$, and one has . Finally, if with ${\mathfrak {t}}\in {\mathfrak {L}}_+$, then one has , and the result follows inductively from the respective identity for $\sigma$ and the admissibility condition [17, (8.19)], see also [5, (6.14)].

If $g\ne {{\textbf {1}}}^*$ we use the fact that ${\mathscr {S}}$ commutes with the co-product (see Lemma 6) below) and the fact that by our convention g vanishes outside of , which by definition of ${\mathscr {S}}$ implies in particular that one has the identity $g{\mathscr {S}}= g$ on . It follows that one has

$$\begin{aligned} M^g {\mathscr {S}}= (g \otimes \text {Id}) \Delta _-^{\text {ex}}{\mathscr {S}}= (g {\mathscr {S}}\otimes {\mathscr {S}}) \Delta _-^{\text {ex}}= (g \otimes {\mathscr {S}}) \Delta _-^{\text {ex}}, \end{aligned}$$

on . Comparing this with $M^g = (g \otimes \text {Id}) \Delta _-^{\text {ex}}$, the result follows by applying the first part of the proof to the right components of these tensor products. $\square$

The second construction we are carrying out in this section concerns differences between shifts of canonical lifts. To this end we consider the sets and and the corresponding extended noise-type sets and . We write elements in and as pairs and quadrupels , respectively.

In order to construct the next operator, it will be helpful to fix for any tree a total order $\preceq$ on ${\hat{L}}(\tau )$. We then define an linear operator by setting for any typed, decorated tree

$$\begin{aligned} {\mathscr {A}}(\tau ,{\mathfrak {t}}):= \sum _{u \in {\hat{L}}(\tau )} (\tau , {\mathfrak {t}}[u]), \end{aligned}$$

where we define for any $u \in {\hat{L}}(\tau )$ the type map by setting

(24)

for any edge $e \in E(\tau )$. Here, ${\mathfrak {t}}_e^{(1)}$ denotes the first component of ${\mathfrak {t}}_e$, so that for any $e \in {\hat{L}} (\tau )$. We write ${\mathscr {A}}^\preceq$ and ${\mathfrak {t}}^\preceq [u]$ if we want to highlight the underlying family of total orderings on the sets $L(\tau )$ for $\tau \in {\mathscr {T}}_-^{\text {ex}}$ used to construct ${\mathscr {A}}$. The point of this total ordering is that we want to expand the difference into a telescoping sum, and the statement below will be valid for any total order of ${\hat{L}}(\tau )$. We will use this telescopic sum later on in order to obtain almost automatically local Lipschitz bounds from uniform bounds. The next Lemma shows in particular that this telescopic sum interacts nicely with the action of renormalization.

Lemma 3

For any $f,h,k \in \Omega _\infty ({\mathfrak {L}}_-)$ write and for the canonical lifts of (f, h) and $(f,h,k,h-k)$ to models in and , respectively. Then, for any one has the identity

(25)

on for any $z \in {{{\textbf {D}}}}$.

Proof

Assume first that $g= {{\textbf {1}}}^*$ and fix . The proof is somewhat complicated by the fact that ${\mathscr {A}}$ is not multiplicative with respect to the tree product due the arbitrary choice of $\preceq$. However, using an argument similar to the one given in the proof of Lemma 2, going inductively in the size of the tree, it is straight forward to see that one has the identity

for any $u \in {\hat{L}}(\tau )$, where we define ${\mathfrak {t}}^{\preceq }_+[u]$ and ${\mathfrak {t}}^{\preceq }_-[u]$ as in (24), but in the case $e=u$ with right hand side replaced by

Now, directly from the definition we get that whenever $u,v \in {\hat{L}}(\tau )$ are adjacent with respect to $\preceq$ and such that $u \preceq v$, then one has that ${\mathfrak {t}}^{\preceq }_+[u] = {\mathfrak {t}}^{\preceq }_-[v]$, so that turns into a telescopic sum, which is equal to

For a general character , note that the first part of the proof implies in particular that is independent of $\preceq$. Since moreover by our convention the character g vanished outside of it follows with an argument almost identically to the one given in Lemma 2 for the shift operator ${\mathscr {S}}$ that ${\mathscr {A}}$ satisfies the following identity

on ⁴. We conclude as in the proof of Lemma 2. $\square$

3.2 Extension of models

We now assume that we are given a partition of the set of noise-types into ${\mathfrak {L}}_- = {\mathfrak {L}}_-^{\text {rand}}\sqcup {\mathfrak {L}}_-^{\text {det}}$. We want to consider noises which are random, centred, stationary and Gaussian for $\Xi \in {\mathfrak {L}}_-^{\text {rand}}$ and deterministic for $\Xi \in {\mathfrak {L}}_-^{\text {det}}$. To this end, we introduce the notation that given a pair we write $Z^{\eta ,f}$ fo the canonical lift of the tupel $\eta \sqcup f$ to a random model. In such a situation, we furthermore want to consider a modification of negative renormalization that only takes into account diverging subtrees $\tau$ which have the property that all leaves $u \in L(\tau )$ have types ${\mathfrak {t}}(u) \in {\mathfrak {L}}_-^{\text {rand}}$. Denoting the set of trees $\tau \in {\mathscr {T}}_-^{\text {ex}}$ with this property by ${\mathscr {T}}_-^{\text {ex}}[{\mathfrak {L}}_-^{\text {rand}}]$, we define the character on trees $\tau \in {\mathscr {T}}_-^{\text {ex}}$ by setting if $\tau \in {\mathscr {T}}^{\text {ex}}_-[{\mathfrak {L}}_-^{\text {rand}}]$, and $g_{\textrm{BPHZ}}^\eta (\tau )=0$ otherwise, and extending this linearly and multiplicatively. Finally, we define the renormalized model by

$$\begin{aligned} {\hat{Z}}_{\textrm{BPHZ}}^{\eta ,f}:= M^{g^\eta _{\textrm{BPHZ}}} Z^{\eta , f}. \end{aligned}$$

We write $L^{\text {rand}}(\tau )$ and $L^{\text {det}}(\tau )$ for the set of $u \in L(\tau )$ with the property that ${\mathfrak {t}}(u) \in {\mathfrak {L}}_-^{\text {rand}}$ and ${\mathfrak {t}}(u) \in {\mathfrak {L}}_-^{\text {det}}$, respectively.

Remark 8

We are mainly going to be interested in the setting where is itself a one-fold extension with noise types ${\mathfrak {L}}_- \sqcup \hat{\mathfrak {L}}_-$, and one has ${\mathfrak {L}}_-^{\text {rand}}= {\mathfrak {L}}_-$ and ${\mathfrak {L}}_-^{\text {det}}= \hat{\mathfrak {L}}_-$. Note that in this case the notation of canonical lifts $Z^{\eta ,f}$ and the BPHZ character $g^\eta _{\textrm{BPHZ}}$ introduced above coincides with the notation introduced in Sect. 3.1.

We recall that, following arguments similar to [17, Thm. 7.8], see also [10, Thm. 2.31], convergence of ${\hat{Z}}^{\eta ^\varepsilon }_{\textrm{BPHZ}}=:({\hat{\Pi }}^{\eta ^\varepsilon }, \hat{\Gamma }^{\eta ^\varepsilon })$ in can essentially be reduced to bounds of the form

$$\begin{aligned} {{{\textbf {E}}}}| ({\hat{\Pi }}_z^{\eta ^\varepsilon } \tau )(\phi ^\lambda _z)|^2 \lesssim \lambda ^{2|\tau |_{\mathfrak {s}}+\kappa } \end{aligned}$$

(26)

uniformly in $\lambda \in (0,1)$, for some $\kappa >0$, any $\tau \in {\mathscr {T}}$ of negative homogeneity and any , compare [17, Thm. 7.8]. These moments can be conveniently represented as a finite sum over BPHZ-renormalized evaluations of graphs, each obtained via a "pairing" of the leaves of two disjoint copies of $\tau$, and the bound (26) follows from bounding each of these contractions separately. This was carried out in [10] by applying a purely analytical BPHZ theorem for (hyper-)graphs. In the sequel we will need to work with a slightly different formulation of this analytical bound and in order to state it we introduce some notation. We begin with a simple lemma.

Lemma 4

Given $w,z \in {{{\textbf {D}}}}$ and a tree $\tau \in {\mathscr {T}}$ there exists a unique locally integrable function $\Lambda _{z;w}\tau : {{{\textbf {D}}}}^{L(\tau )} \rightarrow {{\textbf {R}}}$, smooth away from the big diagonal⁵ and away from w, z, symmetric under any permutation $\sigma$ of $L(\tau )$ with the property that ${\mathfrak {t}}\circ \sigma = {\mathfrak {t}}$, and such that one has

$$\begin{aligned} \Pi ^\eta _w \tau (z) = \int _{{{{\textbf {D}}}}^ {L(\tau )} } dx \, \Lambda _{z;w}\tau (x_{L(\tau )}) \prod _ {u \in L(\tau )} \eta _{{\mathfrak {t}}(u)}(x_{u}). \end{aligned}$$

for any $\eta \in \Omega _\infty$.

Proof

The proof is straightforward using induction over the number of edges of $\tau$, the fact that the canonical lift is an admissible model, and the fact that $\Pi _w^\eta$ is multiplicative for the tree product. $\square$

Furthermore, for any $f \in \Omega _\infty ({\mathfrak {L}}_-^{\text {rand}})$ and ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$ we define the function $\Lambda _{z;w}^{f, {\mathscr {C}}}: {{{\textbf {D}}}}^{ L ^{\text {det}}(\tau ) } \rightarrow {{\textbf {R}}}$ by setting

$$\begin{aligned} \Lambda _{z;w} ^{f,{\mathscr {C}}} \tau ( x _{ L ^{\text {det}}( \tau ) } ):= \int _{{{{\textbf {D}}}}^{ L ( \tau ) } } dy_{ L (\tau ) } \; \Lambda _{z;w} \tau ( y _{ L ( \tau ) } ) \prod _{ u \in L ^ {\text {rand}}(\tau ) } f _{{\mathfrak {t}}(u)} (y_u) \prod _{ u \in L ^ {\text {det}}( \tau ) } {\mathscr {C}}_{{\mathfrak {t}}(u)}(x_u - y_u). \end{aligned}$$

It follows that $\Lambda _{z;w} ^{f, {\mathscr {C}}}$ is symmetric under any permutation $\sigma$ of $L ^{\text {det}}(\tau )$ with the property that ${\mathfrak {t}}\circ \sigma = {\mathfrak {t}}$ on $L^{\text {det}}( \tau )$, so that we can naturally view the domain of definition of $\Lambda _{z;w} ^{f,{\mathscr {C}}}$ as ${{{\textbf {D}}}}^ { \texttt {m}^{\text {det}}(\tau )}$, where $\texttt {m}^{\text {det}}(\tau )$ is the multiset given by $\texttt {m}^{\text {det}}(\tau ):=[L^{\text {det}}(\tau ),{\mathfrak {t}}]$, compare the notation introduced in Sect. 2.1. We will switch frequently between these two pictures in the sequel.

Given a ${\mathfrak {L}}_-^{\text {det}}$-valued multiset $\texttt {m}$ and a kernel ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$, we let be the space given by all functions which can be written in the form

$$\begin{aligned} F(x_\texttt {m}) := \int _{{{{\textbf {D}}}}^\texttt {m}} d{\bar{x}}_{\texttt {m}} G( {\bar{x}}_\texttt {m}) \prod _{u \in \texttt {m}} {\mathscr {C}}_{{\mathfrak {t}}(u)}(x_u - {\bar{x}} _u) \end{aligned}$$

(27)

for some , endowed with the scalar product

$$\begin{aligned} \left\langle F, {\bar{F}} \right\rangle _{\texttt {m}, {\mathscr {C}}} := \int _{ {{{\textbf {D}}}}^{ \texttt {m}} \times {{{\textbf {D}}}}^{ \texttt {m}} } dx _{ \texttt {m}} d {\bar{x}} _{ \texttt {m}} G( x_ \texttt {m}) {\bar{G}} ( {\bar{x}} _ \texttt {m}) \prod _{u \in \texttt {m}} {\mathscr {C}}_{ {\mathfrak {t}}( u ) } ( x_u - {\bar{x}}_{u} ), \end{aligned}$$

(28)

where G and ${\bar{G}}$ are as in (27) for F and $\bar{F}$ respectively, and we write $H_{\texttt {m},{\mathscr {C}}}$ for the closure of under the induced norm. We also write for the linear subspace spanned by trees $\tau \in {\mathscr {T}}^{\text {ex}}$ with the property that one has $\texttt {m}^{\text {det}}(\tau ) = \texttt {m}$, and we note that for any $f \in \Omega _\infty ({\mathfrak {L}}_-^{\text {rand}})$ and any $w,z \in {{{\textbf {D}}}}$ one has

Finally, it follows directly from the definition of the coproduct $\Delta _-^{\text {ex}}$ and the character $g_{\textrm{BPHZ}}^\eta$ that for any one has

for any multiset $\texttt {m}$, so that it makes sense to define the random variable

$$\begin{aligned} \Lambda ^{\eta , {\mathscr {C}}}_{z, {\bar{z}}; w} \tau := \left\langle \Lambda _{z; w} ^{\eta , {\mathscr {C}}} M^{g_{\textrm{BPHZ}}^\eta } \tau , \, \Lambda _{{\bar{z}};w} ^{\eta , {\mathscr {C}}} M^{g_{\textrm{BPHZ}}^\eta } \tau \right\rangle _{\texttt {m}, {\mathscr {C}}} \end{aligned}$$

for any . The following theorem contains the key analytic bound on which the analysis below is bases on.

Theorem 6

For any $C>0$ and any compact $K \subseteq {{{\textbf {D}}}}$ one has the bound

$$\begin{aligned} \sup _{w \in K} {{{\textbf {E}}}}\Big | \int _{ {{{\textbf {D}}}}\times {{{\textbf {D}}}}} dz d{\bar{z}} \, \Lambda ^{\eta ,{\mathscr {C}}}_{z, {\bar{z}}; w} \tau \, \phi ^\lambda _w(z) \phi ^\lambda _w({\bar{z}}) \Big | \lesssim \lambda ^{2 |\tau |_{\mathfrak {s}}+ \theta } \end{aligned}$$

(29)

uniformly over $\lambda \in (0,1)$, ${\mathscr {C}}\in {\mathfrak {C}}( {\mathfrak {L}}_- ^{\text {det}})$ with $\Vert {\mathscr {C}}\Vert _{|\cdot |_{\mathfrak {s}}}\le C$, and with $\Vert \eta \Vert _{|\cdot |_{\mathfrak {s}}}\le C$, for $\theta >0$ small enough and for any $\tau \in {\mathscr {T}}$ of negative homogeneity and any .

Remark 9

The reason for writing the above expression in this unusual way is that we are going to apply Cauchy-Schwarz estimates in the Hilbert space $H_{{\mathfrak {m}},{\mathscr {C}}}$.

Proof

First note that we can replace ${\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}$ with a regularization ${\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}^\varepsilon :={\mathscr {C}}_{{\mathfrak {t}},{\mathfrak {t}}'}*\rho ^{(\varepsilon )}$ for some symmetric, non-negative definite mollifier $\rho$ and then take the limit $\varepsilon \rightarrow 0$, so that it suffices to show (29) for smooth kernels ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$. Furthermore, by definition ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}^{\text {det}}_-)$ is non-negative definite when viewed as an integral operator on $L^2({{{\textbf {D}}}})^{{\mathfrak {L}}_-^{\text {det}}}$, and as a consequence there exists a (unique) Gaussian, centred, stationary noise with the property that

$$\begin{aligned} {{{\textbf {E}}}}[{\bar{\eta }}_{\mathfrak {t}}(\phi ) {\bar{\eta }}_{{\mathfrak {t}}'}(\psi )] = \delta _{{\mathfrak {t}},{\mathfrak {t}}'} \int _{{{{\textbf {D}}}}\times {{{\textbf {D}}}}} dx dy {\mathscr {C}}_{{\mathfrak {t}}}(x-y) \phi (x) \psi (y) \end{aligned}$$

for any choice of test functions and any ${\mathfrak {t}},{\mathfrak {t}}' \in {\mathfrak {L}}_-^{\text {det}}$. Enlarging the probability space $(\Omega ,{{\textbf {P}}})$ if necessary, we can additionally assume that $\Omega =(\Omega _{\text {rand}}\times \Omega _{\text {det}})$ and ${{\textbf {P}}}={{\textbf {P}}}_{\text {rand}}\otimes {{\textbf {P}}}_{\text {det}}$ for some probability measures ${{\textbf {P}}}_{\text {rand}}$ on $\Omega _{\text {rand}}$ and ${{\textbf {P}}}_{\text {det}}$ on $\Omega _{\text {det}}$, respectively, and such that $\eta$ respectively ${\bar{\eta }}$ is a collection of random fields on $(\Omega _{\text {rand}},{{\textbf {P}}}_{\text {rand}})$ respectively $(\Omega _{\text {det}},{{\textbf {P}}}_{\text {det}})$. In particular, one has that $\eta$ and ${\bar{\eta }}$ are independent, and we write . We denote as usual the BPHZ character for $\xi$ by and we denote the BPHZ renormalized canonical lift of $\xi$ to a model by ${\hat{Z}}_{\textrm{BPHZ}}^\xi =({\hat{\Pi }}^\xi , {\hat{\Gamma }}^\xi )$.

