Singular SPDEs in domains with boundaries
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Abstract
We study spaces of modelled distributions with singular behaviour near the boundary of a domain that, in the context of the theory of regularity structures, allow one to give robust solution theories for singular stochastic PDEs with boundary conditions. The calculus of modelled distributions established in Hairer (Invent Math 198(2):269–504, 2014. https://doi.org/10.1007/s0022201405054) is extended to this setting. We formulate and solve fixed point problems in these spaces with a class of kernels that is sufficiently large to cover in particular the Dirichlet and Neumann heat kernels. These results are then used to provide solution theories for the KPZ equation with Dirichlet and Neumann boundary conditions and for the 2D generalised parabolic Anderson model with Dirichlet boundary conditions. In the case of the KPZ equation with Neumann boundary conditions, we show that, depending on the class of mollifiers one considers, a “boundary renormalisation” takes place. In other words, there are situations in which a certain boundary condition is applied to an approximation to the KPZ equation, but the limiting process is the Hopf–Cole solution to the KPZ equation with a different boundary condition.
Mathematics Subject Classification
60H15 35K511 Introduction
The aim of the present article is to provide a framework within the context of this theory, with which one can provide solution theories for initialboundary problems for singular SPDEs. The appropriate spaces of modelled distributions introduced here are flexible enough to account for singular behaviour at the spatial boundary. These are similar to the singularities at the initial time treated in [10] and indeed a similar calculus can be built on them. One could hope that, provided such a generalisation of the abstract calculus is obtained, coupling it with rest of the theory automatically gives solution theories of the same equations that were previously considered without or with periodic boundary conditions, now with for instance Dirichlet or Neumann boundary conditions. However, a subtlelooking but notable difference is that the codimension 2 of the initial time hyperplane is replaced by the codimension 1 of the spatial boundary, and therefore dual elements of spaces of test functions supported away from the boundary which are uniformly ‘locally in \(\mathcal {C}^\alpha \)’ for \(\alpha <1\) have no canonical extensions as bona fide distributions—a simple example for such situation is the function 1 / x, considered as an element of \(\mathcal {D}'({\mathbb {R}}{\setminus }\{0\})\). As elements with (local) regularity less than \(1\) are quite common in applications (unlike elements with regularity less than \(2\)), for each such object one has to make sense of their extensions, in a consistent way so that the sufficient continuity properties are preserved. Although, unlike the rest of the theory, the treatment of this issue is not performed in a systematic way, the methods used to treat the examples discussed in the next section are likely to be relevant to different situations.
1.1 Applications
We now give a few examples of singular SPDEs to which the framework developed in this article can be applied. The proofs of the results stated here are postponed to Sect. 6.
Theorem 1.1
There exists a choice of diverging constants \(C_\varepsilon \) and a random time \(T>0\) such that the sequence \(u^\varepsilon \mathbf {1}_{[0,T]}\) converge in probability to a continuous function u. Furthermore, provided that the constants \(C_\varepsilon \) are suitably chosen, the limit does not depend on the choice of the mollifier \(\rho \).
Remark 1.2
We believe that the choice \(D = (1,1)^2\) is not essential, the restriction to the square case is mostly for the sake of convenience: it is easier to verify our conditions when the explicit form of the Greens function is known.
Remark 1.3
One could easily deal with inhomogeneous Dirichlet data of the type \(u^\varepsilon =g\) on \(\partial D\) by considering the equation for \(u^\varepsilon  {\hat{g}}\), where \({\hat{g}}\) is the harmonic extension of g to all of D.
Remark 1.4
We have chosen to include the arbitrary constant \(\textstyle {1\over 2}\) in front of the term \(\partial _x^2 u\) so that the corresponding semigroup at time t is given by the Gaussian with variance t.
We then have the following analogous result on local solvability.
Theorem 1.5
If \(\rho \) satisfies the condition \(\rho (x,t) = \rho (x,t)\), then the statement of Theorem 1.1 also holds for \(u^\varepsilon \) defined in (1.3).
Remark 1.6
If the additional symmetry on \(\rho \) fails, then an analogous result holds, but an additional drift term appears in general, see for example [13].
Theorem 1.7
Remark 1.8
Even in the symmetric case, one can have \(a \ne 0\), so that one can end up with nonzero boundary conditions in the limit, although one imposes zero boundary conditions for the approximation.
Remark 1.9
The effect of subtracting cx in (1.7) is the same as that of adding a drift term \(2c \partial _x u^\varepsilon \) to the right hand side of (1.6) and changing the boundary condition \({\hat{c}}_\pm \) into \({\hat{c}}_\pm  c\), which is the reason for the form of the constants \(c_\pm \).
Remark 1.10
At first sight, this may appear to contradict the results of [1] where the authors consider the threedimensional parabolic Anderson model in a rather general setting which covers that of domains with boundary. Since this scales in exactly the same way as the KPZ equation (after applying the Hopf–Cole transform), one would expect to observe a similar “boundary renormalisation” in this case. The reason why there is no contradiction with our results is that there is no statement on the behaviour of the renormalisation term \(\lambda ^\varepsilon \) in [1, Thm 1] as a function of position. What our result suggests is that, at least in the flat case, one should be able to take \(\lambda ^\varepsilon \) of the form \(\lambda ^\varepsilon = C_\varepsilon + \mu \), where \(C_\varepsilon \) is a constant and \(\mu \) is some measure concentrated on the boundary of the domain.
Remark 1.11
The recent result [8] is consistent with our result in the sense that it shows that the “natural” notion of solution to (1.4) with homogeneous Neumann boundary condition (i.e. \(c_\pm = 0\)) does not coincide with the Hopf–Cole solution with homogeneous boundary data. In this particular case, one possible interpretation is that, for any fixed time, the solution to the KPZ equation is a forward/backwards semimartingale (in its own filtration) near the right/left boundary point. It is then natural to define the “space derivative” at the boundary to be the derivative of its bounded variation component. When performing the Hopf–Cole transform, one then picks up an Itô correction term, which is precisely what one sees in [8]. Note however that it is not clear at all whether the homogeneous Neumann solution of [8] can be obtained by considering (1.6) with \({\hat{b}}_\pm = 0\) for some mollifier \(\rho \). This is because, with our conventions for units, this corresponds to the Hopf–Cole solution with \(b_\pm = \pm 1\), while in our case one has \(a \le {1\over 2}\) as a consequence of the explicit formula (1.8) for typical choices of the mollifier, i.e. those with \(\rho \ge 0\).
The remainder of the article is structured as follows. After recalling some elements of the theory of regularity structures in Sect. 2, mostly to fix our notations, we introduce in Sect. 3 the spaces of modelled distributions that are relevant for solving singular stochastic PDEs on domains. Section 4 is then devoted to a rederivation of the calculus developed in [10], adapted to these spaces, with an emphasis on those aspects that actually differ in the present context. In Sect. 5, we then “package” these results into a rather general fixed point theorem, which is finally applied to the above examples in Sect. 6.
2 Elements of the theory of regularity structures
First let us summarise the relevant definitions, constructions, and results from the theory of regularity structures that we will need in the sequel.
2.1 Main definitions
Definition 2.1

An index set \(A\subset {\mathbb {R}}\) which is locally finite and bounded from below.

