Abstract
We establish the large scale convergence of a class of stochastic weakly nonlinear reaction–diffusion models on a three dimensional periodic domain to the dynamic \(\Phi ^4_3\) model within the framework of paracontrolled distributions. Our work extends previous results of Hairer and Xu to nonlinearities with a finite amount of smoothness (in particular \(C^9\) is enough). We use the Malliavin calculus to perform a partial chaos expansion of the stochastic terms and control their \(L^p\) norms in terms of the graphs of the standard \(\Phi ^4_3\) stochastic terms.
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References
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Acknowledgements
The authors would like to thank the anonymous referee for the detailed and constructive critique which contributed to improve the overall exposition of the results. Support via SFB CRC 1060 is also gratefully acknowledged.
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Appendices
Appendix A: Paracontrolled analysis and kernel estimations
In this section we first recall the the basic results of paracontrolled calculus first introduced in [6], without proofs. For more details on Besov spaces, Littlewood–Paley theory, and Bony’s paraproduct the reader can refer to the monograph [2]. We then proceed to give some results on the convolution of functions with known singularity and the estimation of finite-chaos Gaussian trees. We refer to Section 10 of [8] and to the nice pedagogic exposition [18] for further details.
1.1 A.1 Notation
Throughout the paper, we use the notation \(a \lesssim b\) if there exists a constant \(c > 0\), independent of the variables under consideration, such that \(a \leqslant c \cdot b\). If we want to emphasize the dependence of c on the variable x, then we write \(a (x) \lesssim _x b (x)\). If f is a map from \(A \subset \mathbb {R}\) to the linear space Y, then we write \(f_{s, t} = f (t) - f (s)\). For \(f \in L^p (\mathbb {T}^d)\) we write \(\Vert f (x)\Vert ^p_{L^p_x (\mathbb {T}^3)} :=\int _{\mathbb {T}^3} |f (x) |^p \mathrm {d}x\).
Given a Banach space X with norm \(\Vert \cdot \Vert _X\) and \(T > 0\), we note \(C_T X = C ([0, T], X)\) for the space of continuous maps from [0, T] to X, equipped with the supremum norm \(||\cdot ||_{C_T X}\), and we set \(C X = C (\mathbb {R}_+, X)\). For \(\alpha \in (0, 1)\) we also define \(C^{\alpha }_T X\) as the space of \(\alpha \)-Hölder continuous functions from [0, T] to X, endowed with the seminorm \(\Vert f\Vert _{C^{\alpha }_T X} = \sup _{0 \leqslant s < t \leqslant T} \Vert f (t) - f (s)\Vert _X / |t - s|^{\alpha }\), and we write \(C^{\alpha }_{{\text {loc}}} X\) for the space of locally \(\alpha \)-Hölder continuous functions from \(\mathbb {R}_+\) to X. For \(\gamma > 0\), \(p \in [1, \infty )\), we define
The space of distributions on the torus is denoted by \(\mathscr {D} ' (\mathbb {T}^3)\) or \(\mathscr {D} '\). The Fourier transform is defined with the normalization
so that the inverse Fourier transform is given by \(\mathscr {F} ^{- 1} v (x) = (2 \pi )^{- 1} \sum _k e^{\iota \langle k, x \rangle } v (k)\). Let \((\chi , \rho )\) denote a dyadic partition of unity such that \({\text {supp}} (\rho (2^{- i} \cdot )) \cap {\text {supp}} (\rho (2^{- j} \cdot )) = \emptyset \) for \(|i - j| > 1\). The family of operators \((\Delta _j)_{j \ge - 1}\) will denote the Littlewood–Paley projections associated to this partition of unity, that is \(\Delta _{- 1} u = \mathscr {F} ^{- 1} \left( \chi \mathscr {F} u \right) \) and \(\Delta _j = \mathscr {F} ^{- 1} \left( \rho (2^{- j} \cdot ) \mathscr {F} u \right) \) for \(j \ge 0\). Let \(S_j = \sum _{i < j} \Delta _i\), and \(K_q\) be the kernel of \(\Delta _q\) so that
For the precise definition and properties of the Littlewood–Paley decomposition \(f = \sum _{q \geqslant - 1} \Delta _q f\) in \(\mathscr {D} ' (\mathbb {T}^3)\), see Chapter 2 of [2]. The Hölder-Besov space \(B^{\alpha }_{p, q} (\mathbb {T}^3, \mathbb {R})\) for \(\alpha \in \mathbb {R}\), \(p, q \in [1, \infty ]\) with \(B^{\alpha }_{p, q} (\mathbb {T}^3, \mathbb {R}) = : \mathscr {C} ^a\) is and equipped with the norm