We first assume that the tree $\tau \in {\mathscr {T}}$ has the property that for any noise type $\Xi \in {\mathfrak {L}}_-^{\text {det}}$ there exists at most one $u \in L(\tau )$ such that ${\mathfrak {t}}(u) = \Xi$. We claim that in this case one has

$$\begin{aligned} {{{\textbf {E}}}}\Big | \int _{ {{{\textbf {D}}}}\times {{{\textbf {D}}}}} dz d{\bar{z}} \, \Lambda ^{\eta ,{\mathscr {C}}}_{z, {\bar{z}}; w} \tau \, \phi ^\lambda _w(z) \phi ^\lambda _w({\bar{z}}) \Big | = {{{\textbf {E}}}}\Big | \int _{ {{{\textbf {D}}}}} dz \hat{\Pi }^\xi _{w} \tau (z) \phi ^\lambda _w(z) \Big |^2, \end{aligned}$$

(30)

from which (29) follows from Theorem 3. In order to see (30) note that from the assumption on $\tau$ and the fact that $\xi _{\mathfrak {t}}$ and $\xi _{{\mathfrak {t}}'}$ are independent for any ${\mathfrak {t}}\in {\mathfrak {L}}_-^{\text {det}}$ and any ${\mathfrak {t}}' \in {\mathfrak {L}}_-^{\text {rand}}$ it follows that $g^\xi _{\textrm{BPHZ}}$ vanishes on subtrees $\sigma \subseteq \tau$ with the property that , which in turn implies that one has the identity

$$\begin{aligned} M^{g_{\textrm{BPHZ}}^\eta } \tau = M^{g_{\textrm{BPHZ}}^\xi } \tau . \end{aligned}$$

A fortiori it follows that one has

$$\begin{aligned} \Lambda ^{\eta ,{\mathscr {C}}}_{z, {\bar{z}}; w} = {{{\textbf {E}}}}^{{{\textbf {P}}}_{\text {det}}} {\hat{\Pi }}^\xi _w(z) {\hat{\Pi }}^\xi _w({\bar{z}}) \qquad \quad {{\textbf {P}}}_{\text {rand}}-a.s., \end{aligned}$$

and (30) follows.

In the general case, define for $\Xi \in {\mathfrak {L}}_-^{\text {det}}$ the number $m(\Xi ) \in {{{\textbf {N}}}}\cup \{0\}$ as the number of noise-type edges $u \in L(\tau )$ with the property that ${\mathfrak {t}}(u) = \Xi$, and let $m:= \max _{\Xi \in {\mathfrak {L}}_-^{\text {det}}} m(\Xi )$. We then consider the m-fold extension of .

We define and , and we define an element by setting

$$\begin{aligned} {\tilde{{\mathscr {C}}}}_{\Xi ,\Xi '}:= {\tilde{{\mathscr {C}}}}_{\Xi ,(\Xi ',k')}:=0 \qquad \text { and }\qquad {\tilde{{\mathscr {C}}}}_{(\Xi ,k),(\Xi ',k')}:= \delta _{k.k'} {\mathscr {C}}_{\Xi ,\Xi '} \end{aligned}$$

for any $\Xi ,\Xi ' \in {\mathfrak {L}}_-$ and any $k,k'\in [m]$. Moreover, we define $\Phi$ as the set of all type maps such that ${\tilde{{\mathfrak {t}}}}(e) = {\mathfrak {t}}(e)$ for any kernel-type edge $e \in K(\tau )$ and such that the following holds.

• For any noise-type edge $u \in L^{\text {rand}}(\tau )$ one has ${\tilde{{\mathfrak {t}}}}(u) = {\mathfrak {t}}(u)$.

• For any noise-type edge $u \in L^{\text {det}}(\tau )$ one has ${\tilde{{\mathfrak {t}}}} (u) = ({\mathfrak {t}}(u), k(u))$ for some $k(u) \in [m]$.

• For any noise-type $\Xi \in {\mathfrak {L}}_-^{\text {det}}$ the map k restricted to $L_\Xi (\tau ):=\{u \in L(\tau ): {\mathfrak {t}}(u) = \Xi \}$ is a bijection from $L_\Xi (\tau )$ onto $[m(\Xi )]$.

We note the following consequences of this definition: For any ${\tilde{{\mathfrak {t}}}} \in \Phi$ one has and $(\tau ,{\tilde{{\mathfrak {t}}}})$ satisfies the assumption of the first part of the proof. It remains to apply the results of the first part to any of the trees $(\tau ,{\tilde{{\mathfrak {t}}}})$ individually, note that one has

$$\begin{aligned} \Lambda _{z; w}^{\eta ,{\mathscr {C}}} \tau (x_{L^{\text {det}}(\tau )}) = \frac{1}{S} \sum _{{\tilde{{\mathfrak {t}}}} \in \Phi } \Lambda _{z;w} ^{\eta ,{\tilde{{\mathscr {C}}}}} (\tau ,{\tilde{{\mathfrak {t}}}}) (x_{L^{\text {det}}(\tau )}), \end{aligned}$$

and use a Cauchy-Schwarz estimate. Here S is a symmetry factor given by $S = \prod _{\Xi \in {\mathfrak {L}}_-^{\text {det}}}{m(\Xi )!}.$ $\square$

In order to continue, we first note a relation between the norms of the spaces ${ H[{\mathfrak {L}}_-^{\text {det}}, {\mathscr {C}}]}$ defined in (18) and $\Vert \cdot \Vert _{{\mathfrak {m}},{\mathscr {C}}}$ defined in (28).

Lemma 5

Let $\texttt {m}$ be any multi-set with values in ${\mathfrak {L}}_-^{\text {det}}$, let ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$, let $h \in H[{\mathfrak {L}}_-^{\text {det}}, {\mathscr {C}}]$, and define ${\mathfrak {h}}\in H_{\texttt {m},{\mathscr {C}}}$ by setting

$$\begin{aligned} {\mathfrak {h}}:= \bigotimes _{u \in \texttt {m}} h_{{\mathfrak {t}}(u)}. \end{aligned}$$

Then one has

$$\begin{aligned} \Vert {\mathfrak {h}}\Vert _{\texttt {m},{\mathscr {C}}} \lesssim (\Vert h\Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]})^{\#\texttt {m}}. \end{aligned}$$

Proof

By definition (28), it suffices to show the statement for $\# \texttt {m}= 1$. In this case write ${\mathfrak {t}}\in {\mathfrak {L}}_-^{\text {det}}$ for the type such that one has $\texttt {m}=\{{\mathfrak {t}}\}$ and note that, writing $\pi _{\mathfrak {t}}:H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}] \rightarrow H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ for the projection given by $(\pi _{\mathfrak {t}}(k))_{{\mathfrak {t}}'}:= k_{{\mathfrak {t}}} \delta _{{\mathfrak {t}},{\mathfrak {t}}'}$ for any $k \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$, we have the equality $\Vert {\mathfrak {h}}\Vert _{\texttt {m},{\mathscr {C}}} = \Vert \pi _{\mathfrak {t}}h\Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]}$. The result now follows from the fact that the $\pi _{\mathfrak {t}}$ is a continuous projection, since both kernel and range of $\pi _{\mathfrak {t}}$ are closed in $H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$. $\square$

Recall that for we defined the multiset $\texttt {m}^{\text {det}}(\tau )=[L^{\text {det}}(\tau ),{\mathfrak {t}}]$, and with this notation we define for any $h \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ the quantities

$$\begin{aligned} {\mathfrak {h}}^h_\tau := \bigotimes _{u \in \texttt {m}^{\text {det}}(\tau )} h_{{\mathfrak {t}}(u)} \qquad \text { and }\qquad [h]_\tau := \Vert {\mathfrak {h}}^h_\tau \Vert _{\texttt {m}^{\text {det}}(\tau ),{\mathscr {C}}}. \end{aligned}$$

(31)

We stress that $[\cdot ]_\tau$ fails to be a semi-norm unless $\#\texttt {m}^{\text {det}}(\tau )=1$. Our ultimate goal is to show that $(\eta ,f) \mapsto {\hat{Z}}^{\eta ,f}_{\textrm{BPHZ}}$ can be extended by mollification to and $f \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ for any kernel ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$. As a preparation for this statement, we show the following result.

Proposition 2

Let be a regularity structure constructed as in Sect. 2.2 satisfying Assumption 2, and let ${\mathfrak {L}}_-={\mathfrak {L}}_-^{\text {det}}$ and ${\mathfrak {L}}_-^{\text {rand}}= \emptyset$. Let moreover ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-)$ be a kernel. Then the map , $h \mapsto Z^h$, extends to a continuous map from $H[{\mathfrak {L}}_-,{\mathscr {C}}]$ into which is locally Lipschitz continuous in the sense that for any $\gamma \in {{\textbf {R}}}$, any compact $K \subseteq {{{\textbf {D}}}}$ and any $R>0$ one has

(32)

uniformly over all $h,k \in H[{\mathfrak {L}}_-,{\mathscr {C}}]$ with $\Vert h\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} \vee \Vert k \Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]}<R$.

Proof

For any fixed $\gamma >0$ and compact $K \subseteq {{{\textbf {D}}}}$ the pseudo metric induces a complete metric space via metric identification, so that it is sufficient to show that one has the bound (32) for any ; note for this that is dense in , and thus any such local Lipschitz map has a unique extension to which is again locally Lipschitz, and this concludes the proof. Following arguments identical to [17, Thm. 10.7], it suffices to show that for any tree of negative homogeneity and any with ${{\,\textrm{supp}\,}}\phi \subseteq B_1(0)$ there exists $\theta >0$ such that one has the bound

$$\begin{aligned} |(\Pi ^{h}_z \tau - \Pi ^k_z \tau )(\phi _z^\lambda )| \lesssim \lambda ^{|\tau |_{\mathfrak {s}}+\theta } \Vert h-k\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} \end{aligned}$$

(33)

uniformly over with $\Vert h\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} \vee \Vert k\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]}\le R$, $z \in K$ and $\lambda \in (0,1)$.

We first show that one has the bound

$$\begin{aligned} |\Pi ^{h}_z \tau (\phi _z^\lambda )| \lesssim \lambda ^{|\tau |_{\mathfrak {s}}+\theta } [h]_\tau . \end{aligned}$$

(34)

uniformly over with $\Vert h \Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} \le R$, $z \in K$ and $\lambda \in (0,1)$. For this we use the identity

(35)

from which a Cauchy-Schwarz estimate on the Hilbert space $H_{ \texttt {m}^{\text {det}}(\tau ), {\mathscr {C}}}$ shows that the left hand side of (34) can be estimated by

$$\begin{aligned} \Vert {\mathfrak {h}}^h_\tau \Vert _{ \texttt {m}(\tau ), {\mathscr {C}}} \, \Vert F_z \Vert _{ \texttt {m}(\tau ), {\mathscr {C}}}, \end{aligned}$$

where . Comparing the second term in this expression with Theorem 6, the estimate (34) follows.

The bound (33) is now an almost immediate consequence of (34) applied to extended regularity structure and Lemma 3 applied for $g = {{\textbf {1}}}^*$. Indeed, first note that one has for any typed tree the identity⁶

where on the right hand side we denote the canonical lift of (0, h) to a model for the onefold extension of , and we write ${\hat{{\mathfrak {t}}}}:= {\mathfrak {t}}$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. By Lemma 3 it follows that one has

Applying (34) to each tree on the right hand side of this identity, we obtain the desired bound. Note for this that Lemma 5 implies in particular that $[h]_\tau \lesssim \Vert h-k\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]}$ uniformly over all $h,k \in H[{\mathfrak {L}}_-,{\mathscr {C}}]$ with $\Vert h\Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]} \vee \Vert k \Vert _{H[{\mathfrak {L}}_-,{\mathscr {C}}]}<R$. $\square$

The main stochastic ingredient for the proof of Theorem 7 below is the following bound, for which we introduce the notation ${\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}} = (\hat{\Pi }^{\eta ,h}, {\hat{\Gamma }}^{\eta ,h})$ for any . We then have the following.

Proposition 3

Let be a regularity structure constructed as in Sect. 2.2 satisfying Assumption 2, and assume that we are given the decomposition ${\mathfrak {L}}_- = {\mathfrak {L}}_-^{\text {det}}\sqcup {\mathfrak {L}}_-^{\text {rand}}$, and let ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$. Then there exists $\kappa >0$ such that one has for any compact $K \subseteq {{{\textbf {D}}}}$, any $p\ge 1$, any $C>0$, and any , and any test function the bound

$$\begin{aligned} \sup _{z \in K} {{{\textbf {E}}}}\Big [ \sup _{ h \in H[{\mathfrak {L}}_-^{\text {det}}, {\mathscr {C}}] } \frac{1}{[h]_\tau } |({\hat{\Pi }}^{\eta ,h}_z \tau ) (\phi ^\lambda _z)| \Big ] ^{2p} \lesssim \lambda ^{2p|\tau |_{\mathfrak {s}}+ 2p\kappa } \end{aligned}$$

(36)

uniformly over $\lambda \in (0,1)$ and with $\Vert \eta \Vert _{|\cdot |_{\mathfrak {s}}} \le C$.

Proof

First note that ${\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}}$ is almost surely well defined for any and any $h \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ by Proposition 2. Furthermore, using Gaussian hypercontractivity it suffices to show this proposition for $p=1$.

We now write ${\mathfrak {h}}^h_\tau \in H_{\texttt {m}^{\text {det}}(\tau ),{\mathscr {C}}}$ for the function in (31), and we use the identity

$$\begin{aligned} {\hat{\Pi }}_w^{\eta , h}\tau (z) = \Pi ^{\eta , h}_w M^{g_{\textrm{BPHZ}}^\eta } \tau (z) = \left\langle {\hat{\Lambda }}^{\eta ,{\mathscr {C}}}_{z; w} M^{g_{\textrm{BPHZ}}^\eta } \tau , \, {\mathfrak {h}}^h_\tau \right\rangle _{\texttt {m}^{\text {det}}(\tau ), {\mathscr {C}}} \end{aligned}$$

so that by Cauchy-Schwarz we obtain the estimate

$$\begin{aligned} |({\hat{\Pi }}^{\eta ,h}_z \tau ) (\phi ^\lambda _z)| \le \Big | \int _{ {{{\textbf {D}}}}\times {{{\textbf {D}}}}} dz d{\bar{z}} \, \Lambda ^{\eta ,{\mathscr {C}}}_{z, \bar{z}; w} \tau \, \phi ^\lambda _w(z) \phi ^\lambda _w({\bar{z}}) \Big | ^{\frac{1}{2} } \Vert {\mathfrak {h}}^h_\tau \Vert _{\texttt {m},{\mathscr {C}}} \end{aligned}$$

The result now follows from Theorem 6. $\square$

We demonstrate the idea of the proof of Proposition 3 with the following example.

Example 5

We consider a situation similar to Example 2, but this time we assume we have two noise types with and , and we write . We want to derive in detail the bound (36) for and $p=1$. First, we can write

where on the right hand side we take the -norm of the map , where $\delta _y$ denotes the $\delta$-distribution centred around y. This is a slight abuse of notation, explicitly this expression is equal to

Now, let be a Gaussian noise independent of $\eta$ and such that . It then follows from the definition of the Hilbert space that we have the identity

From this expression we obtain the bound (36) from [10, Thm. 2.31].

The main result of this section is the following theorem.

Theorem 7

Assume we are given the decomposition ${\mathfrak {L}}_- = {\mathfrak {L}}_-^{\text {det}}\sqcup {\mathfrak {L}}_-^{\text {rand}}$, and let ${\mathscr {C}}\in {\mathfrak {C}}({\mathfrak {L}}_-^{\text {det}})$. Then the map , $(\eta ,h) \mapsto {\hat{Z}}^{\eta , h}_{\textrm{BPHZ}}$ can be extended by mollification to a continuous map from into .

Moreover, for any there exists a null set N such that for any $\omega \in N^c$ one has that the map $h \mapsto {\hat{Z}}^{\eta , h}_{\textrm{BPHZ}}(\omega )$ is locally Lipschitz continuous in the sense for any $\gamma \in {{\textbf {R}}}$, any compact $K \subseteq {{{\textbf {D}}}}$ and any $R>0$ one has

(37)

uniformly over all $h,k \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ with $\Vert h\Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]} \vee \Vert k \Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]}<R$.

Finally, given an approximation of $\eta$ there exists a subsequence $\varepsilon \rightarrow 0$ and a null set N with the property that $\eta ^\varepsilon (\omega ) \rightarrow \eta (\omega )$ and ${\hat{Z}}^{\eta ^\varepsilon ,h}_{\textrm{BPHZ}}(\omega ) \rightarrow {\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}}(\omega )$ for any $h \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ and $\omega \in N^c$.

Remark 10

At this stage we point out a key difference between the present approach and [7]. In the latter the authors obtain a stronger statement as they construct a deterministic continuous extension operator which is such that it maps $({\hat{Z}}^\eta _{\textrm{BPHZ}}(\omega ),h)$ onto ${\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}}(\omega )$ for almost every fixed $\omega$. This however comes at the expense of analytical difficulties that could only be carried out in a very special case. In the current paper in contrast, the analytic difficulties are bypassed by constructing the extension stochastically, which in particular allows us to use the results of [10, Thm. 2.31] to show the necessary analytic estimates. This comes at the expense of a somewhat weaker statement but has the advantage of being immediately completely general.

Remark 11

The proof of this theorem relies on a "noise doubling" strategy. This is very similar to the strategy exploited previously in [22] in the context of rough path theory.

Proof

We first note that as a consequence of Proposition 3 for any and $h \in H[{\mathfrak {L}}_-^{\text {det}}, {\mathscr {C}}]$ there exists a random variable ${\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}}$ taking values in such that for some $\theta >0$ one has

(38)

uniformly in $\varepsilon >0$, where is an approximation of $\eta$. To see this, note that the first bound above is a consequence of Proposition 3 using an argument identical to the one given in [17, Thm. 10.7]. The existence of the extension ${\hat{Z}}^{\eta , h}_{\textrm{BPHZ}}$ and the second bound in (38) follow now along the usual lines, see for instance [17, Sec. 10.4,10.5] or the proof of [10, Thm. 2.34].