A graded vector space \(T=\bigoplus _{\alpha \in A}T_\alpha \) with each \(T_\alpha \) a finitedimensional normed vector space.

A group G of linear operators \(\varGamma :T\rightarrow T\), such that, for all \(\varGamma \in G\), \(\alpha \in A\), \(a\in T_\alpha \), one has \( \varGamma aa=\bigoplus _{\beta <\alpha }T_\beta \).
Definition 2.2
Given a regularity structure and \(\alpha \le 0\), a sector V of regularity \(\alpha \) is a Ginvariant subspace of T of the form \(V=\bigoplus _{\beta \in A}V_{\beta }\) such that \(V_\beta \subset T_\beta \) and \(V_\beta =\{0\}\) for \(\beta <\alpha .\)
With V as above, we will always use the notations \( V_\alpha ^+=\bigoplus _{\gamma \ge \alpha }V_\gamma \) and \(V_{\alpha }^=\bigoplus _{\gamma <\alpha }V_\gamma \), with the convention that the empty direct sum is \(\{0\}\). Some further notations will be useful. For \(a\in T\), its component in \(T_\alpha \) will be denoted either by \(\mathcal {Q}_\alpha a\) or by \((a)_\alpha \) and the norm of \((a)_\alpha \) in \(T_\alpha \) is \(\Vert a\Vert _\alpha \). The projection onto \(T_\alpha ^\) is denoted by \(\mathcal {Q}_\alpha ^\). The coefficient of \(\mathbf {1}\) in a is denoted by \(\langle \mathbf {1},a\rangle \).
In most of the following we consider d, \(\mathscr {T}\), and \(\mathfrak {s}\) to be fixed. We will always assume that our regularity structures contain \(\bar{\mathscr {T}}\) in the sense of [10, Sec. 2.1]. A concise definition of the Hölder spaces of all (noninteger) exponents that are used in the sequel is the following.
Definition 2.3
We shall also use the notation \(\mathcal {B}^r\) for smooth functions \(\varphi \) supported on B(0, 1) and having derivatives up to order r bounded by 1.
Definition 2.4

A map \(\varGamma :{\mathbb {R}}^d\times {\mathbb {R}}^d\rightarrow G\) such that \(\varGamma _{xy}\varGamma _{yz}=\varGamma _{xz}\) for all x, y, \(z\in {\mathbb {R}}^d\).

A collection of continuous linear maps \(\varPi _x: T\rightarrow \mathcal {S}'({\mathbb {R}}^d)\) such that \(\varPi _x=\varPi _y\circ \varGamma _{xy}\) for all x, \(y\in {\mathbb {R}}^d\).
The best proportionality constants in (2.2) are denoted by \(\Vert \varPi \Vert _{\gamma ,\mathfrak {K}}\) and \(\Vert \varGamma \Vert _{\gamma ,\mathfrak {K}}\), respectively.
We shall always assume that all models under consideration are compatible with the polynomials in the sense that \((\varPi _x X^k)(y)=(yx)^k\) for any multiindex k. A central notion of the theory is that of a modelled distribution, spaces of which are defined as follows.
Definition 2.5
Although the spaces \(\mathcal {D}^\gamma \) depend on \(\varGamma \), in many situation, where there can be no confusion about the model, this dependence will be omitted in the notation. The name ‘modelled distribution’ is justified by the following result.
Theorem 2.6
It is clear from (2.4) that the reconstruction operator \(\mathcal {R}\) is local, so in particular one can ‘reconstruct’ modelled distributions that only locally lie in \(\mathcal {D}^\gamma \).
Remark 2.7
While in [10] in the bound (2.4), \(y=x\) is assumed, this version is essentially equivalent: for all \(y\in \mathop {\mathrm {supp}}\psi _x^\lambda \), one can simply rewrite \(\psi _x^\lambda \) as \(\bar{\psi }_y^{2\lambda }\) with some \(\bar{\psi }\in \mathcal {B}^r\).
Let us also note that in the literature the use of the notation \({\vert \vert \vert \cdot \vert \vert \vert }\) is slightly inconsistent: sometimes it is defined as in (2.3), in some other instances it includes the term \(\sup _{x\in \mathfrak {K}}\sup _{l<\gamma }\Vert f(x)\Vert _l\). We will also be guilty of this: while for now, in the unweighted setting, (2.3) is convenient since that is what appears in the bounds for reconstructions like (2.4) above and (2.11) below, the weighted versions of \({\vert \vert \vert \cdot \vert \vert \vert }\) introduced in Sect. 3 do include controls over \(\Vert f(z)\Vert \).
Definition 2.8
A continuous bilinear map \(\star :T\times T\rightarrow T\) is called a product if, for \(a\in T_\alpha \) and \(b\in T_\beta \), one has \(a\star b\in T_{\alpha +\beta }\), and \(\mathbf {1}\star a=a\star \mathbf {1}\) for all \(a\in T\). The products arising in this article will always be associative and commutative, at least on some sufficiently large subspace.
A pair of sectors (V, W) is said to be \(\gamma \)regular with respect to the product \(\star \) if \((\varGamma a)\star (\varGamma b)=\varGamma (a\star b)\) for all \(\varGamma \in G\) and \(a\in V_\alpha \), \(b\in W_\beta \), satisfying \(\alpha +\beta <\gamma \). A sector is called \(\gamma \)regular, if the pair (V, V) is \(\gamma \)regular. Given two Tvalued functions f and \({\bar{f}}\), we also denote by \(f\star _\gamma {\bar{f}}\) the function \(x\rightarrow \mathcal {Q}_\gamma ^(f(x)\star {\bar{f}}(x))\).
The abstract version of differentiation is quite straightforward.
Definition 2.9
Given a sector V, a family of operators \(\mathscr {D}_i: V\rightarrow V\) with \(i=1,\ldots ,d\) is called an abstract gradient if for every i, every \(\alpha \) and every \(a\in V_\alpha \), one has \(\mathscr {D}_i a\in T_{\alpha \mathfrak {s}_i}\) and \(\varGamma \mathscr {D}_i a=\mathscr {D}_i \varGamma a\) for all \(\varGamma \in G\).
The final important operation on modelled distribution is the integration against singular kernels, the aim of which is to ‘lift’ convolutions with Green functions to the abstract setting. The first ingredient is the abstract integral operator.
Definition 2.10

\(\mathcal {I}(V_\alpha )\subset T_{\alpha +\beta }\) for all \(\alpha \in A\).