If f is in \(\mathscr {C} ^{\alpha - \varepsilon }\) for all \(\varepsilon > 0\), then we write \(f \in \mathscr {C} ^{\alpha -}\). For \(\alpha \in (0, 2)\), we define the space \(\mathscr {L} _T^{\alpha } = C^{\alpha / 2}_T L^{\infty } \cap C_T \mathscr {C} ^{\alpha }\), equipped with the norm
The notation is chosen to be reminiscent of
by which we will always denote the heat operator with periodic boundary conditions on \(\mathbb {T}^d\). We also write \(\mathscr {L} ^{\alpha } = C^{\alpha / 2}_{{\text {loc}}} L^{\infty } \cap C \mathscr {C} ^{\alpha }\). When working with irregular initial conditions, we will need to consider explosive spaces of parabolic type. For \(\gamma \geqslant 0\), \(\alpha \in (0, 1)\), and \(T > 0\) we define the norm
and the space \(\mathscr {L} ^{\gamma , \alpha }_T = \left\{ f : [0, T] \rightarrow \mathbb {R}: \Vert f \Vert _{\mathscr {L} ^{\gamma , \alpha }_T} < \infty \right\} \). In particular, we have \(\mathscr {L} ^{0, \alpha }_T = \mathscr {L} ^{\alpha }_T\). We introduce the linear operator \(I : C \left( \mathbb {R}_+, \mathscr {D} ' (\mathbb {T}) \right) \rightarrow C \left( \mathbb {R}_+, \mathscr {D} ' (\mathbb {T}) \right) \) given by
where \((P_t)_{t \geqslant 0}\) is the heat semigroup with kernel \(P_t (x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} \mathbb {I}_{t \geqslant 0}\).
Paraproducts are bilinear operations introduced by Bony [3] in order to linearize a class of non-linear PDE problems. They appear naturally in the analysis of the product of two Besov distributions. In terms of Littlewood–Paley blocks, the product fg of two distributions f and g can be formally decomposed as
where
This decomposition behaves nicely with respect to Littlewood–Paley theory. We call \(f \prec g\) and \(f \succ g\)paraproducts, and \(f \circ g\) the resonant term. We use the notation \(f \preccurlyeq g = f \prec g + f \circ g\). The basic result about these bilinear operations is given by the following estimates, essentially due to Bony [3] and Meyer [14].
When dealing with paraproducts in the context of parabolic equations it would be natural to introduce parabolic Besov spaces and related paraproducts. But to keep a simpler setting, we choose to work with space–time distributions belonging to the scale of spaces \(\left( C_T \mathscr {C} ^{\alpha } \right) _{\alpha \in \mathbb {R}}\) for some \(T > 0\). To do so efficiently, we will use a modified paraproduct which introduces some smoothing in the time variable that is tuned to the parabolic scaling. Let therefore \(\varphi \in C^{\infty } (\mathbb {R}, \mathbb {R}_+)\) be nonnegative with compact support contained in \(\mathbb {R}_+\) and with total mass 1, and define for all \(i \geqslant - 1\) the operator
We will often apply \(Q_i\) and other operators on \(C \mathscr {C} ^{\beta }\) to functions \(f \in C_T \mathscr {C} ^{\beta }\) which we then simply extend from [0, T] to \(\mathbb {R}_+\) by considering \(f (\cdot \wedge T)\). With the help of \(Q_i\), we define a modified paraproduct
for \(f, g \in C \left( \mathbb {R}_+, \mathscr {D} ' (\mathbb {T}) \right) \). We define the commutators \({\text {com}}_1, \overline{{\text {com}}}_1, {\text {com}}_2, {\text {com}}_3\) in Lemma A.7. We write \(\check{P} (t, x) = \frac{1}{(4 \pi t)^{3 / 2}} e^{- \frac{| x |^2}{4 t}} e^{- t} \mathbb {1}_{t \geqslant 0}\) for a modified heat kernel which has the same bounds of the usual heat kernel P(t, x). Let \(Y_{\varepsilon }\) as in (3) and recall that \(C_{\varepsilon }\) is the covariance of \(Y_{\varepsilon }\), i.e. \(C_{\varepsilon } (t, x) =\mathbb {E} (Y_{\varepsilon } (t, x) Y_{\varepsilon } (0, 0))\). We will sometimes write \(Y_{\varepsilon , \zeta } :=Y_{\varepsilon } (t, x)\) for \(\zeta = (t, x) \in \mathbb {R} \times \mathbb {T}^3\). Let \(\sigma _{\varepsilon }^2 = \varepsilon \mathbb {E} [ (Y_{\varepsilon } (0, 0))^2] = \varepsilon C_{\varepsilon } (0, 0)\).