To see (37), let and let be an approximation of $\eta$. Let moreover ${\tilde{H}} \subseteq H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ be a dense and countable subset. It suffices to fix a subsequence $\varepsilon \rightarrow 0$ such that ${{\hat{Z}}}^{\eta ^\varepsilon , h}_{\textrm{BPHZ}} \rightarrow {{\hat{Z}}}^{\eta , h}_{\textrm{BPHZ}}$ almost surely for any $h \in {\tilde{H}}$ as $\varepsilon \rightarrow 0$, and to show that (37) holds almost surely with $\eta$ replaced by $\eta ^\varepsilon$ uniformly over $\varepsilon > 0$ and $h,k \in {\tilde{H}}$ with $\Vert h\Vert _{} \vee \Vert k\Vert \le R$, where we write $\Vert h\Vert :=\Vert h\Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]}$ from now on in order to simplify notation. Upon choosing a sub-subsequence, this follows from the estimate

(39)

uniformly over $\varepsilon >0$. Using again an argument identical to the one given in [17, Thm. 10.7] we see that (39) is implied once we show for any fixed and any the bound

$$\begin{aligned} \sup _{x \in K} {{{\textbf {E}}}}\Big [ { \sup _{\begin{array}{c} h,k \in {\tilde{H}} \\ \Vert h\Vert \vee \Vert k\Vert \le R \end{array}} } \Vert h - k \Vert _{H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]}^{-2p} | (\hat{\Pi }_x^{\eta ^\varepsilon ,h} \tau - {\hat{\Pi }}_x^{\eta ^\varepsilon ,k} \tau ) (\phi _x^\lambda )|^{2p} \Big ] \lesssim \lambda ^{2p|\tau |_{\mathfrak {s}}+ 2p\theta } \end{aligned}$$

(40)

uniformly over $\lambda \in (0,1)$ for any $p\ge 1$.

We will show this by applying (36) to the extended regularity structure constructed in Lemma 3. For this consider the set and we denote by the corresponding extended set of noise-types. The set ${\hat{{\mathfrak {L}}}}_-^J$ comes with a natural decomposition into $\hat{\mathfrak {L}}_-^{J,{\text {det}}}:= {\mathfrak {L}}_-^{\text {det}}\times J$ and ${\hat{{\mathfrak {L}}}}_-^{J,{\text {rand}}}:= {\mathfrak {L}}_-^{\text {rand}}\times J$.

We define and, given $h \in {\tilde{H}}$, we define by setting

Now we note that one has for any the identity , where we set ${\tilde{{\mathfrak {t}}}}:= {\mathfrak {t}}$ if ${\mathfrak {t}}\in {\mathfrak {L}}_-^{\text {rand}}\sqcup {\mathfrak {L}}_+$ and if ${\mathfrak {t}}\in {\mathfrak {L}}_-^{\text {det}}$. By Lemma 3 it follows that one has the identity

$$\begin{aligned} ({\hat{\Pi }}_x^{\eta ^\varepsilon ,h} \tau - {\hat{\Pi }}_x^{\eta ^\varepsilon ,k} \tau ) = ({\hat{\Pi }}^{\eta ^\varepsilon , h, k, h-k}_x {\mathscr {A}}\tau ), \end{aligned}$$

and by definition of ${\mathscr {A}}$ we conclude that (36) implies (40).

The fact that the null set N such that ${\hat{Z}}^{\eta ^\varepsilon ,h}_{\textrm{BPHZ}}(\omega ) \rightarrow {\hat{Z}}^{\eta ,h}_{\textrm{BPHZ}}(\omega )$ for $\omega \in N^c$ can be chosen independently of $h \in H[{\mathfrak {L}}_-^{\text {det}},{\mathscr {C}}]$ is now a direct consequence of the uniform bound of the local Lipschitz constants. $\square$

The main application of the previous theorem is the situation in which the regularity structure is itself a onefold extension with noise types given by ${\mathfrak {L}}_- \sqcup {\hat{{\mathfrak {L}}}}_-$, and one has that ${\mathfrak {L}}^{\text {rand}}_- = {\mathfrak {L}}_-$ and ${\mathfrak {L}}^{\text {det}}_- = {\hat{{\mathfrak {L}}}}_-$. This leads to the following immediate corollary.

Corollary 2

Let and let be an approximation of $\xi$. Then, there exists a subsequence $\varepsilon \rightarrow 0$ and a null set $N \subseteq \Omega$ such that the following holds. For any $\omega \in N^c$ and $h\in H^\eta$ one has that converges to as $\varepsilon \rightarrow 0$, and one has that for any $\omega \in N^c$ the map is locally Lipschitz continuous from $H^{\eta }$ into .

3.3 Shift of models

We recall the map from Sect. 3.1 defined as the identity on ${\mathfrak {L}}$ and mapping ${\hat{\Xi }}$ onto $\Xi$ for any $\Xi \in {\mathfrak {L}}_-$. We extend this map to a projection by defining for any typed tree with type map the tree

and extending linearly to all of . An important role is then played by the following operator, which generalizes the operator defined in (22).

Definition 7

For we denote by the set of trees such that . For any $r \in {{\textbf {R}}}$ we define the linear operator by setting for any tree $\tau \in {\mathscr {T}}^{\text {ex}}$

$$\begin{aligned} {\mathscr {S}}_r \tau := \sum _{\sigma \in {\mathscr {S}}[\tau ]} r^{m(\sigma )}\sigma . \end{aligned}$$

where $m(\sigma ):= \# {\hat{L}}(\sigma )$. We call ${\mathscr {S}}_r$ the shift operator, and we write for the image of the shift operator.

Example 6

On the tree from Example 2, writing and , we obtain the formula .

We extend the shift operator linearly and multiplicatively to the algebras and , as well as to the Hopf algebras and . The following is a simple consequence of the definition.

Lemma 6

For any $r \in {{\textbf {R}}}$ the shift operator ${\mathscr {S}}_r$ commutes with the action of both $\Delta _-^{\text {ex}}$ and $\Delta _+^{\text {ex}}$ on . In particular, its image forms a sector in and ${\mathscr {S}}_r$ commutes with the action of renormalization $M^g$ for any . Moreover, ${\mathscr {S}}_r$ maps into , is multiplicative under the tree product, and commutes with the operation of compositions with smooth functions, integration and differentiation.

Proof

The fact that ${\mathscr {S}}_r$ commutes with the action of the co-products is tedious to verify, but straightforward.

To see that forms a sector, let $\Gamma =(\text {Id}\otimes \gamma )\Delta ^+_{\text {ex}}$ for some character as in [5, (6.13)]. Then one has

for any . Similarly, for any one has

$$\begin{aligned} {\mathscr {S}}_r M^g = (g \otimes {\mathscr {S}}_r) \Delta _{\text {ex}}^- = (g \otimes \text {Id}) ({\mathscr {S}}_r \otimes {\mathscr {S}}_r) \Delta _{\text {ex}}^- = M^g {\mathscr {S}}_r. \end{aligned}$$

Here we used the fact that by definition of the embedding one has that $g = g {\mathscr {S}}_r$ for any .

The remaining statements of the lemma are a simple consequence of the definitions. $\square$

It follows directly from the definition that the operator ${\mathscr {S}}_r$ is one to one, so that we can define its inverse ${\mathscr {S}}_r^{-1}$ on the sector . We now denote by⁷ and the maps given respectively by

Lemma 7

The map is well defined and extends to a locally Lipschitz map from onto . Moreover, writing and , one has the identities

(41)

for any $x,y \in {{{\textbf {D}}}}$.

Proof

First note the right hand side of second identity in (41) makes sense, since by Lemma 6 the image forms a sector, so that it is invariant under .

In order to see that the map is well defined, note first that it is clear from the definition that ${\mathscr {S}}_r^*$ maps admissible ${\varvec{\Pi }}$ onto admissible ${\mathscr {S}}_r^* {\varvec{\Pi }}$, so that it remains to show the required analytic bounds. Noting that the map ${\mathscr {S}}_r$ leaves homogeneities of trees invariant, these analytic bounds follow once we show that the identity (41) holds for any and given by the expressions [5, (6.12)] and [5, (6.13)], respectively, but with ${\varvec{\Pi }}$ replaced by ${\mathscr {S}}_r^* {\varvec{\Pi }}$. But as a direct consequence of Lemma 6 it follows that ${\mathscr {S}}_r$ commutes with the positive twisted antipode $\tilde{{\mathcal {A}}}_+^{\text {ex}}$, so that it follows that one has $f_z({\mathscr {S}}_r^* {\varvec{\Pi }})= f_z({\varvec{\Pi }}){\mathscr {S}}_r$, where $f_z$ is as in [5, (6.12)]. Plugging this into the respective formulae for and , the identities in (41) follow.

Finally, the fact that ${\mathscr {S}}_r^*$ extends to a locally Lipschitz map on follows straight forwardly from the identity (41) and the definition of the metric in the space of models [17, (2.17)]. $\square$

With this notation we have the following Theorem.

Theorem 8

Let $N\subseteq \Omega$ be the null set constructed in Corollary 2. Then one has for any $\omega \in N^c$, any $r\in {{\textbf {R}}}$, and any $h \in H^\xi$ the identity⁸

(42)

As a consequence, there exists a fixed subsequence $\varepsilon \rightarrow 0$ (independent of $\omega$ and h) such that one has ${\hat{Z}}_{\textrm{BPHZ}}^{\xi ^\varepsilon } (\omega + h)\rightarrow {\hat{Z}}_{\textrm{BPHZ}}^{\xi }(\omega + h)$ for any $\omega \in N^c$ and $h \in H^\xi$. Finally, for any $\omega \in N^c$ the maps and given by $h \mapsto {\hat{Z}}_{\textrm{BPHZ}}^\xi (\omega + h)$ and $(h,k) \mapsto {\hat{Z}}_{\textrm{BPHZ}}^{\xi ,h}(\omega +k)$ are locally Lipschitz continuous.

Proof

By Theorem 7 it suffices to show (42) with $\xi$ replaced by $\xi ^\varepsilon$ for any $\varepsilon >0$. For this in turn it is sufficient to show the identity ${\mathscr {S}}^*_r (R^g {\underline{Z}}^{f,h})=R^g Z^{f + rh}$ for any $r\in {{\textbf {R}}}$, any and any , which is a simple consequence of Lemma 6. The rest of Theorem 8 is then a direct consequence of Theorem 7 and the fact that ${\mathscr {S}}_r^*$ is locally Lipschitz. $\square$

3.4 Lifts of abstract fixed point problems

We are going to describe a class of abstract fixed point problems on the spaces that we are going to look at in the sequel.

Let be a sector spanned by a set of trees . Then the space spanned by all trees $\sigma \in {\mathscr {S}}[\tau ]$ for some forms a sector in . More generally assume we are given for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ a sector in spanned by sets of trees , we write for the sector in constructed as above, and we write

Moreover, assume we are given additionally exponents $\gamma =(\gamma _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ and $\eta =(\eta _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$. For any model we then define the space and a system of semi norms for any compact $K \subseteq {{{\textbf {D}}}}$ by

We fix from now on families of sectors and in for ${\mathfrak {t}}\in {\mathfrak {L}}_+$, both spanned by sets of trees, and families of exponents $\gamma =(\gamma _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$, $\eta =(\eta _{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$, ${\bar{\gamma }}=(\bar{\gamma }_{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ and ${\bar{\eta }}=({\bar{\eta }}_{\mathfrak {t}})_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ with $\gamma _{\mathfrak {t}}>0$ and ${\bar{\gamma }}_{\mathfrak {t}}\ge \gamma _{\mathfrak {t}}- |{\mathfrak {t}}|_{\mathfrak {s}}$ and $\eta _{\mathfrak {t}}, {\bar{\eta }}_{\mathfrak {t}}\in {{\textbf {R}}}$. We then recall the following terminology from [17, Sec. 7.3]

Definition 8

Given a model , we call a map locally Lipschitz if for any compact set $K\subseteq {{{\textbf {D}}}}$ and any $R>0$ one has the bound

(43)

uniformly over all with . Given a locally Lipschitz map for any model we call F strongly locally Lipschitz if for any there exists a neighbourhood of Z in such that for any compact set $K\subseteq {{{\textbf {D}}}}$, any $R>0$ one has the bound

(44)

uniformly over all models and and such that . Here, we set ${\bar{K}}:= \{x \in {{{\textbf {D}}}}: \text {dist}(x,K)\le 1\}$.

Following our usual convention, we will drop the dependence on the model Z from the notation whenever there is no room for confusion. We say that F is a strongly locally Lipschitz family for if we want to emphasise the underling sectors. We want to consider a class of strongly locally Lipschitz families that admit lifts to the extended regularity structure as described in the next definition.

Definition 9

Let F be a strongly locally Lipschitz family for . Then we call a family for and $r \in {{\textbf {R}}}$ a lift of F if for any fixed $r \in {{\textbf {R}}}$ the family is a strongly locally Lipschitz family for , one has that is jointly Lipschitz continuous in , i.e. one can strengthen (44) to

(45)

uniformly additionally for $|r|\vee |s|\le R$, and one has the identity

(46)

on for any $r \in {{\textbf {R}}}$ and . Here, on the left hand side of (46) we apply F for the model and on the right hand side we apply for . We call a $C^1$-lift if additionally one has that for any fixed model the map $(r,f) \mapsto F^{(r)}(f)$ is a Fréchet differentiable map from into with strongly locally Lipschitz continuous derivatives. In the case that such a ($C^1$-) lift exists, we say that F admits a ($C^1$-) lift.

3.5 Shift of abstract fixed point problems

In this section, if not explicitly otherwise stated, we make the notational convention that given and $r\in {{\textbf {R}}}$ we write

(47)

We will show how to use lifts of strongly locally Lipschitz continuous non-linearities to relate abstract fixed point problems for the model to abstract fixed point problems for and consequently how to "shift" these fixed point problems in directions of Cameron–Martin functions. We start with the following Lemma.

Lemma 8

Fix $r\in {{\textbf {R}}}$ and let be a sector of regularity $\alpha$. For any $\gamma >0$, $\eta \in {{\textbf {R}}}$ and any the map ${\mathscr {S}}_r$ is a Lipschitz continuous map from into , and provided that $\eta \le \gamma$ and $\alpha \wedge \eta >-{\mathfrak {s}}_0$ one has the identity

(48)

for any .

Finally, ${\mathscr {S}}_r$ maps strongly locally Lipschitz families onto strongly locally Lipschitz families .

Proof

In order to see that ${\mathscr {S}}_r$ maps into , it suffices to use the identity given by Lemma 6 and to note that ${\mathscr {S}}_r$ does not change homogeneities of trees. The identity (48) is a direct consequence from the properties of the reconstruction operator, in particular [17, (3.3)], and the first identity in (41). $\square$

Let now F be a strongly locally Lipschitz family for and let be a lift of F. We assume from now on that the pairs of sectors are chosen such that for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. We also fix a strongly locally Lipschitz family for , and we define for any and $r \in {{\textbf {R}}}$ the function

(49)

so that by Lemma 8.

We consider the fixed point problems for U and given by

(50)

in and respectively for any model and and any $r\in {{\textbf {R}}}$. Since ${\bar{\gamma }}_{\mathfrak {t}}\ge \gamma _{\mathfrak {t}}- |{\mathfrak {t}}|_{\mathfrak {s}}$ and the right hand sides are locally Lipschitz continuous by definition, it follows from [17, Thm. 7.8] (see also [4, Thm. 5.21]) that there exist unique maximal solutions U and to these equations. We denote the maximal time of existence by T(Z) and , respectively. We also define the stopping time $\tau (\omega ):= T({\hat{Z}}^\xi _{\textrm{BPHZ}}(\omega ))$. In this setting, we have the following result.

Proposition 4

Fix and $r \in {{\textbf {R}}}$, and let U and be the unique solutions to the fixed point equations (50) for the models and , respectively. Then one has

(51)

Proof

Since the solution to these fixed point equations are unique, we only need to show that ${\mathscr {S}}_r U$ satisfies the second equation in (50). For this note that

and since it follows directly from the definition that one has the result follows. $\square$

We stress at this point that the function U on the right hand side of (51) depends on r through the model .

Remark 12

In [17, Thm. 7.3] it was shown that if $W^Z$ is locally Lipschitz continuous in the model, then the time of existence $T = T(Z)$ of the solution U is a lower semi-continuous map in the model Z. Proposition 4 shows that the time of existence of is additionally lower semi-continuous in $r \in {{\textbf {R}}}$.

Remark 13

The modelled distribution will be used in order to deal with the initial condition. The assumption that $W^Z$ is locally Lipschitz continuous in the model is unreasonably strong, as it ends up imposing that the initial condition is a locally Lipschitz continuous map of the model.⁹ For the existence of the local $H^\xi$-Fréchet derivative it is sufficient to assume that for any $\omega \in N^c$ the map given by

is locally Lipschitz continuous, which is a trivial consequence of the assumption that the initial condition is locally $H^\xi$-Fréchet differentiable. Under this less restrictive assumption, the statement of Remark 12 is no longer true. However, this assumption is still sufficient to ensure in the same way as above that the time of existence is lower semi-continuous with respect to r, which ultimately ends up ensuring that the time of existence of the solution u is lower semi-continuous with respect to Cameron–Martin shifts. A similar Remark applies to the functions and introduced in Proposition 5 below.

4 The Malliavin derivative

We show in this section that the reconstructed solutions to the abstract fixed point problems considered in Sects. 3.4 and 3.5 admit a local $H^\xi$-Fréchet derivative. In Sect. 4.3 we apply this abstract result to singular SPDEs of the form (1).

4.1 Differentiability of the solution to the abstract fixed point problem

We show that is differentiable in $r \in {{\textbf {R}}}$ with values in . For this let F be a $C^1$-liftable, strongly locally Lipschitz family for a pair of sectors with the property that for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, and let be a $C^1$-lift of F. Let moreover for be a family as in the previous section, and assume that additionally the map defined as in (49) is Fréchet differentiable as a map from ${{\textbf {R}}}$ into for any . Finally, for any let U and be the solutions to (50) for and , respectively. We then have the following.