\(\mathcal {I}a=0\) for all \(a\in V\cap {\bar{T}}\).

\(\mathcal {I}\varGamma a\varGamma \mathcal {I}a\in \bar{T}\) for all \(a\in V\) and \(\varGamma \in G\).
In our applications \(\beta \) will always be 2, but for most of the analysis the one important property required of \(\beta \) is that for each \(\alpha \in A\), \(\alpha +\beta \in \mathbb {Z}\) implies \(\alpha \in \mathbb {Z}\). In particular, under this assumption, \(\mathcal {I}\) does not produce any components in integer homogeneities. The class of kernels we will want to lift is characterised as follows.
Definition 2.11

For all \(n\ge 0\), \(K_n\) is supported on \(\{(x,y):\Vert xy\Vert _\mathfrak {s}\le 2^{n}\}\).

For any two multiindices k and l, \( D_1^kD_2^lK_n(x,y)\lesssim 2^{n(\mathfrak {s}+k+l_\mathfrak {s}\beta )} \), where the proportionality constant only depends on k and l, but not on n, x, y.
 For any two multiindices k and l, \(y\in {\mathbb {R}}^d\), \(i=1,2\), it holds, for all \(n\ge 0\),where the proportionality constant only depends on k and l.$$\begin{aligned} \Big \int _{R^d}(xy)^lD_i^kK_n(x,y)dx\Big \lesssim 2^{\beta n} \end{aligned}$$

For a given \(r>0\), \( \int _{{\mathbb {R}}^d}K_n(x,y)P(y)dy=0 \), for all \(n\ge 0\), \(x\in {\mathbb {R}}^d\), and every polynomial P of (scaled) degree at most r.
Definition 2.12
Theorem 2.13
2.2 Preliminaries

For each m, the set \( \{\varphi _x^{m,\mathfrak {s}}:x\in \varLambda ^m_\mathfrak {s}\}\cup \{\psi _x^{n,\mathfrak {s}}:n\ge m,x\in \varLambda _\mathfrak {s}^n,\psi \in \varPsi \} \) forms an orthonormal basis of \(L^2({\mathbb {R}}^d)\).