1.2 A.2 Basic paracontrolled calculus results
First, let us recall some interpolation results on the parabolic time-weighted spaces \(\mathscr {L} ^{\gamma , \alpha }_T\):
Lemma A.1
For all \(\alpha \in (0, 2)\), \(\gamma \in [0, 1)\), \(\varepsilon \in [0, \alpha \wedge 2 \gamma )\), \(T > 0\) and \(f \in \mathscr {L} ^{\gamma , \alpha }_T\) with \(f (0) = 0\) we have
Let \(\alpha \in (0, 2)\), \(\gamma \in (0, 1),\)\(T > 0\), and let \(f \in \mathscr {L} ^{\alpha }_T\). Then for all \(\delta \in (0, \alpha ]\) we have
Schauder estimates
Lemma A.2
Let \(\alpha \in (0, 2)\) and \(\gamma \in [0, 1)\). Then
for all \(t > 0\). If further \(\beta \geqslant - \alpha \), then
For all \(\alpha \in \mathbb {R}\), \(\gamma \in [0, 1)\), and \(t > 0\) we have
Proofs can be found e.g. in [7]. We need also some well known bounds for the solutions of the heat equation with sources in space–time Lebesgue spaces.
Lemma A.3
Let \(\beta \in \mathbb {R}\) and \(f \in L^p_T B^{\beta }_{p, \infty }\), then for every \(\kappa \in [0, 1]\) we have \(I f \in C_T^{\kappa / q} \mathcal {\mathscr {C} }^{\beta + 2 (1 - \kappa ) - (2 - 2 \kappa + d) / p}\) with
with \(\frac{1}{q} + \frac{1}{p} = 1\). Moreover, for every \(\gamma< \gamma ' < 1 - 1 / p\) and every \(0< \alpha < (2 - 5 / p + \beta ) \wedge 2\) we have
Proof
We only show the second inequality as the first one is easier and obtained with similar techniques. Let \(u = I f\), we have
which allows us to bound \(\Vert I f \Vert _{\mathcal {M}^{\gamma }_T \mathscr {C} ^{\alpha }}\). In order to estimate \(\Vert t \mapsto t^{\gamma '} I f \Vert _{C^{\alpha / 2}_T L^{\infty }}\) we write
We can estimate the first term as
For the third term we have
We obtain then if \(2^{2 i} | t - s | \leqslant 1\)
and if \(2^{2 i} | t - s | > 1\) we just use the trivial estimate
Therefore, for every \(\kappa \in [0, 1]\):
Choosing \(\kappa / q = \alpha / 2\) we obtain the desired estimate. \(\square \)
Estimates on Bony’s paraproducts and commutators
Lemma A.4
For any \(\beta \in \mathbb {R}\) we have
and for \(\alpha < 0\) furthermore
For \(\alpha + \beta > 0\) we have
A natural corollary is that the product fg of two elements \(f \in \mathscr {C} ^{\alpha }\) and \(g \in \mathscr {C} ^{\beta }\) is well defined as soon as \(\alpha + \beta > 0\), and that it belongs to \(\mathscr {C} ^{\gamma }\), where \(\gamma = \min \{\alpha , \beta , \alpha + \beta \}\). We will also need a commutator estimation:
Lemma A.5
Let \(\alpha > 0\), \(\beta \in \mathbb {R}\), and let \(f, g \in \mathscr {C} ^{\alpha }\), and \(h \in \mathscr {C} ^{\beta }\). Then
We collect in the following lemma various estimates for the modified paraproduct \(f \prec \!\!\!\prec g\), proofs are again in [7].