Proposition 5

Under the assumption at the beginning of this section, for any and any there exists such that the map is $C^1$ as a map from $(-r_0,r_0)$ into and its derivative satisfies the fixed point equation

(52)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. Moreover, the function is strongly locally Lipschitz continuous in the sense that for any and any there exists a neighbourhood of of such that one has

(53)

uniformly over and $r,s \in (-r_0+\theta , r_0-\theta )$ for any $\theta >0$.

Finally, can be chosen as the supremum over all $r_0>0$ such that for any $s \in (-r_0, r_0)$, and with this choice for any fixed T the function is lower semicontinuous in .

Proof

We fix for the first part of the proof a model . By definition one has that is a Fréchet differentiable map from into for any model . In order to see that is Fréchet differentiable, we make use of the implicit function theorem¹⁰ on the map given by

in a neighbourhood of . Note that by (50) one has . Since is a bounded linear operator from into by Proposition 8 it also Fréchet differentiable, and since by assumption one has that is Fréchet differentiable as well, it follows that $\Phi$ is a $C^1$ map from into . We now show that the derivative of $\Phi$ with respect to is an isomorphism for any . By definition, the derivative is a bounded linear operator between these spaces, which is given by

(54)

so that we are left to show that this expression is invertible. This is equivalent to solving, for any fixed the equation for , which is in the form of a fixed point problem and admits a unique solution by [17, Thm. 7.8]. This follows from the fact that the map from into is linear and continuous, and thus it is also Lipschitz continuous.

It now follows that there exists a neighbourhood of and $r_0>0$ and a (unique) $C^1$-function with such that

By uniqueness of solutions to the fixed point problem (50) we infer that one has necessarily , so that it follows in particular that is $C^1$ in $(-r_0,r_0)$. In order to see the identity (52) for the derivative, note that at this point all functions appearing in (50) are Fréchet differentiable in $r\in (-r_0, r_0)$, so that (52) follows by differentiating the right hand side of this identity.

In order to see (53) it suffices to show local Lipschitz continuity in r and separately. The former follows from arguments identically to above, which shows the stronger statement of Fréchet differentiability of $V^{(r)}$ in r. For the latter, by [17, Thm. 7.8] it suffices to show that is strongly locally Lipschitz continuous between the spaces and . This however follows by combining the fact that by definition the Frechet derivatives and are strongly locally Lipschitz continuous and the solution is strongly locally Lipschitz continuous as a consequence of [17, Thm. 7.8].

For the last part of the theorem assume that $r_0$ has been chosen maximally, and assume that for some $r_1>r_0$ one has that for all $r\in (-r_1, r_1)$. We can then redo the arguments in the first part of the proof with replaced by to obtain $s_0>0$ such that is $C^1$ as a map from $(-s_0, s_0)$ into , which shows that $U^{(r)}$ is a $C^1$ map on $(-r_0, r_0 + s_0)$. A similar argument shows that the lower bound can be improved and yields a contradiction. The lower semicontinuity of $r_0$ is now a consequence of the lower semicontinuity of in . $\square$

4.2 Local H-differentiability of the solution

As a consequence of Proposition 5 we can show that the reconstructed solution map is Gateaux differentiable in $H^\xi$ directions.

Lemma 9

Let $\omega \in N^c$ and let $h \in H^\xi$. Then for any $T< \tau (\omega )$ there exists $r_0>0$ such that the map

$$\begin{aligned} r \mapsto u(\omega + r h) \end{aligned}$$

with values in , where $\alpha _{\mathfrak {t}}$ denotes the regularity of the sector , is Fréchet differentiable differentiable on $(-r_0,r_0)$.

Proof

Let be as in Proposition 5. By Theorem 8 and Proposition 4 one has that

Since is $C^1$ with values in by Proposition 5, the result follows from the fact that for any fixed model the reconstruction operator is a bounded linear map on and thus Fréchet differentiable. $\square$

Finally, we can show the following.

Theorem 9

For any $T>0$ the solution u restricted on $(0,T)\times {{\textbf {T}}}^d$ and conditioned on the event $\{ \tau > T \}$ with values in is locally $H^\xi$-Fréchet differentiable in the sense of Definition 5.

Proof

For fixed $\omega \in N^c$ and $h \in H^\xi$ let be as in Proposition 5. Then one has the identity

for $r \in (-r_0(\omega ,h), r_0(\omega ,h) )$, where denotes the derivative of $U^{(r)}$ in the direction of r as in Proposition 5 for the model . Since is locally Lipschitz continuous by Corollary 2, it follows that for any fixed $\omega \in N^c$ the map $h \mapsto r_0(\omega , h)$ is lower semi-continuous. Since furthermore one has $r_0(\omega , 0)=+\infty$, there exists $\mu >0$ and a ball $B_\mu (0) \subseteq H^\xi$ around the origin such hat one has $r_0(\omega ,h)>1$ for any $h \in B_\mu (0)$. Now it follows from (42) that for any $h,k \in B_\mu (0)$ one has

so that it follows in particular from Proposition 5 that $h \mapsto u(\omega +h)$ is Gateaux differentiable in $B_\mu (0)$ with Gâteaux derivative given by

so that it remains to show that this expression is continuous in $(h,k) \in H^\xi \times H^\xi$. This follows from the fact that is strongly Lipschitz continuous, the map is locally Lipschitz continuous and by (53) one has that is strongly locally Lipschitz continuous. $\square$

4.3 Application to subcritical SPDEs

We now apply the result of the previous section to abstract fixed point problems arising from singular SPDEs and conclude the proof of Theorem 1. That is, we show that under the assumption introduced in Theorem 1, the solution u to the singular SPDE (1) admits a local H-Fréchet derivative, and in this situation we can furthermore derive a "tangent equation" (7) for this Fréchet derivative, which is informally given by differentiating the original equation with respect to the noise. The precise meaning of (7) is that $v_h$ can be written as a limit $v_h = \lim _{\varepsilon \rightarrow 0} v_h^\varepsilon$, where the random smooth function $v_h^\varepsilon = (v_{{\mathfrak {t}},h}^\varepsilon )_{{\mathfrak {t}}\in {\mathfrak {L}}_+}$ is the unique classical solutions to the system of equations

(55)

with initial condition $v_{{\mathfrak {t}},h}^\varepsilon (0) = D_{h} u_{{\mathfrak {t}},0}^\varepsilon$. It is not hard to see that $v_h^\varepsilon$ is the $H^\xi$-Fréchet derivative $D_h u^\varepsilon$ of the solution $u^\varepsilon$ to the regularized and renormalized Eq. (5) in the direction of $h \in H^\xi$. Note that both $\psi ^\varepsilon$ and are locally $H^\xi$-Fréchet differentiable (for the former this follows by assumption, while for the latter this follows from the explicit definition of ${\mathscr {S}}_{\rho ,\varepsilon }^-(\xi )$ in [4, (A.10), (5.10)], which imply in particular that takes values in some inhomogeneous Wiener chaos), so that the same is true for .

Remark 14

The tangent equation (7) is in the form of a singular SPDE, however it does not fall under the setting of [4] since it involves a source $h_\Xi$ which is deterministic and in general not smooth. The fact that $h_\Xi$ is not necessarily smooth was the main reason that the analysis of Sect. 3.2 was necessary in the first place. However, if $h_\Xi$ happens to be smooth for any $\Xi \in {\mathfrak {L}}_-$, then one can treat the tangent equation directly in the framework of [4], and in this case the regularized and renormalized equation derived in [4] coincides with (55).

The solution u constructed in [4] is given as the reconstruction of (c.f. [4, Prop. 5.22]), where $\tilde{U}_{\mathfrak {t}}$ is the constant modelled distributions in explicitly given in [4, (5.10)]. We now introduce an abstract differentiation operator as the derivative of ${\mathscr {S}}_r$ at $r=0$, so that one has

With this notation we can see that the $H^\xi$-Fréchet derivative of the function in the direction of $h \in H^\xi$ is given by . The fact that $u_{\mathfrak {t}}$ is locally $H^\xi$-Fréchet differentiable is now equivalent to showing that is locally $H^\xi$-Fréchet differentiable, which at this stage is an application of Theorem 9 to the abstract fixed point problem [4, (5.16)] for ${\bar{U}}$. The main step here is to show that the right hand side of this fixed point problem admits $C^1$ lifts, and since this is largely a technical issue, we postpone the proof to Appendix B below.

It remains to derive the tangent equation. For this consider a regularization of $\xi$ given by $\xi ^\varepsilon _\Xi := \xi _\Xi * \rho ^{(\varepsilon )}$ and let $h_\Xi ^\varepsilon := h_\Xi *\rho ^{(\varepsilon )}$, let $v_{h,{\mathfrak {t}}}^\varepsilon := D_h u_{{\mathfrak {t}}}^\varepsilon$ be the local $H^\xi$-Fréchet derivative of the solution $u_{\mathfrak {t}}^\varepsilon$ to (5) in the direction of $h \in H^\xi$. The fact that one has $v_{h,{\mathfrak {t}}} = \lim _{\varepsilon \rightarrow 0 }v_{h,{\mathfrak {t}}}^\varepsilon$ follows simply from the fact that both sides of this equation are given as the reconstruction of the modelled distribution , where denotes the solution to the abstract tangent equation (52), for the model and , respectively, and the fact that the latter converges to the former as $\varepsilon \rightarrow 0$ by Theorem 7. It is now sufficient to show that $v_{h,{\mathfrak {t}}}^\varepsilon$ solves (55). But in the regularized case the map $r \mapsto \xi ^\varepsilon (\omega + rh)$ is smooth with derivative given by $\partial _r |_{r=0} \xi ^\varepsilon (\omega + rh) = h^\varepsilon$. Furthermore, $(r,x) \mapsto \xi ^\varepsilon (\omega +rh)(x)$ is a smooth function ${{\textbf {R}}}\times {{{\textbf {D}}}}\rightarrow {{\textbf {R}}}^{{\mathfrak {L}}_-}$ and since both F and $\Upsilon$ are smooth, the former by assumption and the latter by construction, c.f. (16), it follows readily from standard Schauder estimates that the map $(r,x) \mapsto u^\varepsilon (\omega +rh)(x)$ is smooth as well. This is sufficient to argue that we are allowed to commute the differentiation operators $\partial _t$ and with $\partial _r$ in (5), and since per definitionem one has that $v_{h,{\mathfrak {t}}}^\varepsilon = \partial _r|_{r=0} u_{\mathfrak {t}}^\varepsilon (\omega + rh)$, we obtain (55) by a direct computation.

5 Density of solutions to singular SPDEs

In this section we always fix a time $T>0$. We want to derive conditions such that random variables of the form $X= \left\langle u,\phi \right\rangle :=\sum _{{\mathfrak {t}}\in {\mathfrak {L}}_+} \left\langle u_{\mathfrak {t}}, \phi _{\mathfrak {t}} \right\rangle$, for some tuple of test functions , conditioned on the event $\{\tau >T \}$ admit a density with respect to Lebesgue measure. By Theorem 1 the random variable X is locally $H^\xi$-Frechet differentiable with derivative in direction of $h \in H^\xi$ given by $\left\langle v_{h}, \phi \right\rangle$, where $v_{h}$ solves (7). By the Bouleau Hirsch criterion, Corollary 1, we are lead to study non-degeneracy of this local $H^\xi$-Fréchet derivative.

For simplicity we make in this section the following additional assumption.

Assumption 4

We assume that the following is satisfied.

• The renormalization constants are given by the BPHZ character $c_\tau ^\varepsilon = g_{\textrm{BPHZ}}^\varepsilon (\tau )$.

• For any $\Xi \in {\mathfrak {L}}_-$ and any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ one has that $F_{\mathfrak {t}}^\Xi =F_{\mathfrak {t}}^\Xi (u)$ and $F_{\mathfrak {t}}=F_{\mathfrak {t}}(u)$ depend only on the solution (and not on its derivatives).

• For any noise type ${\mathfrak {t}}\in {\mathfrak {L}}_-$, any $\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^F$ and any $\varepsilon >0$ with the property that $c_\tau ^\varepsilon \ne 0$ one has either $\Upsilon ^\tau _{\mathfrak {t}}= \Upsilon ^\tau _{\mathfrak {t}}(u)$ depends only on the solution (and not on its derivatives), or $\Upsilon ^\tau _{\mathfrak {t}}(u, \nabla u, \ldots ) = \partial _i u_{\mathfrak {t}}$ for some $i\in \{0, \ldots , d \}$. We write $\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^{F,{ \circ }}$ in the first case and $\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^{F,i}$ for $i\in \{0, \ldots d \}$ in the second case.

The first assumption is merely a convenience and could be easily dropped with a little more algebraic effort later on, compare Remark 20 below. The second and the third assumption greatly simplify the computation; we believe that the statements below are still true without these assumptions, but the proofs given in this paper do not seem to easily generalize to this case. The case $\Upsilon ^\tau _{\mathfrak {t}}(u, \nabla u, \ldots ) = \partial _i u_{\mathfrak {t}}$ ensures we cover the $\Phi ^4_{4-\varepsilon }$ equation, see [4, Sec. 2.8.2]. Under Assumption 4 Eq. (55) simplifies to

(56)

We denote by the dual operator to , which is again a differential operator involving only spatial derivatives, and we consider the equation dual to (56), which is a backward random PDE given by

(57)

on $(0,T)\times {{\textbf {T}}}^d$ with final condition $w_{{\mathfrak {t}},\phi }^\varepsilon (T,\cdot )= 0$. The following lemma is a straight-forward computation.

Lemma 10

Let $T>0$ and let for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. Then for any $h \in L^2({{{\textbf {D}}}}) ^{{\mathfrak {L}}_-}$ one has the identity

$$\begin{aligned} \left\langle v_h^\varepsilon , \phi \right\rangle _{L^2({{{\textbf {D}}}})} = \left\langle \sum _{\Xi \in {\mathfrak {L}}_-} F^\Xi (u^\varepsilon ) h_\Xi ^\varepsilon , w_\phi ^\varepsilon \right\rangle _{L^2({{{\textbf {D}}}})} + \left\langle v^\varepsilon _h(0,\cdot ), w^\varepsilon _\phi (0, \cdot ) \right\rangle _{L^2({{\textbf {T}}}^d)} \end{aligned}$$

(58)

between random variables conditioned on the event $\{ T < \tau _\varepsilon \}$.

5.1 A regularity structure adapted to the dual equation

Our goal is to understand the behaviour of $w^\varepsilon _{{\mathfrak {t}},\phi }$ in the limit $\varepsilon \rightarrow 0$. To this end we want to interpret $w^\varepsilon _{{\mathfrak {t}},\phi }$ as the reconstructed solution to an abstract fixed point problem, which can be viewed as the dualization of the abstract tangent equation (52). The equation for w can be written in its mild formulation, and it is not hard to see that the Green’s function for is given by $x\mapsto G_{\mathfrak {t}}(-x)$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, where $G_{\mathfrak {t}}$ is the Greens function for . It follows that the kernel types present in our regularity structure are not rich enough to encode the dual equation, so that as a first step we are lead to build an extension of the regularity structure in which one can consider abstract fixed point problems associated to $G_{\mathfrak {t}}(-\cdot )$.

Remark 15

Note that in Sect. 3.1 we considered an extension of the noise-types ${\mathfrak {L}}_-$, whereas in this section we will consider an extension of the kernel-types ${\mathfrak {L}}_+$. The extension constructed in Sect. 3.1 plays no role in this section, so that from now on we use the symbol for the regularity structure constructed below, and we refer to this structure as the extended regularity structure form now on.

To this end we extend the set of kernel types to a set , where ${ {\mathfrak {L}}_+'}:= \{{ {\mathfrak {t}}'}: {\mathfrak {t}}\in {\mathfrak {L}}_+\}$ is a disjoint copy of ${\mathfrak {L}}_+$, and we let $|{ {\mathfrak {t}}'}|_{\mathfrak {s}}:=|{\mathfrak {t}}|_{\mathfrak {s}}$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. One should think of ${ {\mathfrak {t}}'}$ as representing the "dual" integral operator to ${\mathfrak {t}}$. In particular, it will represent the Greens function for a parabolic differential operator going backward in time.

Given the extended set of types we extend ${\text {reg}}$ to a function by setting ${\text {reg}}( { {\mathfrak {t}}'}):= \theta$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and some $\theta >0$ small enough, and we define an extension of the rule R by allowing any kernel type ${\mathfrak {t}}\in {\mathfrak {L}}_+$ to be replaced by ${ {\mathfrak {t}}'}$, and additionally allowing an arbitrary number of types ${ {\mathfrak {L}}_+'}$. To be more precise, we define and , with , and we set

(59)

(60)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, and as usual for ${\mathfrak {t}}\in {\mathfrak {L}}_-$. Here we define for any ${\mathfrak {t}}\in {\mathfrak {L}}$, for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and

where the sum runs over all with .

Remark 16

The fact that we allow for arbitrary instead of just $M=0$ in (59) has the advantage that satisfies [4, Ass. 3.12] as soon as R satisfies this assumption. This simply ends up ensuring that one can build arbitrary products of $U_{\mathfrak {t}}$ for any with ${\text {reg}}({\mathfrak {t}})>0$. As was already remarked below [4, Ass. 3.12], this is not a restriction at all, since any subcritical rule can be trivially extended in such a way that this assumption holds, and we will assume from now that [4, Ass. 3.12] holds for R and thus also for .