For every \(\psi \in \varPsi \) and polynomial P of degree at most r, one has \( \int \psi (x)P(x)dx=0 \).
Lemma 2.14
Proof
Next we recall some results on extending dual elements of a space of smooth functions that are supported away from a submanifold, to distributions, at least locally. This is essentially the content of [10, Prop. 6.9], but we slightly reformulate the statements in order to fit the needs of Sect. 4.3 below better.
Proposition 2.15
Proof
3 Definition of \(\mathcal {D}_P^{\gamma ,w}\) and basic properties
Remark 3.1
It might be at first sight surprising to have not two, but three different orders of singularity. While in the subsequent calculus the use of exponent \(\mu \) will become clear, it is worth mentioning a simple example when the singularities at the different boundaries do not in any way determine the one at the intersection: Consider the solution of \(\partial _t u=\varDelta u\), \(u_0\equiv 1\), with 0 Dirichlet boundary conditions on some domain D. Then, while away from the “corner” \(\{(0,x):x\in \partial D\}\), all derivatives of u are continuous up to both the temporal and the spatial boundaries, the kth derivative exhibits a blowup of order \(k_\mathfrak {s}\) at the corner.
Definition 3.2
This notation is slightly ambiguous since the knowledge of P does of course not imply the knowledge of \(P_0\) and \(P_1\). One should therefore really interpret the instance of P appearing in \(\mathcal {D}_{P}^{\gamma ,w}\) as meaning \(P = \{P_0,P_1\}\) rather than \(P = P_0 \cup P_1\), which is used whenever we view P as a subset of \({\mathbb {R}}^d\). It will also sometimes be useful to consider functions in \(\mathcal {D}_{P}^{\gamma ,w}\) that are slightly better behaved when approaching one of the two boundaries. This is the purpose of the following definition.
Definition 3.3
We shall assume throughout the article that these exponents satisfy \(\eta \vee \sigma \vee \mu \le \gamma \).
Remark 3.4
Denoting the regularity of the sector V by \(\alpha \), the definition is set up so that, when \(\mu \le \alpha \) and there exists an x with \(x_{P_0}\sim x_{P_1}\sim 1\) and \( \sup _{l<\gamma }\Vert f(x)\Vert _{l}\sim 1 \), then the first term in (3.1) bounds the second and third. For \(\mu >\alpha \), one would actually need to add \(x_{P_1}^{(\mu l)\wedge 0}\) to the denominator in the second term and \(x_{P_0}^{(\mu l)\wedge 0}\) in the third. As this would make the calculations significantly longer, we omit this modification and deal with the slight difficulties arising from this restriction later.
Proposition 3.5
Proof
We prove separately for \(\mathfrak {K}_i=\mathfrak {K}\cap \{x_{P_i}\le x_{P_{1i}}\}\).
For \(\mathfrak {K}_1\), further introducing \(\mathfrak {K}_1^n=\mathfrak {K}_1\cap \{2^{n}\le x_{P_0}\le 2^{n+1}\}\), the bounds for \(\mathfrak {K}_1^n\) in place of \(\mathfrak {K}\) follow immediately from Lemmas 6.5 and 6.6, [10], uniformly in n. Since there is no dependence on n in the bounds, and for any pair \((x,y)\in (\mathfrak {K}_1)_P\), the indices \(n_x\) and \(n_y\) for which \(x\in \mathfrak {K}_1^{n_x}\), \(y\in \mathfrak {K}_1^{n_y}\), differ by at most 1, the estimates carry through for \(\mathfrak {K}_1\).
Proposition 3.6
Proof
The fact that \({\hat{f}}\) is locally in \(\mathcal {D}^\sigma \) then follows, since on \(\{\delta \le x_{P_0}\le x_{P_1}\}\), f actually belongs to \(\mathcal {D}^\gamma \), so its projection \({\hat{f}}\) belongs to \(\mathcal {D}^\sigma \), and \(\delta >0\) was arbitrary. \(\square \)
Remark 3.7
4 Calculus of the spaces \(\mathcal {D}_P^{\gamma ,w}\)
In order to reformulate our stochastic PDEs as fixed point problems in \(\mathcal {D}_P^{\gamma ,w}\), one first needs to know how the standard operations like multiplication, differentiation, or convolution with singular kernels, act on these spaces. The aim of this section is to recover the calculus of [10] in the present context.
Remark 4.1
This of course means that repetition of arguments to a certain degree is inevitable. We shall try to minimise the overlap and concentrate on the aspects that are different due to the additional weights and don’t just follow trivially from [10]. This in particular applies to the continuity statements: since the space of models is not linear, boundedness of the operations do not imply their continuity. However, in practice they usually follow from the same principles, with an added level of notational inconvenience. We therefore only give the complete proof of continuity for the multiplication, after which the reader is hopefully convinced that obtaining the other similar continuity results is a lengthy but straightforward combination of the corresponding arguments in [10] and the treatment of the additional weights as described in the ‘boundedness’ part of the corresponding statements. Alternatively, the continuity statements can also be obtained by using the trick introduced in the proof of [12, Prop. 3.11], which allows to some extent to “linearise” the space of models.
Remark 4.2
4.1 Multiplication
Lemma 4.3
Proof
For \(T_2\), we use (4.7) with \(h=f_1\), and then proceed just like for \(T_1\), with the role of the indices reversed.
The bound for the term \(T_5\) goes similarly to \(T_3\), with the indices reversed, and so does \(T_4\), with the only difference that the prefactor of the sum is \(\Vert \varGamma \bar{\varGamma }\Vert _{\gamma _1+\gamma _2}{\vert \vert \vert f_1 \vert \vert \vert }_{\gamma _1,w_1}\). \(\square \)
4.2 Composition with smooth functions
Lemma 4.4
Let V be a sector of regularity 0 with \(V_0 = \langle \mathbf {1} \rangle \) that is \(\gamma \)regular with respect to the product \(\star \) and furthermore \(V\star V\subset V\).
Let \(f_1,\ldots ,f_n\in \mathcal {D}_P^{\gamma ,w}(V)\) with \(w=(\eta ,\sigma ,\mu )\) such that \(\eta ,\sigma ,\mu \ge 0.\) Let furthermore \(F:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a smooth function. Then \(\hat{F}_\gamma (f)\) belongs to \(\mathcal {D}_P^{\gamma ,w}(V)\). Furthermore, \(\hat{F}_\gamma :\mathcal {D}_P^{\gamma ,w}\rightarrow \mathcal {D}_P^{\gamma ,w}\) is locally Lipschitz continuous in any of the seminorms \(\Vert \cdot \Vert _{\gamma ,w;\mathfrak {K}}\) and \({\vert \vert \vert \cdot \vert \vert \vert }_{\gamma ,w;\mathfrak {K}}\).
Remark 4.5
If two modelled distributions f, \({\bar{f}}\) are such that \(f\bar{f}\in \mathcal {D}_{P,\{1\}}^{\gamma ,w}\), then clearly \({\hat{F}}_\gamma (f)\hat{F}_\gamma ({\bar{f}})\) also has 0 limit at \(P_1{\setminus } P_0\). In this case the analogous Lipschitz bound for \({\hat{F}}\) in the seminorms Open image in new window also holds.
Remark 4.6
One can use the same construction as in [12, Prop. 3.11] to obtain local Lipschitz continuity when comparing two modelled distributions modelled on two different models.
Proof
We only give a sketch of the proof, as the majority of the argument is exactly the same as that of the proof of Theorem 4.16 and Proposition 6.12 in [10]. We prove the main estimates which are somewhat different due to the additional weights and refer the reader to [10] to confirm that these indeed imply the theorem.