Remark 17

It might appear more natural to set for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, in which case ${\mathfrak {t}}$ and ${ {\mathfrak {t}}'}$ would be end up to be completely interchangeable in the extended regularity structure constructed from . The present formulation is more restrictive and leads to a smaller regularity structure, but as we shall see, this structure is still rich enough to lift the Eq. (57) for w to an abstract fixed point problem. The reason we choose the present formulation is that the natural sector in which the solution to this abstract fixed point problem takes values in is function-like, compare Lemma 15 below.

Example 7

Continuing Example 1 of the stochastic heat equation, the set is equal to for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and contains those multisets with the property that ${\mathfrak {m}}(\Xi ,0)\le 1$ and ${\mathfrak {m}}({\mathfrak {l}},k) = 0$ for any ${\mathfrak {l}}\in {\mathfrak {L}}$ and $k \in {{{\textbf {N}}}}^{d+1} \backslash \{0\}$.

An example where and do not coincide is given by the $\Phi ^4_3$ Eq. (3). In this case, writing again ${\mathfrak {L}}_+=\{{\mathfrak {t}}\}$ and ${\mathfrak {L}}_-=\{\Xi \}$, the set is given by the set of all multisets which can be written as

$$\begin{aligned} \emptyset ,\, [(\Xi ,0)],\, [({ {\mathfrak {t}}'},0),\ldots ],\, [({\mathfrak {t}},0),({ {\mathfrak {t}}'},0),\ldots ],\, \ldots ,\,\text { or }\, [({\mathfrak {t}},0),({\mathfrak {t}},0),({\mathfrak {t}},0), ({ {\mathfrak {t}}'},0),\ldots ], \end{aligned}$$

where $({ {\mathfrak {t}}'},0),\ldots$ stands for an arbitrary number of types $({ {\mathfrak {t}}'},0)$. On the other hand, is given by all multisets which can be written as

$$\begin{aligned} \emptyset ,\, [({ {\mathfrak {t}}'},0),\ldots ],\, [({\mathfrak {t}},0),({ {\mathfrak {t}}'},0),\ldots ], \,\text { or }\, [({\mathfrak {t}},0),({\mathfrak {t}},0), ({ {\mathfrak {t}}'},0),\ldots ]. \end{aligned}$$

In order to continue, we recall from [5, Rem. 5.17] that given $\tilde{\theta }>0$ one can assume without loss of generality that the function ${\text {reg}}$ satisfies the bound

(61)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, where ${\mathscr {T}}_{\mathfrak {t}}^F$ denotes the set of trees $\tau$ such that and $\tau$ is ${\mathfrak {t}}$-non-vanishing as in Sect. 2.3.

Remark 18

There are some subtleties here, since in [5, Rem. 5.17] this identity was only shown with ${\mathscr {T}}_{\mathfrak {t}}^F$ replaced by the larger set ${\mathscr {T}}_{\mathfrak {t}}$. In general (61) might simply not be true, since the rule might be chosen larger then necessary to deal with the singular SPDE at hand. However, this problem can easily be circumvented by assuming without loss of generality that R is given as the completion of the "naive" rule $R_{\text {naive} }$, which is defined in such a way that the set of trees $\tau \in {\mathscr {T}}_{\mathfrak {t}}$ that strongly conform to $R_{\text {naive} }$ coincide with the set of trees $\tau \in {\mathscr {T}}_{\mathfrak {t}}$ that are ${\mathfrak {t}}$-non-vanishing. One can then apply [5, Rem. 5.17] to $R_{\text {naive} }$ in order to obtain (61).

We assume from now on that (61) holds for some ${\tilde{\theta }}>0$ small enough (to be determined later), and with this convention we have the following lemma.

Lemma 11

Assume that $\theta >0$ is small enough and that (61) holds. Then the rule is a subcritical rule with respect to ${\text {reg}}$. In particular, there exists a subcritical completion of defined via [5, Prop. 5.21], which we again denote by , and we can define the extended regularity structure as in [5, Sec. 5.5].

Proof

In order to see that is subcritical, note first that for ${\mathfrak {t}}\in {\mathfrak {L}}_+$ one has

Let now ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and $N \in R({ {\mathfrak {t}}'})$, and let ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and $k\in N^{d+1}$ such that $N \sqcup \{({\mathfrak {l}},k)\} \in R({\mathfrak {t}})$. By (61) we can choose for any kernel-type ${\mathfrak {j}}\in {\mathfrak {L}}_+$ a tree such that ${\text {reg}}( {\mathfrak {j}}) > | \tau _{ {\mathfrak {j}}} | _{\mathfrak {s}}+ | {\mathfrak {j}}|_{\mathfrak {s}}- {\tilde{\theta }}$ for some ${\tilde{\theta }}>0$ small enough. We now consider the tree

It follows that $\tau$ strongly conforms to R (c.f. [5, Def. 5.8]) and it follows from the definition of in [5, Def. 5.26] that $\tau \in {\mathscr {T}}_{\mathfrak {t}}^F$. We also define the subtree ${\tilde{\tau }}$ of $\tau$ by setting

Then ${\tilde{\tau }}\in {\mathscr {T}}_{\mathfrak {t}}^F$ is a proper subtree of $\tau$ with identically root, and trees like this satisfy $|{\tilde{\tau }}|_{\mathfrak {s}}> - |{\mathfrak {t}}|_{\mathfrak {s}}+ 2\theta$ for $\theta >0$ small enough by [4, Ass. 2.13].

On the other hand, provided that ${\tilde{\theta }}$ is smaller then $\frac{\theta }{\# N}$, one has ${\text {reg}}(N)> |{\tilde{\tau }}|_{\mathfrak {s}}- \theta > -|{\mathfrak {t}}|_{\mathfrak {s}}+\theta = -|{\mathfrak {t}}|_{\mathfrak {s}}+ {\text {reg}}({ {\mathfrak {t}}'})$, and this concludes the proof. $\square$

For $X\in \{G,K,R\}$ we write $X_{{ {\mathfrak {t}}'}}(z):= X_{{\mathfrak {t}}}(-z)$, so that in particular $G_{{ {\mathfrak {t}}'}}$ is the Greens function for and the compactly supported kernels $K_{{ {\mathfrak {t}}'}}$ satisfy [17, Ass. 5.1 Ass. 5.4]. Given the kernel assignment $K_{\mathfrak {l}}$ for any we write for the set of smooth, reduced, admissible models for and we write for the closure of this set. For $f \in \Omega _\infty ({\mathfrak {L}}_-)$ and we write and for the canonical lift of f and the BPHZ-renormalized canonical lift of $\eta$, respectively, defined as usual via [5, Rem. 6.12] and [5, Thm. 6.17]. We also write for the space of modelled distributions as in Sect. 2.2.5 with replaced by . We will later on need to work with modelled distributions that are only defined on a domain of the form $(\theta ,T)\times {{\textbf {T}}}^d$ for some $T>\theta >0$, and we write for the space of functions that satisfy (13).

5.2 An abstract fixed point problem for the dual equation

Given , with as in Proposition 9, and we want to consider the abstract point problem in for some families ${ \gamma '}_{ { {\mathfrak {t}}'}}>0$, for ${\mathfrak {t}}\in {\mathfrak {L}}_+$, and some sector , given by

(62)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. The purpose of this section it to find the right sectors and exponents for this fixed point problem to be well posed. In order to unify notation, we define

$$\begin{aligned} F_{{ {\mathfrak {t}}'}}( u , w ) := DF_{\mathfrak {t}}(u) w \quad \text { and } \quad F_{{ {\mathfrak {t}}'}}^\Xi ( u , w ) := DF_{\mathfrak {t}}^\Xi ( u ) w, \end{aligned}$$

(63)

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, $\Xi \in {\mathfrak {L}}_-$ and $u,w \in {{{\textbf {D}}}}^{{\mathfrak {L}}_+}$. With this convention $F_{\mathfrak {l}}$ and $F_{\mathfrak {l}}^\Xi$ are well defined for any and $\Xi \in {\mathfrak {L}}_-$. We will sometimes write $F^\bullet _{\mathfrak {t}}:= F_{\mathfrak {t}}$ to avoid case distinctions.

Remark 19

In [4, Sec.3–5] the authors were working in a more general setting, in the sense that they allowed for derivatives hitting noises and noises being multiplied together. Additionally, they were considering non-linearities that can depend on the extended decoration ${\mathfrak {o}}$. Our setting can easily be embedded into this more general setting, by defining ${\tilde{F}}$ via

$$\begin{aligned} {\tilde{F}}_{\mathfrak {t}}^{0,0}:= F_{\mathfrak {t}}\quad \text { and } \quad \tilde{F}_{\mathfrak {t}}^{{{{\textbf {I}}}}_\Xi , 0 }:= F_{\mathfrak {t}}^\Xi \end{aligned}$$

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and $\Xi \in {\mathfrak {L}}_-$, and $F_{\mathfrak {t}}^{\mathfrak {l}}= 0$ otherwise. Whenever we refer to results from [4] in the sequel we will make these identification implicitly.

In the sequel we will need results of [4] applied to . In order to do so, we need the following technical Lemma.

Lemma 12

Assume that $\theta >0$ is small enough. Then obeys in the sense of [4, Def. 3.10].

Proof

We use [4, Prop. 3.13]. In our setting, which is a bit simplified compared to [4], the second conditions of [4, Prop. 3.13] reduces to the statement

(64)

(65)

for any , $\Xi \in {\mathfrak {L}}_-$ and where . For ${\mathfrak {t}}\in {\mathfrak {L}}_+$ this follows from the respective assumption on $R({\mathfrak {t}})$ and the definition of in (59).

Let now that such that $D^\alpha F_{{ {\mathfrak {t}}'}} \ne 0$. Assume first that $\alpha \in {{{\textbf {N}}}}^{{\mathfrak {L}}_+ \times {{{\textbf {N}}}}^{d+1}}$. By definition of $F_{{ {\mathfrak {t}}'}}$ in (63) it follows that there exists ${\mathfrak {l}}\in {\mathfrak {L}}_+$ such that

$$\begin{aligned} D^{\alpha \sqcup \{({\mathfrak {l}},0)\} } F_{\mathfrak {t}}\ne 0, \end{aligned}$$

and by the first part of the proof this implies . From (60) we infer that , as required. Assume now that $\alpha \notin {{{\textbf {N}}}}^{{\mathfrak {L}}_+ \times {{{\textbf {N}}}}^{d+1}}$. Then it follows that there exists a (unique) ${\mathfrak {l}}\in {\mathfrak {L}}_+$ such that $\alpha _{({ {\mathfrak {l}}'},0)} \ne 0$, and we can write $\alpha =: \bar{\alpha }\sqcup \{({ {\mathfrak {l}}'},0) \}$. It follows that one has

$$\begin{aligned} 0 \ne D^\alpha F_{{ {\mathfrak {t}}'}} = D^{{\bar{\alpha }} \sqcup \{ ({\mathfrak {l}},0) \}} F_{\mathfrak {t}}, \end{aligned}$$

and thus . Again by (60) we infer that , and since satisfies [4, Ass. 3.12] (or directly from the definition), we infer that , as required.

The claim concerning $F_{\mathfrak {l}}^\Xi$ for any and $\Xi \in {\mathfrak {L}}_-$ follows in the same way. $\square$

In analogue to (15), given we say that a tree is ${\mathfrak {l}}$-non-vanishing for F if

$$\begin{aligned}\bigg ( \partial ^{{\mathfrak {n}}(\rho _\tau )} \prod _{ e \in K(\tau ), e^\downarrow = \rho _\tau } D_{({\mathfrak {t}}(e), {\mathfrak {e}}(e))} \bigg ) F^\Xi _{\mathfrak {t}}\ne 0\end{aligned}$$

and $\tau _e$ is ${\mathfrak {t}}(e)$-non-vanishing for any $e \in K(\tau )$ with $e^\downarrow = \rho _\tau$. Here we set $\Xi := {\mathfrak {t}}(e)$ if there exists a (necessarily unique) noise-type edge $e \in K(\tau )$ with $e^\downarrow = \rho _\tau$ and $\Xi := \bullet$ otherwise, and $\tau _e$ denotes the largest sub-tree of $\tau$ with root $e^\uparrow$. We define , and in analogue to Sect. 2.3, so that one has

(note that in particular for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$), and the sets and consist of those trees such that $|\tau |_{\mathfrak {s}}\le 0$ and $|\tau |_{\mathfrak {s}}<0$, respectively. We also set for any . With this notation we define for any ${\mathfrak {l}}\in { {\mathfrak {L}}'}_+$ the sectors

(66)

We write similar to above and . We now have the following analogue to [4, Lem. 5.9].

Lemma 13

For any the spaces and form sectors in . Moreover, for any and and any $\Xi \in {\mathfrak {L}}_-$ one has that

$$\begin{aligned} DF_{\mathfrak {t}}(U) W \qquad \text { and } \qquad DF_{\mathfrak {t}}^\Xi (U) W \Xi \end{aligned}$$

are elements of for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$.

Proof

This is the content of [4, Lem. 5.9]. $\square$

In the sequel we need to understand structure of the sets for ${\mathfrak {t}}\in {\mathfrak {L}}_+$. For this we introduce the following notation. Given a tree and a node $u \in N(T)$ we write

$$\begin{aligned} {{\mathfrak {D}}_u(\tau )}:= (T^{{\mathfrak {n}},{\mathfrak {o}}}_{\mathfrak {e}}, {\tilde{{\mathfrak {t}}}}_u), \end{aligned}$$

where is given by

$$\begin{aligned} {\tilde{{\mathfrak {t}}}}_u (e) := {\left\{ \begin{array}{ll} { {\mathfrak {t}}(e)'} \quad &{} \text { if }e\text { lies on the shortest path from }u\text { to }\rho _T \\ {\mathfrak {t}}(e) \quad &{} \text { otherwise.} \end{array}\right. } \end{aligned}$$

(67)

It follows from Lemma 11 that one has for any and any $u \in N(\tau )$. Given additionally an edge $e \in K(\tau )$ with $e^\downarrow = u$, then we write ${ {\mathfrak {D}}_u^e(\tau ) }$ for the tree obtained from ${{\mathfrak {D}}_u(\tau )}$ by removing e from the edge set, and removing furthermore all edges ${\tilde{e}} \in E({{\mathfrak {D}}_u(\tau )})$ and vertices ${\tilde{u}} \in V({{\mathfrak {D}}_u(\tau )})$ with the property that e lies on the shortest path from ${\tilde{e}}$ respectively ${\tilde{u}}$ to the root $\rho _\tau$. It is clear that one obtains another decorated, typed tree in this way by simply restricting the corresponding maps to $N({ {\mathfrak {D}}_u^e(\tau ) })$ and $E({ {\mathfrak {D}}_u^e(\tau ) })$, respectively, and since is a normal rule, one has . We now have the following Lemma.

Lemma 14

Assume that Assumption 4 holds. Then for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ the set agrees with the set of trees ${ {\mathfrak {D}}_u^e(\tau ) }$ for $\tau \in {\mathscr {T}}^F_{\mathfrak {t}}$, $u \in N(\tau )$ and $e \in K(\tau )$ such that $e^\downarrow = u$.

Proof

Let first $\tau \in {\mathscr {T}}^{F}_{{\mathfrak {t}}}$ and let $u \in N(\tau )$ and $f \in K(\tau )$ be such that $f^\downarrow = u$. It follows from the definition of in (60) that implies and by completeness of the rule one also has . It thus remains to show that ${ {\mathfrak {D}}_u^f(\tau ) }$ is ${ {\mathfrak {t}}'}$-non-vanishing. Proceeding inductively in the number of edges of $\tau$, it suffices to show that (15) does not vanish identically for the root ${\tilde{\rho }}:=\rho ({ {\mathfrak {D}}_u^f(\tau ) })$. For this let $\tilde{\mathfrak {t}}_u$ be as in (67) and let and denote the set of edges $e \in K(\tau )$ and $e \in K({ {\mathfrak {D}}_u^f(\tau ) })$ such that $e^\downarrow = \rho _\tau$ and $e^\downarrow = {\tilde{\rho }}$, respectively. In case that $u = \rho _\tau$, one has ${\tilde{{\mathfrak {t}}}}_u = {\mathfrak {t}}$ and by definition of ${ {\mathfrak {D}}_u^f(\tau ) }$ one has , so that it follows that

(68)

Since this expression is linear in $\partial _k w_{\mathfrak {l}}$, in order to see that this expression does not vanish identically, it suffices to find one pair $({\mathfrak {l}},k) \in {\mathfrak {L}}_+ \times {{{\textbf {N}}}}^{d+1}$ such that the coefficient of $\partial _l w_{\mathfrak {l}}$ is non vanishing. We choose $({\mathfrak {l}},k)=({\mathfrak {t}}(f),{\mathfrak {e}}(f))$ and we note that the corresponding coefficient in (68) is equal to

which does not vanish identically by assumption. In case $u \ne \rho _\tau$ one has , and there exists a unique edge such that ${\bar{e}}$ lies on the unique shortest path from $\rho _\tau$ to u. It follows that ${\tilde{{\mathfrak {t}}}}_u({\bar{e}}) = { {\mathfrak {t}}'}({\bar{e}})$ and ${\mathfrak {t}}(e) = {\tilde{{\mathfrak {t}}}}_u(e)$ for any , and using the fact that $D_{({ {\mathfrak {l}}'},k)} F_{{ {\mathfrak {t}}'}}^\Xi (u, \nabla u, \ldots ; w, \nabla w, \ldots ) = D_{ ({\mathfrak {l}},k) } F^\Xi _{\mathfrak {t}}(u, \nabla u, \ldots )$ for any ${\mathfrak {t}},{\mathfrak {l}}\in {\mathfrak {L}}_+$, we obtain

(69)

which does not vanish identically by assumption.