4.3 Reconstruction
Recall that, since reconstruction is a local operation, there exists an element \(\tilde{\mathcal {R}}f\) in the dual of smooth functions supported away from P such that the bound (2.4) is satisfied if \(\lambda \ll x_{P_0}\wedge x_{P_1}\). A natural guess for the target space of the extension of the reconstruction operator acting on \(\mathcal {D}^{\gamma ,w}_P(V)\) would be \(\mathcal {C}^{\eta \wedge \sigma \wedge \mu \wedge \alpha }\). While this certainly does hold, we need some finer control over the behaviour at the different boundaries. To this end, we introduce weighted versions of Hölder spaces as follows.
Definition 4.7
 (a)For any \(x\in \{x_{P_0}\le 2x_{P_1}\}\), \(\lambda \in (0,1]\) satisfying \(2\lambda \le x_{P_1}\), and every \(\psi \in \mathcal {B}^r\), where \(r=\lceil a_0+1\rceil \),$$\begin{aligned} u(\psi _x^{\lambda })\lesssim x_{P_1}^{a_{\cap }a_0}\lambda ^{a_0}. \end{aligned}$$(4.14)
 (b)For any \(x\in \{x_{P_1}\le 2x_{P_0}\}\), \(\lambda \in (0,1]\) satisfying \(2\lambda \le x_{P_0}\), and every \(\psi \in \mathcal {B}^r\), where \(r=\lceil a_1+1\rceil \),$$\begin{aligned} u(\psi _x^{\lambda })\lesssim x_{P_0}^{a_{\cap }a_1}\lambda ^{a_1}. \end{aligned}$$(4.15)
Proposition 4.8
Let \(u \in \mathcal {D}'({\mathbb {R}}^d{\setminus } (P_0\cap P_1))\) be such that the bounds (4.14) and (4.15) are satisfied. Then, provided \(a_\wedge >\mathfrak {m}\), there exists a unique distribution \(u'\in \mathcal {C}^{a}_P\) that agrees with u on test functions supported away from \(P_0\cap P_1\).
Proof
 (i)
The \(\phi _i^{(\lambda )}\) are supported on \(\{x: x_{P_i}\ge 4\lambda ,2 x_{P_i}\ge x_{P_{1i}}\}\).
 (ii)
If \(x\in {\mathbb {R}}^d\) is such that \(d_\mathfrak {s}(x,P_0\cap P_1)\ge (c1)\lambda ,\) then \( \phi _0^{(\lambda )}(x)+\phi _1^{(\lambda )}(x)=1\).
 (iii)
For any multiindex k, the bound \(D^k\phi _i^{(\lambda )}(x)\lesssim \lambda ^{k_\mathfrak {s}}\) is satisfied for all \(x\in {\mathbb {R}}^d\).
Theorem 4.9
Proof
By virtue of Proposition 4.8, it suffices to extend \(\tilde{\mathcal {R}} f\) to an element of \(\mathcal {D}'({\mathbb {R}}^d {\setminus } (P_0\cap P_1))\) in such a way that (4.14) and (4.15) hold with the desired exponents.
One can similarly construct \(\tilde{\mathcal {R}}_1 f \in \mathcal {D}'({\mathbb {R}}^d {\setminus } P_0)\) such that \( (\tilde{\mathcal {R}}_1f)(\psi _x^\lambda )\lesssim \lambda ^{\sigma \wedge \alpha }x_{P_1}^{\mu (\sigma \wedge \alpha )} \) holds in the symmetric situation. Since \(\tilde{\mathcal {R}}_0 f\) and \(\tilde{\mathcal {R}}_1 f\) agree on the intersection of their domains, they can be pieced together to get the claimed extension of \(\tilde{\mathcal {R}} f\). The proof of continuity is again analogous and is omitted here. \(\square \)
Keeping in mind that our goal will be to apply this calculus for singular SPDEs with boundary conditions on some domain D, \(P_1\) will typically stand for \({\mathbb {R}}\times \partial D\). With a parabolic scaling we have \(\mathfrak {m}_1=1\) and so condition (4.17), in particular requiring \(\sigma \wedge \alpha >1\) is rather strict and will often be violated. In these situations, a \(\mathcal {C}^{(\eta \wedge \alpha ,\sigma \wedge \alpha ,\mu )}_P\) extension \(\tilde{\mathcal {R}} f\) is not unique and hence sometimes it will be more suggestive to write \(\hat{\mathcal {R}} f\) for particular choices of such extensions. On some occasions this choice will be made ‘by hand’, but there is also another generic situation when a canonical choice can be made, as follows.
Theorem 4.10
Proof
First notice that such a \(\hat{\mathcal {R}} f\) has to be unique: any two extensions of \(\tilde{\mathcal {R}} f\) differ by a distribution concentrated on P, which, due to the conditions on the exponents and the constraint (4.24), has to vanish.
4.4 Differentiation
Lemma 4.11
Let \(\mathscr {D}\) be an abstract gradient and let \(f\in \mathcal {D}_P^{\gamma ,w}(V)\), where \(\gamma >\mathfrak {s}_i\) and \(w=(\eta ,\sigma ,\mu )\in {\mathbb {R}}^3\). Then \(\mathscr {D}_{i} f\in \mathcal {D}_P^{\gamma \mathfrak {s}_i,(\eta \mathfrak {s}_i,\sigma \mathfrak {s}_i,\mu \mathfrak {s}_i)}\).
This lemma is a direct consequence of the definition of abstract gradients, and since the proof is a trivial modification of that of [10, Prop 5.28], it is omitted here.
4.5 Integration against singular kernels
As seen above, in certain situations the distribution \(\mathcal {R}f\) is not uniquely defined as there might be many distributions \(\zeta \) with the appropriate regularity that extend \(\tilde{\mathcal {R}} f\). For any such \(\zeta \), let us denote by \(\mathcal {N}_\gamma ^\zeta f\) and \(\mathcal {K}^\zeta _\gamma f\) the modelled distributions defined analogously to \(\mathcal {N}_\gamma f\) and \(\mathcal {K}_\gamma f\), but with \(\mathcal {R}f\) replaced by \(\zeta \).
Lemma 4.12
Fix \(\gamma > 0\), \(w = (\eta ,\sigma ,\mu )\), let V be a sector of regularity \(\alpha \), and set \(a = (\eta \wedge \alpha ,\sigma \wedge \alpha ,\mu )\).
(ii) If \(f\in \mathcal {D}_{P,\{1\}}^{\gamma ,w}\) and the coordinates of w satisfy (4.23), then choosing \(\hat{\mathcal {R}} f\) in the above in place of \(\zeta \), the same conclusions hold, but with the definition of \(\bar{\sigma }\) in (4.33) replaced by \(\bar{\sigma }=\sigma +\beta \).
Proof
The argument showing that \(\mathcal {N}^\zeta _\gamma f\) (and therefore \(\mathcal {K}^\zeta _\gamma f\)) is actually welldefined is exactly the same as in [10]. Also, the fact that the required bounds trivially hold for components of \((\mathcal {K}^\zeta _\gamma f)(x)\) and \((\mathcal {K}^\zeta _\gamma f)(x)\varGamma _{yx}(\mathcal {K}^\zeta _{\gamma }f)(y)\), whose homogeneity is noninteger, does not change in our setting.
Turning to bounding \(\Vert \mathcal {K}^\zeta _\gamma f(x)\varGamma _{xy}\mathcal {K}_\gamma ^\zeta f(y)\Vert \), recall that we need only consider pairs (x, y) where \(2\Vert xy\Vert _\mathfrak {s}\le x,y_{P_0}\le x,y_{P_1}\). As before, this implies \(x_{P_i}\sim y_{P_i}\sim x,y_{P_i}\).
The proof of continuity again goes in an analogous way and is omitted here.
(ii) In the \(f\in \mathcal {D}_{P,\{1\}}^{\gamma ,w}\) case, when repeating the above arguments, one should only pay attention in order to get the improved exponent \(\bar{\sigma }=\sigma +\beta \) in place of \((\sigma \wedge \alpha )+\beta =\alpha +\beta \). This improvement is the consequence of the improved bound on \(\Vert f(x)\Vert _l\) near \(P_1\), thanks to Proposition 3.5, and of the improved regularity of \(\hat{\mathcal {R}} f\) when tested against functions centred on \(P_1\), thanks to (4.24). \(\square \)
Remark 4.13
4.6 Integration against smooth remainders with singularities at the boundary
From this point on we move to a more concrete setting, and in particular \(P_0\) and \(P_1\) will play different roles. We shall view \({\mathbb {R}}^d\) as \({\mathbb {R}}\times {\mathbb {R}}^{d1}\), denoting its points by either z or by (t, x), where \(t\in {\mathbb {R}}\), \(x\in {\mathbb {R}}^{d1}\). Furthermore we assume that \(P_0\) is given by \(\{(0,x):x\in {\mathbb {R}}^{d1}\}\)
Definition 4.14