Conversely, let . It follows from the fact that $F_{{ {\mathfrak {l}}'}}^\Xi$ is linear in $(w, \nabla w, \ldots )$ for any ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and $\Xi \in {\mathfrak {L}}_-$ that there exists a (unique) vertex $\mu \in N(\sigma )$ such that ${\mathfrak {t}}(e) \in { {\mathfrak {L}}'}_+$ if and only if e lies on the unique shortest path from $\rho _\sigma$ to $\mu$. Let be the set of edges $e \in K(\sigma )$ such that $e^\downarrow = \mu$, and define by setting ${\mathfrak {j}}:= {\mathfrak {t}}(u^\downarrow )$ if $u \ne \rho _\sigma$, and ${\mathfrak {j}}:= { {\mathfrak {t}}'}$ otherwise. By definition of the rule in (60) it follows that there exists $({\mathfrak {l}},k) \in {\mathfrak {L}}_+ \times {{{\textbf {N}}}}^{d+1}$ such that one has¹¹

Choose an arbitrary tree ${\tilde{\tau }} \in {\mathscr {T}}_{{\mathfrak {l}}}^F$ and define now the typed, decorated tree $(T^{{\tilde{{\mathfrak {n}}}}}_{{\tilde{{\mathfrak {e}}}}},{\mathfrak {l}})$ by connecting $\rho ({\tilde{\tau }})$ to $\mu$ via an edge ${\bar{e}}$ such that ${\mathfrak {l}}({\bar{e}}) = {\mathfrak {l}}$ and ${\tilde{{\mathfrak {e}}}}({\bar{e}}) = k$, and where ${\tilde{{\mathfrak {n}}}}$, ${\tilde{{\mathfrak {e}}}}$ and ${\mathfrak {l}}$ extend the decorations and type-maps of $\sigma$ and ${\tilde{\tau }}$ otherwise. It then follows that and one has $\sigma = { {\mathfrak {D}}_\mu ^{{\bar{e}}}(\tau ) }$. The fact that $\tau$ if ${\mathfrak {t}}$-non-vanishing follows by reversing the arguments of the first part of the proof. $\square$

A particular consequence of Lemma 14 is that we can give a direct proof of the fact that the sectors are function like. Note that such a statement would also follow directly from the analysis [4] and the fact that ${\text {reg}}({ {\mathfrak {t}}'})>0$.

Lemma 15

For any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and any one has $|\tau |_{\mathfrak {s}}>- (|{\mathfrak {t}}|_{\mathfrak {s}}\vee {\mathfrak {s}}_0)$. In particular, the regularity of the sector is larger then $-(|{\mathfrak {t}}|_{\mathfrak {s}}\vee {\mathfrak {s}}_0)$, and the sector is function-like.

Proof

This is a direct consequence of Lemma 14, the fact that $|{{\mathfrak {D}}_u(\tau )}|_{\mathfrak {s}}= |\tau |_{\mathfrak {s}}$ and [4, Ass. 2.13]. $\square$

For ${\mathfrak {t}}\in {\mathfrak {L}}_+$ let $\alpha _{{ {\mathfrak {t}}'}}\le 0$ denote the regularity of the sector and for ${ \gamma '}>0$ let ${ \gamma '}_{{ {\mathfrak {t}}'}}:= \alpha _{{ {\mathfrak {t}}'}}+\gamma +|{\mathfrak {t}}|_{\mathfrak {s}}$.

Corollary 3

Assume that $\gamma >0$ is large enough such that $\gamma _{\mathfrak {t}}>{ \gamma '}_{{ {\mathfrak {t}}'}}>0$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. Then for any $\theta >0$ and any and any tupel , the fixed point problem (62) has a unique solution .

Proof

We first note that as a corollary from the proof of [4, Lem. 5.9], in particular [4, (5.15)], it follows that for any , any ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and any $\Xi \in {\mathfrak {L}}_-$ one has and are elements of .

Moreover, combing Lemma 15 and Lemma 13, it follows that both and take values in a sector of regularity bigger then $-|{\mathfrak {t}}|_{\mathfrak {s}}$. Consequently, using the results of [17, Sec. 6] (see Proposition 8), one has that

$$\begin{aligned} W \mapsto DF_{\mathfrak {t}}(U) W +\sum _{\Xi \in {\mathfrak {L}}_-} DF_{\mathfrak {t}}^\Xi (U) W \Xi + \phi _{\mathfrak {t}}\end{aligned}$$

is a locally Lipschitz continuous map from to for some $\kappa >0$ small enough. At this point the unique existence of a solution to (62) follows directly from [17, Thm. 7.8]. $\square$

5.3 Identifying the solution to the dual equation

We fix from now on a regularization $\xi ^\varepsilon$ of $\xi$, and we write for any $\varepsilon >0$. We also write for the solution of (62) constructed in Corollary 3 for the model with $U = {\bar{U}} + {\tilde{U}}$ given as in Sect. B (recall that and is the solution to (1)). As above we denote by the BPHZ-character of $\xi ^\varepsilon$ (for the extended regularity structure ) and we let . Our goal is to link the abstract dual equation (62) to the dual tangent equation (57). In a first step we can use the machinery of [4] to derive an equation for the reconstructed solution to the abstract fixed-point problem (62). This equation will be automatically of the form (57), but it is a-priori unclear whether the renormalization constants that one obtains in these two ways coincide (or at least differ by something of order 1 in a suitable sense). This however is necessary if we want to take the limit $\varepsilon \rightarrow 0$ in the model. Thus, in order to continue, we introduce the following assumption that makes sure that the dual renormalization constants are given by what we would naively expect.

Assumption 5

For any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ one has the identity

(70)

Remark 20

The simplicity of Assumption 5 is the main reason for assuming that $c^\varepsilon _\tau$ is given by the BPHZ character. In general, in order to pass to the limit $\varepsilon \rightarrow 0$ in $u^\varepsilon$, one could choose $c_\tau ^\varepsilon = (h \circ g^\varepsilon _{\textrm{BPHZ}})(\tau )$ where is an arbitrary fixed group element and $\circ$ denotes the group product in the renormalization group . In order to treat this more general situation, one would need to show that (70) above implies a similar relation with $g^\varepsilon _{\textrm{BPHZ}}$ replaced by $h\circ g^\varepsilon _{\textrm{BPHZ}}$ and replaced by for some character determined by h. We refrain from doing so for simplicity.

With this assumption, the following Proposition is a straight-forward application of [4, Thm. 5.7], which provides a convenient link between reconstructed solutions to abstract fixed point problems (for smooth models) and renormalized random PDEs.

Proposition 6

Assume that Assumptions 4 and 5 hold. Then for any $\varepsilon >0$ the smooth function $w_{{\mathfrak {t}},\phi }^\varepsilon$ given by

(71)

solves (57).

Proof

We are going to apply [4, Thm. 5.7]. First note that we are indeed in the setting of this theorem, since by Lemma 12 one has that the right hand side of (62) obeys , the assumption on $\eta$ follows trivially, since we stay away from the initial time, the condition on $\gamma$ can always be achieved by increasing $\gamma$ if necessary, and the fact that ${{{\textbf {I}}}}_+ DF_{\mathfrak {t}}(U)W$ and ${{{\textbf {I}}}}_+ DF_{\mathfrak {t}}^\Xi (U)W \Xi$ take values in for some ${\bar{\gamma }}>0$ and ${\bar{\eta }}>-{\mathfrak {s}}_0$ follows from Corollary 3. Note also that in the proof of [4, Thm. 5.7] the equation is derived via its mild formulation and the equation in its derivative form is only obtained in the last step, so that is all arguments go through in the time reversed setting verbatim. Denoting by $\tilde{w}_{{\mathfrak {t}},\phi }^\varepsilon$ the right hand side of (71), it follows now from [4, Thm. 5.7] that one has

for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$. Here, the function $(MF)_{\mathfrak {l}}^\Xi$ for and $\Xi \in {\mathfrak {L}}_-\sqcup \{\bullet \}$ was defined in [4, (3.9)], and is given by

(72)

while $(MF)_{{ {\mathfrak {t}}'}}^\Xi (u^\varepsilon , w^\varepsilon ) = F_{{ {\mathfrak {t}}'}}^\Xi (u^\varepsilon , w^\varepsilon ) = DF_{\mathfrak {t}}^\Xi (u^\varepsilon ) w^\varepsilon$.

On the other hand, the right hand side of (57) can be written in the form

$$\begin{aligned}{} & {} \sum _{\Xi \in {\mathfrak {L}}_-} DF^\Xi _{\mathfrak {t}}(u^\varepsilon ) w^\varepsilon _\phi \xi ^\varepsilon _\xi + \phi _{\mathfrak {t}}+ DF_{\mathfrak {t}}(u^\varepsilon )w^\varepsilon _\phi + \sum _{\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^{F,{ \circ }}} c^\varepsilon _\tau D\Upsilon _{\mathfrak {t}}^\tau (u^\varepsilon ) w_{\phi }^\varepsilon - \sum _{i=1}^d \sum _{\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^{F,i}} c^\varepsilon _\tau \partial _i w_{{\mathfrak {t}},\phi }^\varepsilon , \nonumber \\{} & {} \quad = \sum _{ \tau \in {\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F} \frac{ g^\varepsilon _{\textrm{BPHZ}}(\tau ) }{S(\tau )} D\Upsilon _{\mathfrak {t}}^F[\tau ]( u^\varepsilon ) ( w^\varepsilon , \nabla w^\varepsilon ). \end{aligned}$$

(73)

We conclude by applying Assumption 5. $\square$

Using Proposition 6 and performing the limit $\varepsilon \rightarrow 0$ in (58) now gives the following corollary.

Corollary 4

Assume that Assumptions 4 and 5 holds, and let . Then one has the identity

$$\begin{aligned} \left\langle v_h , \phi \right\rangle _{L^2}&= \left\langle \sum _{\Xi \in {\mathfrak {L}}_-} F^\Xi (u) h_\Xi , w_\phi \right\rangle _{L^2} + \left\langle v_h(0,\cdot ), w_\phi (0, \cdot ) \right\rangle _{L^2({{\textbf {T}}}^d)} \end{aligned}$$

(74)

Assumption 5 is not straight-forward to show in general. However, an important special case in which we can show directly that Assumption 5 holds is the case that we consider only a single equation, that is, in case that $\#{\mathfrak {L}}_+=1$.

Proposition 7

Under Assumption 4 assume that ${\mathfrak {L}}_+=\{{\mathfrak {t}}\}$. Then Assumption 5 holds.

Proof

With the aid of Lemmas 16 - 19 below, we obtain the following chain of equalities.

$$\begin{aligned} \sum _{ \tau \in {\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F} \frac{ g^\varepsilon _{\textrm{BPHZ}}(\tau ) }{S(\tau )} D\Upsilon _{\mathfrak {t}}^F[\tau ]&= \sum _{ \tau \in {\tilde{{\mathscr {T}}}}^F_{\mathfrak {t}}}\frac{ g^\varepsilon _{\textrm{BPHZ}} (\tau ) }{ S(\tau ) } \sum _{u \in N(\tau )} \Upsilon _{{\tilde{{\mathfrak {t}}}}}^F[ {{\mathfrak {D}}_u(\tau )} ] \end{aligned}$$

(75)

(76)

(77)

Note that the summand in (76) vanishes whenever , and otherwise one has by Lemma 18, so that is well defined. $\square$

5.4 Existence of densities

Assumption 6

We assume that the smooth functions and the solution u have the property that $(\sum _{{\mathfrak {t}}\in {\mathfrak {L}}_+} F_{\mathfrak {t}}^\Xi (u) w_{\mathfrak {t}})_{\Xi \in {\mathfrak {L}}_-} \ne 0$ on $(0,\tau )\times {{\textbf {T}}}^d$ for any $w \in {{\textbf {R}}}^{{\mathfrak {L}}_+}\backslash \{0\}$ almost surely.

We now have the following theorem, the proof of which is at this stage a generalization of the proof of [13, Prop. 5.3].

Theorem 10

Under Assumptions 4 to 6, assume that additionally $H^\xi$ is such that whenever $f \in L^2((0,T)\times {{\textbf {T}}}^d)$ such that $\int _0^T \int _{{{\textbf {T}}}^d} f h\, dtdx = 0$ for every , then $f=0$. Let $T>0$ and let $\phi ^i$, $i\le n$ be a collection of linearly independent test function . Then, the ${{\textbf {R}}}^{n}$-valued random variable

$$\begin{aligned} \left( \left\langle u, \phi ^1 \right\rangle , \cdots , \left\langle u, \phi ^n \right\rangle \right) \end{aligned}$$

conditioned on the event $\{ T < \tau \}$ admits a density with respect to Lebesgue measure.

Proof

Let $X:=( \left\langle u_{\mathfrak {t}}, \phi ^1_{\mathfrak {t}} \right\rangle , \cdots , \left\langle u_{\mathfrak {t}}, \phi ^n_{\mathfrak {t}} \right\rangle ) _{{\mathfrak {t}}\in {\mathfrak {L}}_+}$. By Theorem 1 we know that X is locally $H^\xi$-differentiable, so that by the Bouleau–Hirsch criterion, Corollary 1, we are left to show that DX is almost surely of full rank. Assume first that $n=1$. Then by Theorem 1 and Corollary 4 one has for any $h \in H^\xi$ the identity

$$\begin{aligned} D_h X = \left\langle v_{h}, \phi \right\rangle = \sum _{\Xi \in {\mathfrak {L}}_-} \left\langle h_\Xi , \sum _{{\mathfrak {t}}\in {\mathfrak {L}}_+} F^\Xi _{\mathfrak {t}}(u) w_{\phi ,{\mathfrak {t}}} \right\rangle + \left\langle v_h(0,\cdot ), w_\phi (0, \cdot ) \right\rangle _{L^2({{\textbf {T}}}^d)}. \end{aligned}$$

Using the assumption that $H^\xi$ is dense in $L^2({{{\textbf {D}}}})^{{\mathfrak {L}}_-}$, it suffices to show that one has

$$\begin{aligned} \sum _{{\mathfrak {t}}\in {\mathfrak {L}}_+} F^\Xi _{\mathfrak {t}}(u) w_{\phi ,{\mathfrak {t}}} \ne 0, \end{aligned}$$

which together with Assumption 6 is equivalent to showing that $w_{\phi ,{\mathfrak {t}}} \ne 0$ for at least one ${\mathfrak {t}}\in {\mathfrak {L}}_+$. On the other hand, by assumption there exists ${\mathfrak {t}}\in {\mathfrak {L}}_+$ such that $\phi _{\mathfrak {t}}\ne 0$, and it follows directly from (62) that $W_{{ {\mathfrak {l}}'}} \ne 0$ for at least one ${\mathfrak {l}}\in {\mathfrak {L}}_+$. It thus suffices to argue that whenever W is a solution to (62) on some time interval $(\theta ,T)$ such that the reconstruction vanishes on $(\theta ,T)\times {{\textbf {T}}}^d$, then this implies that one also has $W=0$ on $(\theta ,T)\times {{\textbf {T}}}^d$. Since W takes values in a function-like sector by Lemma 15, one hat , and thus by [17, Prop. 3.29] it suffices to show that $\left\langle W_{{ {\mathfrak {t}}'}}, \tau \right\rangle =0$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and any non-polynomial tree . Assume this was not the case, and let ${\mathfrak {l}}\in {\mathfrak {L}}_+$ and be the tree of minimal homogeneity such that $\left\langle W_{{ {\mathfrak {l}}'}}, \hat{\tau } \right\rangle \ne 0$. It follows in particular from Lemma 15 that $DF_{\mathfrak {l}}(U)$ and $D F_{\mathfrak {l}}^\Xi (U)\Xi$ take values in a sector of regularity $\alpha _{\mathfrak {l}}>-|{\mathfrak {l}}|_{\mathfrak {s}}$. Plugging this in the fixed point equation (62) implies that

which gives the desired contradiction.

The case $n>1$ can readily be reduced to the case $n=1$. To see this, assume that the statement holds in the one dimensional case. Note that the Nullset outside which the one dimensional conclusion holds is fixed (it coincides with the Nullset on which the extended model converges). In particular, this is the same Nullset for every $\phi$. Fix and assume now that DX is not almost surely of full rank. This implies in particular that there exists (random) $\lambda _{\mathfrak {t}}^i \in {{\textbf {R}}}$ for ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and $i\le n$ such that $\lambda$ is not identically zero and

$$\begin{aligned} \sum _{i \le n} \sum _{{\mathfrak {t}}\in {\mathfrak {L}}_+} \lambda ^i_{\mathfrak {t}}v_{h,{\mathfrak {t}}}(\phi _{\mathfrak {t}}^i)=0, \end{aligned}$$

which in turn implies that one has $\left\langle v_h, \psi \right\rangle =0$ where $\psi _{\mathfrak {t}}:= \sum _{i \le n} \lambda ^i_{\mathfrak {t}}\phi _{\mathfrak {t}}^i \ne 0$, in contradiction to the assumption that the statement holds for $n=1$. Note that the coefficients $\lambda$ are random in this proof; this is were we use the fact that the Null set does not depend on the test function. $\square$

Appendices

Continuity of maps between spaces of modelled distributions

Proposition 8

Let $V, {\bar{V}}$ be sectors in of regularity $\alpha ,{\bar{\alpha }}$ respectively. The one has the following.

• Multiplication. Let $\gamma , {\bar{\gamma }}>0$ and $\eta ,{\bar{\eta }} \in {{\textbf {R}}}$ and let ${\hat{\gamma }}:= (\gamma + {\bar{\alpha }})\wedge ({\bar{\gamma }}+\alpha )$ and ${\hat{\eta }}:= (\eta + {\bar{\alpha }}) \wedge ({\bar{\eta }}+\alpha )\wedge (\eta + {\bar{\eta }})$. Assume furthermore that ${\hat{\gamma }} \ge 0$ and that (V, W) is ${\hat{\gamma }}$-regular (c.f. [18, Def. 4.6]) and denote by the tree product. Then one has

  

  is locally Lipschitz continuous.