\(Z_n\) is supported on \(\{(z,z')=((t,x),(t',x')):\,z_{P_1}+z'_{P_1}+tt'^{1/\mathfrak {s}_0}\le C 2^{n}\}\), where C is a fixed constant depending only on the domain D.
 For any (ddimensional) multiindices k and l,where the proportionality constant may depend on k and l, but not on n, z, \(z'\).$$\begin{aligned} D_1^kD_2^lZ_n(z,z')\lesssim 2^{n(\mathfrak {s}+k+l_\mathfrak {s}\beta )}, \end{aligned}$$
The relevance of this definition is illustrated by the following example, which shows that if we consider a heat kernel on a domain obtained by the reflection principle, then it can always be decomposed into an element of \(\mathscr {K}_\beta \) and an element of \(\mathscr {Z}_{\beta ,P}\).
Example 4.15

We have a decomposition \(G^0=K^0+R^0\), where \(K^0\in \mathscr {K}_\beta \), while \(R^0\) is a globally smooth function.

For any two multiindices k and l and any number a, there exists a constant \(C_{k,l,a}\) such that it holds that \( D_1^kD_2^lR^0(z,z')\le C_{k,l,a}(xx'\vee 1)^a \).
Lemma 4.16
If u furthermore satisfies \(\langle u,\psi _z^\lambda \rangle \lesssim \lambda ^{\bar{a}_1}z_{P_0}^{a_\cap {\bar{a}}_1}\) for \(z\in P_1{\setminus } P_0\) and \(2\lambda \le z_{P_0}\) with some \({\bar{a}}_1\ge a_1\), then the conclusions hold with the definition of \(\sigma \) replaced by \(\sigma ={\bar{a}}_1+\beta \).
Proof
Remark 4.17
The mapping \(u\rightarrow \mathcal {Q}_{\gamma +\beta }^v\), where v is as in (4.42), will also be denoted by \(Z_\gamma \). As all models that we consider act the same on polynomials, the usual continuity estimates are in this case direct consequences of the above result.
Remark 4.18
One can easily verify that the action of \(\mathcal {K}_\gamma \) and \(Z_\gamma \) are compatible in the following sense: take \(f\in \mathcal {D}_P^{\gamma ,w}\) and an extension \(\zeta \) of \(\tilde{\mathcal {R}} f\) as in Lemma 4.12 (i). Then \(Z_{\gamma +\beta }\zeta \in \mathcal {D}_{P}^{\gamma +\beta ,{\bar{w}}}\), where \({\bar{w}}\) is as in Lemma 4.12 (i).
5 Solving the abstract equation
5.1 Nonanticipative kernels
First of all, this allows us to improve our conditions on \(\mu \).
Proposition 5.1
 (i)In the setting of Lemma 4.12 (i), suppose that K is nonanticipative, that f is of the form \(\mathbf {R}^{D}_+g\) for some \(g\in \mathcal {D}_P^{\gamma ,w}\), and that \(\zeta \) annihilates test functions supported on negative times. Let furthermore \(\varepsilon >0\) such that \(\mathfrak {m}_0\beta +\varepsilon >0\) and assume \(a_\wedge +\mathfrak {m}_0\ge 0\). Then, modifying the condition on \(\bar{\mu }\) from (4.33) tothe conclusions of Lemma 4.12 (i) still hold.$$\begin{aligned} \bar{\mu }\le a_\wedge +\beta \varepsilon , \end{aligned}$$
 (ii)The analogous statement holds for Lemma 4.12 (ii), where the modified condition on \(\bar{\mu }\) reads as$$\begin{aligned} \bar{\mu }\le \eta \wedge \mu \wedge \alpha +\beta \varepsilon . \end{aligned}$$
 (iii)In the setting of Lemma 4.16, suppose that Z is nonanticipative and that u annihilates test functions supported on negative times and let \(\varepsilon >0\) be as above. Then, modifying the condition on \(\mu \) from (4.43) tothe conclusion of Lemma 4.16 still hold.$$\begin{aligned} \mu \le a_\wedge +\beta \varepsilon , \end{aligned}$$
Proof
The proof of (ii) goes in the same way, and, in light of Remark 4.18, so does that of (iii). \(\square \)
The other important consequence of the nonanticipativity of our kernel is the following shorttime control.
Lemma 5.2
If we are instead in the situation of Proposition 5.1 (ii), then the analogous statement holds, with \(\zeta \) replaced by \(\hat{\mathcal {R}} f\), and hence the last term on the righthand side of (5.3) can be omitted.
Proof
The corresponding results hold for the singular remainder as well.
Lemma 5.3
Proof
5.2 On initial conditions
The class of admissible initial conditions depends on the particular choice of the kernel in that in addition to the regularity, some boundary behaviour may be required. In the setting of Example 4.15, which is general enough to cover all of our examples, this can be formalised as follows.
Lemma 5.4
Proof
5.3 The fixed point problem
At this point everything is in place to solve the abstract equations that will arise as ‘lifts’ of equations similar to the ones in Sect. 1.1. As the notation is already quite involved, we refrain from the full generality concerning the kernel \(K+Z\) and the scaling \(\mathfrak {s}\) and only state the result in a form that is sufficient to treat nonlinear perturbations of the stochastic heat equation with some boundary conditions. Our main goal is to formulate a fixed point argument that is just general enough to cover the examples mentioned in the introduction, as well as some related problems.
Our setup will involve families of Banach spaces depending on some parameter \(\tau >0\) (which will represent the time over which we solve our equation). We will henceforth talk of a “timeindexed space \(\mathcal {V}\)” for a family \(\mathcal {V}= \{\mathcal {V}_\tau \}_{\tau > 0}\) of Banach spaces as well as contractions \(\pi _{\tau '\leftarrow \tau }:\mathcal {V}_\tau \rightarrow \mathcal {V}_{\tau '}\) for all \(\tau ' < \tau \) with the property that \(\pi _{\tau ''\leftarrow \tau '} \circ \pi _{\tau '\leftarrow \tau } = \pi _{\tau ''\leftarrow \tau }\). We consider \(\mathcal {V}\) itself as a Fréchet space whose elements are collections \(\{v_\tau \}_{\tau > 0}\) satisfying the consistency condition \(v_{\tau '} = \pi _{\tau '\leftarrow \tau } v_\tau \) and with the topology given by the collections of seminorms \(\Vert \cdot \Vert _\tau \) inherited by the spaces \(\mathcal {V}_\tau \). We will write \(\pi _\tau :\mathcal {V}\rightarrow \mathcal {V}_\tau \) for the natural projection.
Given a bounded and piecewise \(\mathcal {C}^1\) domain \(D \subset {\mathbb {R}}^{d1}\), a typical example of a timeindexed space is given by the space \(\mathcal {V}= \mathcal {D}^{\gamma ,w}_P\) with \(\pi _\tau \) given by the restriction to \([0,\tau ] \times D\) and norms \(\Vert \cdot \Vert _\tau \) given by \({\vert \vert \vert \cdot \vert \vert \vert }_{\gamma ,w;D_\tau }\), where \(D_\tau = [0,\tau ]\times D\). Similarly, we write again \(\mathcal {C}^w_P\) for the timeindexed space consisting of distributions on \({\mathbb {R}}^d\) which vanish outside of \({\mathbb {R}}_+ \times D\), endowed with the norms of Definition 4.7, but restricted to test functions \(\psi \), points x and constants \(\lambda \) such that the support of \(\psi _x^\lambda \) lies in \((\infty ,\tau ]\times {\mathbb {R}}^{d1}\).
Given two timeindexed spaces \(\mathcal {V}\) and \(\bar{\mathcal {V}}\), we call a map \(A :\mathcal {V}\rightarrow \bar{\mathcal {V}}\) ‘adapted’ if there are maps \(A_\tau :\mathcal {V}_\tau \rightarrow \bar{\mathcal {V}}_\tau \) such that \(\pi _\tau A = A_\tau \pi _\tau \). If A is linear, we will furthermore assume that the norms of \(A_\tau \) are uniformly bounded over bounded subsets of \({\mathbb {R}}_+\). Similarly, we call A “locally Lipschitz” if each of the \(A_\tau \) is locally Lipschitz continuous and, for every \(K>0\) and \(\tau >0\), the Lipschitz constant of \(A_{\tau '}\) over the centred ball of radius K in \(A_{\tau '}\) is bounded, uniformly over \(\tau ' \in (0,\tau ]\).

Fix \(d\ge 2\), \(\beta =2\), the scaling \(\mathfrak {s}=(2,1,\ldots ,1)\) on \({\mathbb {R}}^d=\{(t,x):t\in {\mathbb {R}},x\in {\mathbb {R}}^{d1}\}\), and a regularity structure \(\mathscr {T}\).