• Differentiation. Let $i\le d$ and let $\gamma >{\mathfrak {s}}_i$. Then

  

  with is locally Lipschitz continuous.

• Integration. Let ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and assume that $\eta <\gamma$ and $\eta \wedge \alpha >-|{\mathfrak {t}}|_{\mathfrak {s}}$. Then if $\gamma + |{\mathfrak {t}}|_{\mathfrak {s}}\notin {{{\textbf {N}}}}$ and $\eta + |{\mathfrak {t}}|_{\mathfrak {s}}\notin {{{\textbf {N}}}}$ one has

  

  with is locally Lipschitz continuous.

• Composition with smooth functions. Let V be function-like and $\gamma$-regular and let $F:{{\textbf {R}}}^n \rightarrow {{\textbf {R}}}$ be smooth. Define ${\hat{F}}:V_1 \times \ldots V_n \rightarrow V$ (c.f. [17, (4.2)]) by

  $$\begin{aligned} {\hat{F}}(\tau ):= \sum _{\alpha \in {{{\textbf {N}}}}^n} \frac{D^\alpha F (\langle \tau , {{\textbf {1}}}\rangle )}{\alpha !} (\tau - \langle \tau , {{\textbf {1}}} \rangle )^{\star \alpha }. \end{aligned}$$

  (78)

  Then if $0 \le \eta \le \gamma$ one has that

  

  for some sector W is locally Lipschitz continuous.

Proof

See [18, Sec. 6]. $\square$

Lift of the abstract fixed point problems coming from singular SPDEs

We show in this section that the right hand side of the fixed point problems considered in [4] admit $C^1$ lifts to the extended regularity structure. In order to state our results in a clean way, we introduce some notation from [4]. To begin with, we fix the subspace of considered in [4, Def. 5.1] with metric given by

It is clear that this metric is stronger then for any $\gamma >0$ and $K\subseteq {{{\textbf {D}}}}$ compact. From this it follows easily that all statements derived above holds true with replaced by .

Let now be the constant function defined in [4, (5.10)] and recall that from [4, Prop. 5.18] that is strongly Lipschitz continuous.

Recall from [4, Prop. 5.22] that under Assumption 1 there exists a solution u to the singular SPDE (1) and it is given as the reconstruction of a modelled distribution , which in turn can be written as $U = {\tilde{U}} + {\bar{U}}$ with ${\tilde{U}}$ as above and ${\bar{U}}$ satisfies the fixed point equation

(79)

Since ${\tilde{U}}$ is a constant modelled distribution its reconstruction is trivially locally $H^\xi$-Fréchet differentiable, and it remains to show that the same is true for ${\bar{U}}$. This will follow from the general result of Theorem 9, once we show that the right hand side of (79) admits a $C^1$ lift. The main statement that we will show in this section is thus the following.

Proposition 9

In the notation of [4, Sec. 5.5], let $\gamma _{\mathfrak {t}}:= \gamma + {\text {reg}}({\mathfrak {t}})$, $\eta _{\mathfrak {t}}:=\eta + {\text {ireg}}({\mathfrak {t}})$, ${\bar{\gamma }}_{\mathfrak {t}}:= \gamma _{\mathfrak {t}}- |{\mathfrak {t}}|_{\mathfrak {s}}+ \kappa _{\mathfrak {t}}$ and ${\bar{\eta }}_{\mathfrak {t}}:= \eta _{\mathfrak {t}}+ {\bar{n}}_{\mathfrak {t}}$. Define moreover the sector and . Let finally $H_{\mathfrak {t}}$ be the non-linearity of [4, Lem. 5.19], given by

Then H is strongly locally Lipschitz and admits a $C^1$ lift.

First note that in [4, Lem. 5.19] and the discussions below the proof of [4, Lem. 5.19] it was shown that the non-linearity H is strongly locally Lipschitz continuous.

Recall from [4, (3.7)] that $F_{\mathfrak {t}}^{\mathfrak {l}}$ are given as composition with smooth functions $f_{\mathfrak {t}}^{\mathfrak {l}}$ so that one has

$$\begin{aligned} F_{\mathfrak {t}}^{\mathfrak {l}}(U) = \sum _{\alpha \in {{{\textbf {N}}}}^{{\mathfrak {L}}_+\times {{{\textbf {N}}}}^{d+1}}} \frac{D^\alpha f^{\mathfrak {l}}_{\mathfrak {t}}(\langle {{\textbf {U}}}, {{\textbf {1}}} \rangle )}{\alpha !} ({{\textbf {U}}} - \langle {{\textbf {U}}}, {{\textbf {1}}} \rangle )^\alpha , \end{aligned}$$

for some , where we adopted the notation ${{\textbf {U}}} = ({{\textbf {U}}}_{{\mathfrak {t}},k})_{({\mathfrak {t}},k)\in {\mathfrak {L}}_+ \times {{{\textbf {N}}}}^{d+1}}$ where ${{\textbf {U}}}_{{\mathfrak {t}},k}:= D^k U_{\mathfrak {t}}$, and we write $\langle {{\textbf {U}}}, {{\textbf {1}}} \rangle := (\langle {{\textbf {U}}}_{{\mathfrak {t}},k}, {{\textbf {1}}} \rangle )_{({\mathfrak {t}},k) \in {\mathfrak {L}}_+\times {{{\textbf {N}}}}^{d+1}}$. We then have a natural candidate for the lift of $H_{\mathfrak {t}}$ which is given by

(80)

Here, given an extended model we write for the composition of the smooth function $f^{\mathfrak {t}}_{\mathfrak {l}}$ with in the model .

First note that (46) is a direct consequence of the definitions. We now sketch the proof that is a strongly locally Lipschitz map from into . Since the proof is very similar to the one given in [4, Lem. 5.19], we will not go into too much detail. The proof essentially boils down to an application of the results of [17, Sec. 6], which we have summarized in Proposition 8, and the only thing to notice is that the arguments given in the proof of [4, Lem. 5.19] carry over to to verbatim.

The main part of the proof consists in showing that the map is Fréchet differentiable. The strategy for this is to strengthen the results of [17, Sec. 6] and show that the respective operations considered there are actually not just Lipschitz continuous but $C^1$.

Proposition 10

Under the assumptions of Proposition 8 one has that the operations of multiplication, differentiation, integration and composition are $C^1$ between the respective spaces. Moreover, one has the following identity for the Fréchet derivative of the operation of composition with smooth functions

(81)

where $e_k$ denotes the k-th unit vector in ${{\textbf {R}}}^n$ and $V_k$ denotes the k-th component of V,

Proof

The fact that multiplication, differentiation and integration are $C^\infty$ follows simply from the fact that these operations are continuous and (multi-)linear.

We now show Fréchet differentiability of composition with smooth functions, together with (81). For this we write

$$\begin{aligned}{} & {} {\hat{F}}(V) - {\hat{F}}(U) - \sum _{k \le n}\widehat{D^{e_k} F}(U)(V_k-U_k) \nonumber \\{} & {} \quad = \sum _{\alpha } \frac{1}{\alpha !} \Big ( D^\alpha F({\bar{V}}) \tilde{V}^\alpha - D^\alpha F({\bar{U}}) {\tilde{U}}^\alpha - \sum _{k \le n} D^{\alpha + e_k} F({\bar{U}}) {\tilde{U}}^\alpha (V-U) \Big )\nonumber \\ \end{aligned}$$

(82)

where we write ${\bar{U}}:= \langle U,{{\textbf {1}}}, \rangle$ and ${\tilde{U}}:= U - {\bar{U}}$. We now define for $x \in {{{\textbf {D}}}}$ and $\alpha \in {{{\textbf {N}}}}^{d+1}$ the function $g_\alpha$ by setting

$$\begin{aligned} g_\alpha (t):= D^\alpha F({\bar{U}} + t {\bar{W}})({\tilde{U}} + t \tilde{W})^\alpha \end{aligned}$$

with $W:=V-U$. A direct computation shows that (82) can be re-written into

$$\begin{aligned} \sum _{\alpha } \frac{1}{\alpha !} (g_\alpha (1) - g_\alpha (0) - g_\alpha '(0))&= \sum _{\alpha } \frac{1}{\alpha !} \int _0^1 (1-t) g_\alpha ''(t) dt. \end{aligned}$$

Now note that one has

$$\begin{aligned} \sum _{\alpha } \frac{1}{\alpha !} g_\alpha ''(t) = \sum _{\alpha } \sum _{k,l\le n} \frac{1}{\alpha !} D^{\alpha +e_k + e_l} F({\bar{U}} + t{\bar{W}}) ({\tilde{U}} + t {\tilde{W}})^\alpha W^2, \end{aligned}$$

from which we infer that (82) can be re-written into

$$\begin{aligned} \sum _{k,l\le n} \int _0^1 (1-t) \widehat{F_{k,l}} (U + tW) W^2, \end{aligned}$$

where $F_{k,l}:=D^{e_k+e_l} F$. Now using that is a Lipschitz continuous map, we can estimate the norm of this expression by , which proves the claim. $\square$

With this proposition, the proof that is Fréchet differentiable is now straight forward, and consists in redoing the steps of [4, Lem. 5.19], replacing all statements made about Lipschitz continuity with statements about Fréchet differentiability.

The case of a single equation

In the entire section we assume that ${\mathfrak {L}}_+=\{{\mathfrak {t}}\}$ contains a unique element. Our goal is to derive the identities necessary to show Proposition 7.

The first lemma we are going to prove shows that functional derivatives of the original counter-terms are of the same form as the counter-terms of the dual equation.

Lemma 16

Under Assumption 4, one has for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and any $\tau \in {\tilde{{\mathscr {T}}}}_{{\mathfrak {t}}}^F$ the identity

$$\begin{aligned} D\Upsilon _{ {\mathfrak {t}}} ^F [ \tau ](u, \nabla u)(w, \nabla w) = \sum _{ \mu \in N(\tau ) } \Upsilon _{ { {\mathfrak {t}}'}} ^F [{{\mathfrak {D}}_\mu (\tau )}] ( u, w, \nabla w ). \end{aligned}$$

(83)

Remark 21

Note that under Assumption 4 the function $D\Upsilon ^F_{\mathfrak {t}}[\tau ](u, \nabla u)(w, \nabla w)$ does really only depend on $u, w, \nabla w$.

Proof

We only show the statement in the case that $\tau =T^{{\mathfrak {n}},{\mathfrak {o}}}_{\mathfrak {e}}\in {\mathscr {T}}^{F,{ \circ }}_{{\mathfrak {t}},-}$. Under Assumption 4 the case that $\tau \in {\mathscr {T}}^{F,i}_{{\mathfrak {t}},-}$ for some $i\in \{ 0, \ldots , d \}$ follows easily. We first claim that ${\mathfrak {n}}\equiv 0$. Indeed, otherwise $\Upsilon ^F_{\mathfrak {t}}[\tau ]$ contains a factor of the form

$$\begin{aligned} \partial ^n \Big ( \prod _{i\le m} D_{({\mathfrak {t}}_i,k_i)} \Big ) F_{\mathfrak {l}}^\Xi (u) = \sum _{{\mathfrak {l}}\in {\mathfrak {L}}_+} D_{({\mathfrak {l}},0)} \Big ( \Big (\partial ^{{\bar{n}}} \prod _{i\le m} D_{({\mathfrak {t}}_i,k_i)} \Big ) F_{\mathfrak {l}}^\Xi (u) \Big ) \partial _l u_{{\mathfrak {l}}} \end{aligned}$$

with $n = {\bar{n}} + e_l \in {{{\textbf {N}}}}^{d+1}$ non-zero, where $e_l$ denotes the l-th unit vector on ${{{\textbf {N}}}}^{d+1}$ for some $0 \le l\le d$. But since this factor does not depend on $\partial _l u_{\mathfrak {l}}$ explicitly by assumption, it must vanish identically, in contradiction to the assumption that $\tau$ is ${\mathfrak {t}}$-non-vanishing. Moreover, since F only depends on the solution u and not on its derivatives, one also has that ${\mathfrak {e}}\equiv 0$. Now, denoting by $e_j$ for $j=1,\ldots ,n$ the distinct edges $e \in K(\tau )$ with $e^\downarrow = \rho _T$ and by $\tau _j$ the unique maximal subtree of $\tau$ such that $\rho (\tau _j) = e_j^\uparrow$, it follows by definition that one has

$$\begin{aligned} D\Upsilon _{\mathfrak {t}}^F[\tau ] (u) w= & {} \sum _{i=1}^n D \Upsilon _{{\mathfrak {t}}_i}^F[\tau _i](u)w \prod _{j\le n, j\ne i} \Upsilon _{{\mathfrak {t}}_j}^F[\tau _j] (u) \bigg ( \prod _{j=1}^n D_{({\mathfrak {t}}_j,0)} (u) \bigg ) F_{\mathfrak {t}}^\Xi (u) \nonumber \\{} & {} \quad + \prod _{j\le n} \Upsilon _{{\mathfrak {t}}_j}^F[\tau _j] (u) D\bigg ( \prod _{j=1}^n D_{({\mathfrak {t}}_j,0)} (u) \bigg ) F_{\mathfrak {t}}^\Xi (u) w \end{aligned}$$

(84)

where we set ${\mathfrak {t}}_j:= {\mathfrak {t}}(\tau _j)$, and $\Xi ={\mathfrak {t}}(f)$ if there exists a (necessarily unique) edge $f \in L(\tau )$ with $f^\downarrow = \rho (\tau )$, and $\Xi =\bullet$ otherwise. We proceed inductively in the number of kernel-type edges of $\tau$. For $\# K(\tau ) = 0$ the statement holds trivially, so assume from now on that $\# K(\tau ) \ge 1$. It then follows from the induction hypothesis that we can write

$$\begin{aligned} D\Upsilon _{{\mathfrak {t}}_i}^F[\tau _i](u)w = \sum _{\mu \in N(\tau _i)} \Upsilon _{{ {\mathfrak {t}}_i'}}^F[{{\mathfrak {D}}_\mu (\tau _i)}](u,w) \end{aligned}$$

for any $i \le n$. Moreover, by definition of $F_{{ {\mathfrak {t}}'}} ^\Xi$ in (63), it follows that $D_{({\mathfrak {t}}_i,0)} F_{\mathfrak {t}}^\Xi = D_{({ {\mathfrak {t}}_i'},0)}F_{{ {\mathfrak {t}}'}}^\Xi$, and hence

$$\begin{aligned} \Upsilon _{{ {\mathfrak {t}}_i'}}^F[{{\mathfrak {D}}_\mu (\tau _i)}](u,w) \prod _{j\le n, j\ne i} \Upsilon _{{\mathfrak {t}}_j}^F[\tau _j] (u) \bigg ( \prod _{j=1}^n D_{({\mathfrak {t}}_j,0)} \bigg ) F_{\mathfrak {t}}^\Xi (u) = \Upsilon _{{ {\mathfrak {t}}'}}^F [{{\mathfrak {D}}_\mu (\tau )}](u,w). \end{aligned}$$

Since $\bigsqcup _{i\le n} N(\tau _i) = N(\tau ) \backslash \{\rho _\tau \}$, it remains to note that ${{\mathfrak {D}}_{\rho _\tau }(\tau )}=\tau$ and

$$\begin{aligned} \Upsilon _{{ {\mathfrak {t}}'}}^F[\tau ](u,w) = \prod _{j\le n} \Upsilon _{{\mathfrak {t}}_j}^F[\tau _j] (u) \bigg ( \prod _{j=1}^n D_{({\mathfrak {t}}_j,0)} (u) \bigg ) D F_{\mathfrak {t}}^\Xi (u) w \end{aligned}$$

by definition (63). $\square$

Next, we derive a useful identity for the symmetry factors appearing in (73) and (72). In order to state it, we introduce the set ${\mathfrak {D}}({\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F):=\{ {{\mathfrak {D}}_u(\tau )}: \tau \in {\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F, u \in N(\tau ) \}$.

Lemma 17

Let G be an additive group and for fixed ${\mathfrak {t}}\in {\mathfrak {L}}_+$ let $f: {\mathfrak {D}}({\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F) \rightarrow G$ be any map such that $f(\tau ) = 0$ for any . Then one has

(85)

Proof

Note first that by Lemma 14 the set is included in the set ${\mathfrak {D}}({\tilde{{\mathscr {T}}}}_{\mathfrak {t}}^F)$, so that the right hand side of (85) makes sense. Since moreover f vanishes outside of by definition, it follows that we can rewrite the left hand side of (85) into

where $m(\tau )\in {{{\textbf {N}}}}$ is a symmetry factor given by

(86)

It remains to show the identity , which we show inductively in the number of kernel type edges of $\tau$. If $\# K(\tau )=0$ or the identity is trivial, so that we exclude these cases in the sequel. In case that is planted and (86) holds for ${\tilde{\tau }}$, this identity also holds for $\tau$ since $m(\tau ) = m({\tilde{\tau }})$, $S(\tau ) = S({\tilde{\tau }})$ and . It remains to treat the case that is of the form with $n\ge 1$, $p_i \ge 1$ and $({\mathfrak {t}}_i,k_i,\tau _i)\ne ({\mathfrak {t}}_j, k_j, \tau _j)$ for $i\ne j$, $k\in {{{\textbf {N}}}}^{d+1}$ and $\Xi \in {\mathfrak {L}}_- \sqcup \{\bullet \}$, and such that (86) holds for for any $i \le n$. By assumption there exists such that , and we assume without loss of generality that $\tau$ is of the form

which can always be achieved by simply rearranging the order of the triples $({\mathfrak {t}}_i,k_i,\tau _i)$. In this case one has

On the other hand, we obtain

and the proof is finished, noting that one has the identity $m(\tau ) = p_1 m(\tau _1)$. $\square$

Identifying the renormalization constants takes a bit more work. We start with some intuition. The trees , which appear in the regularity structure used to solve the dual tangent equation, contain kernel-type edges which correspond to the Green’s function for the dual equation, which is given by $x \mapsto G(-x)$, where G is the Green’s function of the original equation. Reversing a kernel for an edge e in the evalution ${\varvec{\Pi }} \sigma$ is equivalent to reversing the direction of the corresponding e itself. Fortunately (since the dual equation is linear and does not feed back into the original equation), all trees that produce non-vanishing counter-terms for the dual equation are such that there exists an increasing (in the tree order) sequence of edge $e_1, \ldots , e_r$ for some $r \ge 0$ such that $e^{\downarrow }_{i+1} = e^{\uparrow }_i$ for any $i < r$ (i.e. the edges are connected) and the set $\{e_i: 1 \le i \le r\}$ contains exactly those edges with dual type. Inverting the direction of these edges is therefore equivalent to moving the root from $\rho (\sigma )=e_1^\downarrow$ to $e_r^\uparrow$. We finally change the type of the edges that have been reversed to the original kernel. We create a new tree in this way with identical renormalisation constant to the tree we started with (apart from symmetry factors, see below), but without dual types. This will allow us to match renormalisation constants of the dual equation with renormalisation constants of the tangent equation.