Let \(\gamma \), \(\gamma _0\) be two positive numbers satisfying \(\gamma <\gamma _0 + 2\) and let V be a sector of regularity \(\alpha \le 0\) and such that \({\bar{T}} \subset V\).

Set \(P_0=\{(0,x):x\in {\mathbb {R}}^{d1}\}\) and \(P_1=\{(t,x):t\in {\mathbb {R}},x\in \partial D\}\), where D is a domain in \({\mathbb {R}}^{d1}\) with a piecewise \(\mathcal {C}^1\) boundary, satisfying the cone condition.

We assume that we have an abstract integration map \(\mathcal {I}\) of order 2 as well as nonanticipative kernels \(K\in \mathscr {K}_2\) and \(Z\in \mathscr {Z}_{2,P}\). We then construct the operator \(Z_\gamma \) and, for every admissible model \((\varPi ,\varGamma )\), the operator \(\mathcal {K}_\gamma \) as in Sects. 4.5 and 4.6.

We fix a family \(((\varPi ^\varepsilon ,\varGamma ^\varepsilon ))_{\varepsilon \in (0,1]}\) of admissible models converging to \((\varPi ^0,\varGamma ^0)\) as \(\varepsilon \rightarrow 0\).
 We fix a collection of timeindexed spaces \(\mathcal {V}_\varepsilon \) with \(\varepsilon \in [0,1]\) endowed with adapted linear maps \(\hat{{\mathcal {R}}}^\varepsilon :\mathcal {V}_\varepsilon \rightarrow \bigoplus _{i=0}^n\mathcal {C}^{w_i}_P\) and \(\iota _\varepsilon :\mathcal {V}_\varepsilon \rightarrow \bigoplus _{i=0}^n\mathcal {D}^{\gamma _0,w_i}_P(V_i,\varGamma ^\varepsilon )\), where \(V_i\) are sectors of regularity \(\alpha _i\), satisfying \(\mathcal {I}(V_i) \subset V\) and \(w_i\in {\mathbb {R}}^3\). Finally, we assume that for every \(\varepsilon \in [0,1]\) and every \(v \in \mathcal {V}_\varepsilon \), one hasfor any \(\psi \in \mathcal {C}_0^\infty ({\mathbb {R}}^d {\setminus } P)\). Denote \(\tilde{\mathcal {C}}=\bigoplus _{i=0}^n\mathcal {C}^{w_i}_P\) and \(\tilde{\mathcal {D}}=\bigoplus _{i=0}^n\mathcal {D}^{\gamma _0,w_i}_P(V_i,\varGamma ^\varepsilon )\), which are themselves timeindexed spaces equipped with the natural norms.$$\begin{aligned} \bigl (\tilde{\mathcal {R}}\mathbf {R}^D_+\iota _\varepsilon v\bigr )(\psi ) = \bigl (\hat{{\mathcal {R}}}^\varepsilon v\bigr )(\psi ) \end{aligned}$$(5.4)
 We fix a collection of timeindexed spaces \(\mathcal {W}_\varepsilon \) of modelled distributions such that the linear mapsare bounded from \(\mathcal {V}_\varepsilon \) into \(\mathcal {W}_\varepsilon \) with a bound of order \(\tau ^\theta \) for some \(\theta >0\) for its restriction to time \(\tau \in (0,1]\), uniformly over \(\varepsilon \in [0,1]\).$$\begin{aligned} \mathcal {P}_\gamma ^{(\varepsilon )} v = \sum _{i=0}^n \bigl (\mathcal {K}_\gamma ^{(\hat{\mathcal {R}}^\varepsilon v)_i} (\mathbf {R}_+^D\iota _\varepsilon v)_i + Z_\gamma (\hat{\mathcal {R}}^\varepsilon v)_i \bigr ), \end{aligned}$$