As a preparation, we introduce some notation. Given a rooted tree T with vertex set V(T), edge set E(T) and root $\rho _T$, we can define for any $\nu \in V(T)$ another tree $\Phi _\nu (T)$ with identical edge and vertex sets, but where we set $\rho (\Phi _\nu (T)):= \nu$. Given additionally a type map ${\mathfrak {t}}: E(T) \rightarrow {\mathfrak {L}}$ and decorations ${\mathfrak {e}}: E(T) \rightarrow {{{\textbf {N}}}}^{d+1}$ and ${\mathfrak {n}}: V(T) \rightarrow {{{\textbf {N}}}}^{d+1}$ we obtain another typed, decorated tree $(\Phi _\nu (T)^{\tilde{\mathfrak {n}}}_{\mathfrak {e}},{\mathfrak {t}})$ by simply letting the maps ${\mathfrak {e}}$, ${\mathfrak {n}}$, and ${\mathfrak {t}}$ unchanged. For a tree $\tau = T^{{\mathfrak {n}}}_{\mathfrak {e}}$ we also define $\hat{\Phi }_u \tau$ by setting

$$\begin{aligned} {\hat{\Phi _u}} \tau := \sum _{ {\mathfrak {m}}: N(\tau ) \rightarrow {{{\textbf {N}}}}^{d+1} } (-1) ^{|{\mathfrak {m}}|} \left( {\begin{array}{c}{\mathfrak {n}}\\ {\mathfrak {m}}\end{array}}\right) (\Phi _u T) ^{{\mathfrak {n}}- {\mathfrak {m}}+ \sum {\mathfrak {m}}{{{\textbf {I}}}}_{\rho (\tau )}}_{\mathfrak {e}}. \end{aligned}$$

(87)

Let now be a tree and let . Then by Lemma 14 there exists $\nu \in N(\tau )$ such that $\sigma = {{\mathfrak {D}}_\nu (\tau )}$. Note now that we can naturally identify the node set $N(\tau )$ of $\tau$ with the node set $N(\sigma )$ of $\sigma$. If we do this identification, then the vertex $\nu$ is distinguished in $\sigma$ by the property that it is the unique maximal vertex (with respect to the tree order) that has the property that ${\mathfrak {t}}(u^\downarrow ) \in { {\mathfrak {L}}'}_+$. In particular, the node $\nu \in N(\sigma )$ is uniquely determined by $\sigma$, and as a consequence it makes sense to define

$$\begin{aligned} {\hat{\Phi }} \sigma := {\hat{\Phi _\nu }}(\sigma ). \end{aligned}$$

The point of this definition is that for any $\tau \in \tilde{\mathscr {T}}_{{\mathfrak {t}}}^F$ and $\nu \in N(\tau )$ we can show an identity between the renormalization constants associated to $\tau$ and ${\hat{\Phi }}{{\mathfrak {D}}_\nu (\tau )}$, compare Lemma 19 below.

Before we state any precise statement, we need to deal with the subtlety that it is in general not the case that for any and any $\nu \in N(\tau )$. However, we have the following lemma.

Lemma 18

Assume that Assumption 4 holds and that ${\mathfrak {L}}_+:=\{{\mathfrak {t}}\}$. Then, the map ${\hat{\Phi }}$ is an involutory bijection from onto itself, and for any tree one has the identity $\Upsilon _{{ {\mathfrak {t}}'}} ^F[ \Phi \tau ] = \Upsilon _{ { {\mathfrak {t}}'}}^F[ \tau ]$ and $S(\Phi \tau )=S(\tau )$.

Proof

It suffices to show $\Upsilon _{{ {\mathfrak {t}}'}} ^F[ \Phi \tau ] = \Upsilon _{ { {\mathfrak {t}}'}}^F[ \tau ]$. The fact that $\Phi$ maps into itself is then a consequence of the definition of the latter and Lemma 14. Moreover, since $\Phi$ is involutory by definition, preserves homogeneity, and the set of with $|\tau |_{\mathfrak {s}}= \gamma$ for some fixed $\gamma \in {{\textbf {R}}}$ is finite, it is also a bijection. The identity $S(\Phi \tau )=S(\tau )$ follows easily from the definition.

Concerning the identity $\Upsilon _{{ {\mathfrak {t}}'}} ^F[ \Phi \tau ] = \Upsilon _{ { {\mathfrak {t}}'}}^F[ \tau ]$, we use the expression

$$\begin{aligned} \Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau ](u,w) = \prod _{\mu \in N(\tau )} \Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \mu ]:= \prod _{ \mu \in N(\tau ) } \partial ^{{\mathfrak {n}}(\mu )} \bigg ( \prod _{j=1}^{n[\mu ]} D_{({\mathfrak {t}}_j[\mu ],0)} \bigg ) F_{{\mathfrak {t}}[\mu ]}^{\Xi [\mu ]}(u,w). \end{aligned}$$

Let $\nu \in N(\tau )$ be the unique vertex such that . It then follows that for any $\mu \notin \{ \rho _\tau , \nu \}$ the vertex $\mu$ has the same incoming edge types (i.e. types ${\mathfrak {t}}(e)$ with $e^\downarrow = \mu$) and the same outgoing type (i.e. the type ${\mathfrak {t}}(e)$ with $e^\uparrow = \mu$) when viewed as an element of $N(\tau )$ and $N({\hat{\Phi \tau }})$, respectively, and moreover the same polynomial label ${\mathfrak {n}}(\mu )$, so that it follows that one has $\Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \mu ] = \Upsilon _{{ {\mathfrak {t}}'}}^F [ {\hat{\Phi }} \tau , \mu ]$. Moreover, if $e_1, \ldots , e_n$ are the incoming edges of $\nu$ and e is the outgoing edge of $\nu$ when viewed as an element of $N(\tau )$, then $e_1, \ldots , e_n, e$ are the incoming edges of $\nu$ when viewed as an element of $N({\hat{\Phi \tau }})$, and by construction one has ${\mathfrak {t}}(e)={ {\mathfrak {t}}'}$. Assume first that . It then follows that ${\mathfrak {n}}\equiv 0$, and using the fact that $F_{{ {\mathfrak {t}}'}}^{\Xi [\nu ]}(u,w)$ is linear in $w_{ {\mathfrak {t}}}$, so that $(D_{({ {\mathfrak {t}}'},0)} F_{{ {\mathfrak {t}}'}}^{\Xi [\nu ]}(u_{\mathfrak {t}},w_{\mathfrak {t}}))w_{\mathfrak {t}}= F_{{ {\mathfrak {t}}'}}^{\Xi [\nu ]}(u_{\mathfrak {t}}, w_{\mathfrak {t}})$, we obtain

$$\begin{aligned} (\Upsilon _{{ {\mathfrak {t}}'}}^F [ {\hat{\Phi }} \tau , \nu ](u_{\mathfrak {t}})) w_{{\mathfrak {t}}}&= \bigg (\bigg ( D_{({ {\mathfrak {t}}'},0)}\prod _{j=1}^{n[\nu ]} D_{({\mathfrak {t}}_j[\nu ],0)} \bigg ) F_{{ {\mathfrak {t}}'}}^{\Xi [\nu ]}(u_{\mathfrak {t}},w _{\mathfrak {t}}) \bigg ) w_{{\mathfrak {t}}} \end{aligned}$$

(88)

$$\begin{aligned}&= \bigg ( \prod _{j=1}^{n[\nu ]} D_{({\mathfrak {t}}_j[\nu ],0)} \bigg ) F_{{ {\mathfrak {t}}'}}^{\Xi [\nu ]}(u _{\mathfrak {t}}, w _{\mathfrak {t}}) \end{aligned}$$

(89)

$$\begin{aligned}&= \Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \nu ](u_{\mathfrak {t}}, w_{\mathfrak {t}}). \end{aligned}$$

(90)

An identical calculation shows that one has $\Upsilon _{{ {\mathfrak {t}}'}}^F [ \Phi \tau , \rho _\tau ](u_{\mathfrak {t}}, w_{\mathfrak {t}}) \!=\! (\Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \rho _\tau ](u_{\mathfrak {t}}))w_{\mathfrak {t}}$, and this concludes the proof.

Finally, if , then one has

$$\begin{aligned} \Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \rho _\tau ](u_{\mathfrak {t}}, w_{\mathfrak {t}}) \equiv c \qquad \text { and }\qquad \Upsilon _{{ {\mathfrak {t}}'}}^F [ \tau , \nu ](u_{\mathfrak {t}}, w_{\mathfrak {t}}) = \partial ^{i} w_{\mathfrak {t}}, \end{aligned}$$

and a similar identity with the roles of $\nu$ and $\rho _\tau$ reversed holds for ${\hat{\Phi \tau }}$. $\square$

Remark 22

In the proof we use that ${\mathfrak {L}}_+$ contains only one element. We argue that for any $\mu \notin \{ \rho _\tau , \nu \}$ the vertex $\mu$ has the same incoming edge types and same outgoing edge type when viewed as an element of $N(\tau )$ and $N({\hat{\Phi \tau }})$, respectively. This would no longer be true in general for any $\mu$ which lies in between $\rho _\tau$ and $\nu$ if ${\mathfrak {L}}_+$ contains more than one element.

Finally, we show the following lemma, which is the reason for introducing the map ${\hat{\Phi }}$.

Lemma 19

For any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, any $\tau \in {\mathscr {T}}_{{\mathfrak {t}},-}^F$ and any node $\nu \in N(\tau )$ with the property that one has the identity

We point out that Lemma 19 does neither require Assumption 4 nor the assumption that $\#{\mathfrak {L}}_+ = 1$.

Proof

In this proof we use the notation that given a kernel assignment satisfying [17, Ass. 5.1, Ass. 5.4] and a smooth Gaussian field , we write ${\varvec{\Pi }}^{\eta ,L}$ and $g_{\textrm{BPHZ}}^{\eta ,L}$ for the canonical evaluation and the BPHZ character constructed as in [5, Rmk. 6.12] and [5, (6.24)] for the kernel assignment L and the noise $\eta$. It follows that if we set $L_{\mathfrak {t}}:= L_{{ {\mathfrak {t}}'}}:= K_{\mathfrak {t}}$ for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$, then the effect of ${\mathfrak {D}}_\nu$ is not seen on the analytic level, and we obtain for any $\tau \in {\mathscr {T}}_{\mathfrak {t}}^F$ and any node $\nu \in N(\tau )$ the identity ${\varvec{\Pi }}^{\eta ,K} \tau = {\varvec{\Pi }}^{\eta ,L}{\mathfrak {D}}_\nu (\tau )$, and similarly $g_{\textrm{BPHZ}}^{\eta ,K} \tau = g_{\textrm{BPHZ}}^{\eta ,L} ({\mathfrak {D}}_\nu (\tau ))$.

For the proof of Lemma 19, we are thus left to show that for any ${\mathfrak {t}}\in {\mathfrak {L}}_+$ and any one has the identity

(91)

We first deal with the issue that the image of under ${\hat{\Phi }}$ does in general not coincide with , which is due to the fact that if $\tau$ is a tree that strongly conforms to the rule , its image under ${\hat{\Phi }}$ might not. We will circumvent this issue by working in the Hopf algebra defined in [5, (4.10)] for the type set . Actually, it suffices for us to work in the reduced Hopf algebra , where is obtained from by identifying any trees that only differ by the extended decoration and additionally factoring out any trees $\tau = (T^{\mathfrak {n}}_{\mathfrak {e}},{\mathfrak {t}})$ with the property that there exists $e \in E(T)$ such that ${\mathfrak {t}}(e) \in {\mathfrak {L}}_-$ and $e^\uparrow$ is either not a leaf or one has ${\mathfrak {n}}(e^\uparrow )\ne 0$ (or both). Following [5, Rem. 4.16], this leads to the following space.

Definition 10

We denote by the unital algebra freely generated by typed, rooted, decorated trees $\tau = (T^{\mathfrak {n}}_{\mathfrak {e}},{\mathfrak {t}})$ such that $\tau \ne \bullet$ and such that , ${\mathfrak {n}}: N(\tau ) \rightarrow {{{\textbf {N}}}}^{d+1}$ and ${\mathfrak {e}}: E(\tau ) \rightarrow {{{\textbf {N}}}}^{d+1}$, and such that $e^\uparrow$ is a leave of T for any noise type edge $e \in L(\tau )$.

By [5, Prop. 3.30], this space becomes a Hopf algebra when endowed with the co-product $\Delta _1$ defined in [5, Def. 3.3]. We denote this co-product on simply by $\Delta$.

Definition 11

We define the ideal generated by all trees such that $|\tau |_{\mathfrak {s}}>0$, and we define the factor algebra , with canonical embedding .

It straight forward to see that is a Hopf ideal, so that is a factor Hopf algebra. The following Proposition follows exactly as [5, Prop. 6.5].

Proposition 11

There exists a unique algebra homomorphism with the property that

on .

We now define a subspace with the property that ${\hat{\Phi }}$ is well defined on and an involutory bijection.

Definition 12

We define (respectively ) as the unital sub algebra generated by all trees (respectively ) with the property that there exists a node $u \in N(\tau )$ such that for any edge $e \in E(\tau )$ one has ${\mathfrak {t}}(e) \in { {\mathfrak {L}}_+'}$ if and only if e lies on the unique path from u to the root $\rho _\tau$.

It is readily checked from the definition of the co-product $\Delta$ and the operation $\Phi$ that is closed under $\Delta$, in the sense that , so that is a Hopf algebra, and is such that ${\hat{\Phi }} \circ {\hat{\Phi }} = \text {Id}$.

On (respectively ) we define the character $g^{\eta ,K}$ (respectively $g^{\eta ,K}_{\textrm{BPHZ}}$) by setting $g^{\eta ,K}\tau := {{{\textbf {E}}}}{\varvec{\Pi }}^{\eta ,K}\tau$ (respectively ) for any tree $\tau$ and extending this linearly and multiplicatively. Note that and , and on these subspaces this notation is consistent, in the sense that one has and on and , respectively. We are thus left to show that $g^{\eta ,L}_{\textrm{BPHZ}} = g^{\eta ,K}_{\textrm{BPHZ}} \circ {\hat{\Phi }}$ on . We now note that directly from the definition one has the identity on , and since ${\hat{\Phi }}$ is an involutory bijection on , it follows that it suffices to show

(92)

on . We now identify ideals and with the property that $g^{\eta ,M} _-$ vanishes on for any smooth Gaussian field $\eta$ and any kernel assignment M, and such that the canonical embedding ${ {\mathfrak {i}}_-^{\text {ex}}}$ restricts to an embedding .

We start with a definition which is completely analogous to the ideal defined in [18, (2.16)–(2.18)] for Feynman diagrams.

Definition 13

We denote by (respectively ) the ideals generated by all elements which are can be written in form (93), (94), or (95) for some tree (respectively ), where

• For any node $u \in N(\tau ) \backslash \{\rho (\tau )\}$ and any $i \le d$

  $$\begin{aligned} \sum _{ \begin{array}{c} e \in E(\tau ) \\ e^\downarrow = u \end{array} } T ^{\mathfrak {n}}_{{\mathfrak {e}}+ e_i{{{\textbf {I}}}}_{e}} - T^{\mathfrak {n}}_{{\mathfrak {e}}+ e_i{{{\textbf {I}}}}_{u^\downarrow }} + {\mathfrak {n}}(u) T^{{\mathfrak {n}}- e_i{{{\textbf {I}}}}_u}_{\mathfrak {e}}\end{aligned}$$

  (93)

• For any $i \le d$

  $$\begin{aligned} \sum _{ \begin{array}{c} e \in E(\tau ) \\ e^\downarrow = \rho (\tau ) \end{array} } T ^{\mathfrak {n}}_{{\mathfrak {e}}+ e_i {{{\textbf {I}}}}_{e}} + \sum _{ u \in N(\tau ) } {\mathfrak {n}}(u) T^{{\mathfrak {n}}- e_i {{{\textbf {I}}}}_u}_{\mathfrak {e}}\end{aligned}$$

  (94)

• One has

  

  (95)

We also define the factor algebras

Here we write $e_i \in {{{\textbf {N}}}}^{d+1}$ for the i-th unit vector.

With a proof identical to [18, Prop. 2.12], we obtain the following.

Lemma 20

The ideal is a Hopf ideal in , so that in particular is a factor Hopf algebra. Moreover, one has , so that in particular the space forms a left co-module over .

Now note that by definition one has for any , so that it remains to show that $g_{\textrm{BPHZ}}^{\eta ,K}$ vanishes on . It follows readily from Lemma 20 and the recursive identity for the twisted antipode that one has . It thus remains to show that $g^{\eta ,K}$ vanishes identically on . This however follows identically to [18, Property 4]. $\square$