For \(\varepsilon \in [0,1]\), we fix a collection of adapted locally Lipschitz continuous maps \(F_\varepsilon :\mathcal {D}^{\gamma ,w}_P(V,\varGamma ^\varepsilon ) \rightarrow \mathcal {V}_\varepsilon \).
 There are ‘distances’ \({\vert \vert \vert \cdot ;\cdot \vert \vert \vert }_{\mathcal {W};\tau }\) (possibly also depending on \(\varepsilon \)) defined on \(\mathcal {W}_\varepsilon \times \mathcal {W}_0\) that are compatible with the maps \(F_\varepsilon \) and \(\mathcal {P}_\gamma \) in the sense that, for \(u\in \mathcal {V}_\varepsilon \), \(v\in \mathcal {V}_0\), and \(\tau \in (0,1]\), one hasas \(\varepsilon \rightarrow 0\). Similarly, uniformly over modelled distributions \(f\in \mathcal {W}_\varepsilon \), \(g\in \mathcal {W}_0\) bounded by an arbitrary constant C and uniformly over \(\tau \in (0,1]\), one has$$\begin{aligned} \tau ^{\theta }{\vert \vert \vert \mathcal {P}^{(\varepsilon )}_\gamma u;\mathcal {P}^{(0)}_\gamma v \vert \vert \vert }_{\mathcal {W};\tau }\lesssim {\vert \vert \vert \iota _\varepsilon u; \iota _0v \vert \vert \vert }_{\tilde{\mathcal {D}}; D_\tau } + \Vert \hat{{\mathcal {R}}}^\varepsilon u \hat{{\mathcal {R}}}^0v\Vert _{\tilde{\mathcal {C}};D_\tau } +o(1), \end{aligned}$$as \(\varepsilon \rightarrow 0\).$$\begin{aligned} {\vert \vert \vert \iota _\varepsilon F_\varepsilon (f); \iota _0 F_0(g) \vert \vert \vert }_{\tilde{\mathcal {D}}; D_\tau } + \Vert \hat{{\mathcal {R}}}^\varepsilon F_\varepsilon (f) \hat{{\mathcal {R}}}^0 F_0(g)\Vert _{\tilde{\mathcal {C}};D_\tau } \lesssim {\vert \vert \vert f;g \vert \vert \vert }_{\mathcal {W};\tau } + o(1) , \end{aligned}$$(5.5)
Remark 5.5
The reader may wonder what the point of this rather complicated setup is. By choosing for \(\mathcal {V}_\varepsilon \) a direct sum of spaces of the type defined in Sect. 3, it allows us to decompose the right hand side of our equation into a sum of terms with wellcontrolled behaviour at the boundary. This gives us the flexibility to exploit different features of each term to control the corresponding “reconstruction operator” \(\hat{{\mathcal {R}}}^\varepsilon _i\). For example, in the case of 2D gPAM, the term \({\hat{f}}_{ij}(u) \star \mathcal {D}_i(u)\star \mathcal {D}_j(u)\) can be reconstructed because the corresponding weight exponents are sufficiently large, the term \(({\hat{g}}(u)  g(0)\mathbf {1}) \star \varXi \) can be reconstructed because it vanishes on the boundary, and the term \(g(0)\varXi \) can be reconstructed because it corresponds to (a constant times) white noise, multiplied by an indicator function.
We then have the following result.
Theorem 5.6
Proof
By assumption \(\mathcal {P}^{(\varepsilon )}_{\gamma _0}\) is an adapted linear map from \(\mathcal {V}_\varepsilon \) to \(\mathcal {W}_\varepsilon \) with control on its norm that is uniform over \(\varepsilon \in [0,1]\). It has the additional property that, when restricted to time \(\tau \), its operator norm is bounded by \(\mathcal {O}(\tau ^\theta )\) for some exponent \(\theta >0\), uniformly in \(\varepsilon \). Combining this with the uniform local Lipschitz continuity of the maps \(F_\varepsilon \), it is immediate that, for every \(C> 2\Vert v\Vert _{\mathcal {W};1}\), there exists \(\tau \in (0,1]\) such that the right hand side of (5.6) is a contraction and therefore admits a unique fixed point in the centred ball of radius C in \(\mathcal {W}_\varepsilon \).
To show that this is the unique fixed point in all of \(\mathcal {W}_\varepsilon \) is also standard: assume by contradiction that there exists a second fixed point \({\bar{u}}\) (which necessarily has norm strictly greater than C). Then, for every \(\tau ' < \tau \), the restrictions of both u and \({\bar{u}}\) are fixed points in \(\mathcal {W}_\varepsilon \). However, since the norm of \(A_\varepsilon \) is bounded by \(\mathcal {O}(\bar{\tau }^\theta )\), one has uniqueness of the fixed point in a ball of radius \({\bar{C}}(\tau ')\) of \(\mathcal {W}_\varepsilon \) with \(\lim _{\tau ' \rightarrow 0} {\bar{C}}(\tau ') = \infty \), so that one reaches a contradiction by choosing \(\tau '\) small enough. The continuity of the solution map at (v, 0) then follows immediately from (5.5). \(\square \)
6 Singular SPDEs with boundary conditions
The next three subsections are devoted to the proofs of Theorems 1.1, 1.5, and 1.7, respectively. We do rely on the results of the corresponding statements without boundary conditions from [9, 10], in particular the specific regularity structures, models, and their convergence do not change in our setting. Therefore we only specify details about these objects to the extent that is sufficient to cover the new aspects of our setting.
6.1 2D gPAM with Dirichlet boundary condition
The regularity structure for the equation (1.1) is built as in [10, Sec 8], and the models \((\varPi ^\varepsilon ,\varGamma ^\varepsilon )_{\varepsilon \in [0,1]}\) as in [10, Sec 10], and we will use the notations from there without further ado. We use the periodic model with sufficiently large period: if the truncated heat kernel \(K^0\) is chosen to have support of diameter 1, then the periodic model on \([2,2]^2\) suffices, since convolution with \(K^0\) and with its periodic symmetrisation agrees on \([1,1]^2\). The homogeneity of the symbol \(\varXi \) is denoted by \(1\kappa \), where \(\kappa \in (0,(1/3)\wedge \delta ){\setminus }\mathbb {Q}\), with \(\delta \) being the regularity of the initial condition.
Remark 6.1
6.2 KPZ equation with Dirichlet boundary condition
The construction of the regularity structure and models (as before, with a sufficiently large period) for the KPZ equation can be for example found in [7, Sec. 15]. The homogeneity of the symbol \(\varXi \) is now denoted by \(3/2\kappa \), where \(\kappa \in (0,(1/8)\wedge \delta ){\setminus }\mathbb {Q}\), with \(\delta \) being the regularity of the initial condition.
It remains to define \(\hat{\mathcal {R}}^\varepsilon \mathbf {R}^D_+(\mathscr {D}\varPsi )^{\star 2}.\) Recall that \(\tilde{\mathcal {R}}\) stands for the local reconstruction operator and that the issue with the singularity of low order is that \(\tilde{\mathcal {R}} \mathbf {R}^{D}_+(\mathscr {D}(G_\gamma \varXi ))^{\star 2}\) does not have a canonical extension as a distribution in \(\mathcal {C}^{12\kappa }\). Of course, for the approximating models this is just a bounded function, so it could even be extended as an element of \(\mathcal {C}^0\), but these extensions may not converge in the \(\varepsilon \rightarrow 0\) limit. Therefore some modification of these natural extensions are required at the boundary.
Remark 6.2
This process is very similar to the situation when one takes the sequence of distributions \(1/(x+\varepsilon )\). This sequence of course does not converge to any distribution as \(\varepsilon \rightarrow 0\), but \(1/(x+\varepsilon )+2\log (\varepsilon )\delta _0\) does, in \(\mathcal {C}^{1\rho }\) for any \(\rho >0\). Moreover, the limiting distribution agrees with 1 / x on test functions supported away from 0.
6.3 KPZ equation with Neumann boundary condition
Proposition 6.3
With the above notations, one has \(C_0(x) = 0\) for every \(x \ne 0\). Furthermore, for every \(\kappa \in (0,1)\), there exists a constant C such that, for \(x \ge C\varepsilon \), one has the bound \(C_0^\varepsilon (x) \le C\varepsilon ^{1\kappa } x^{\kappa 2}\).
Proof
Finally, regarding \(J_3\), the product is supported on \({\mathbb {R}}\times [\varepsilon ,\varepsilon ]\) and each factor is bounded by \((s+x^2)^{1}\) there, so that the corresponding integral is again bounded as in (6.19), thus concluding the proof. \(\square \)
Applying again Theorem 5.6, combined with the results of [13] regarding the convergence of the corresponding admissible models, we conclude that, for any choice of \(b_\pm \), the solution to (6.25) (which is precisely the same as (1.7) provided that the constant \(C_\varepsilon \) is adjusted in the appropriate way) converges locally as \(\varepsilon \rightarrow 0\) to a limit which depends on the choice of \(b_\pm \) but is independent of the choice of mollifier \(\rho \). It remains to show that this limit coincides with the Hopf–Cole solution to the KPZ equation with Neumann boundary data given by \(b_\pm \). This follows by considering the special case \(\rho (t,x) = \delta (t)\hat{\rho }(x)\), which is covered by the above proof, the only minor modification being the proof of convergence of the corresponding admissible model to the same limit, which can be obtained in a way very similar to [9, 10]. As already mentioned at the end of Sect. 1.1, one has \(a = c = 0\) in this case, so that in particular \({\hat{b}}_\pm = b_\pm \). In this case, we can apply Itô’s formula to perform the Hopf–Cole transform and obtain convergence to the corresponding limit by classical means [5], which concludes the proof.
6.3.1 Expression for the drift term
Notes
Acknowledgements
Open access funding provided by Institute of Science and Technology (IST Austria).
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