Lemma A.6
-
a)
For any \(\beta \in \mathbb {R}\) and \(\gamma \in [0, 1)\) we have
$$\begin{aligned} t^{\gamma } \Vert f \prec \!\!\!\prec g (t) \Vert _{\mathscr {C} ^{\beta }} \lesssim \Vert f\Vert _{\mathcal {M}^{\gamma }_t L^{\infty }} \Vert g (t) \Vert _{\mathscr {C} ^{\beta }}, \end{aligned}$$(69)for all \(t > 0\), and for \(\alpha < 0\) furthermore
$$\begin{aligned} t^{\gamma } \Vert f \prec \!\!\!\prec g (t) \Vert _{\mathscr {C} ^{\alpha + \beta }} \lesssim \Vert f\Vert _{\mathcal {M}^{\gamma }_t \mathscr {C} ^{\alpha }} \Vert g (t) \Vert _{\mathscr {C} ^{\beta }} . \end{aligned}$$(70) -
b)
Let \(\alpha , \delta \in (0, 2)\), \(\gamma \in [0, 1)\), \(T > 0\), and let \(f \in \mathscr {L} _T^{\gamma , \delta }\), \(g \in C_T \mathscr {C} ^{\alpha }\), and \(\mathscr {L} g \in C_T \mathscr {C} ^{\alpha - 2}\). Then
$$\begin{aligned} \Vert f \prec \!\!\!\prec g \Vert _{\mathscr {L} ^{\gamma , \alpha }_T} \lesssim \Vert f \Vert _{\mathscr {L} ^{\gamma , \delta }_T} \left( \Vert g \Vert _{C_T \mathscr {C} ^{\alpha }} + \left\| \mathscr {L} g \right\| _{C_T \mathscr {C} ^{\alpha - 2}} \right) . \end{aligned}$$(71)
We introduce various commutators which allow to control non-linear functions of paraproducts and also the interaction of the paraproducts with the heat kernel.
Lemma A.7
-
a)
For \(\alpha , \beta , \gamma \in \mathbb {R}\) such that \(\alpha + \beta + \gamma > 0\) and \(\alpha \in (0, 1)\) there exists bounded trilinear maps
$$\begin{aligned} {\text {com}}_1, \overline{{\text {com}}}_1 : \mathscr {C} ^{\alpha } \times \mathscr {C} ^{\beta } \times \mathscr {C} ^{\gamma } \rightarrow \mathscr {C} ^{\alpha + \beta + \gamma }, \end{aligned}$$such that for smooth f, g, h they satisfy
$$\begin{aligned} {\text {com}}_1 (f, g, h)= & {} (f \prec g) \circ h - f (g \circ h). \end{aligned}$$(72)$$\begin{aligned} \overline{{\text {com}}}_1 (f, g, h)= & {} (f \prec \!\!\!\prec g) \circ h - f (g \circ h). \end{aligned}$$(73) -
b)
Let \(\alpha \in (0, 2)\), \(\beta \in \mathbb {R}\), and \(\gamma \in [0, 1)\). Then the bilinear maps
$$\begin{aligned} {\text {com}}_2 (f, g)&: =&f \prec g - f \prec \!\!\!\prec g. \end{aligned}$$(74)$$\begin{aligned} {\text {com}}_3 (f, g)&: =&\left[ \mathscr {L} , f \prec \!\!\!\prec \right] g :=\mathscr {L} (f \prec \!\!\!\prec g) - f \prec \!\!\!\prec \mathscr {L} g. \end{aligned}$$(75)have the bounds
$$\begin{aligned} t^{\gamma } \Vert {\text {com}}_2 (f, g) (t) \Vert _{\alpha + \beta } \lesssim \Vert f \Vert _{\mathscr {L} ^{\gamma , \alpha }_t} \Vert g (t) \Vert _{\mathscr {C} ^{\beta }}, \qquad t > 0. \end{aligned}$$(76)as well as
$$\begin{aligned} t^{\gamma } \Vert {\text {com}}_3 (f, g) (t) \Vert _{\alpha + \beta - 2} \lesssim \Vert f \Vert _{\mathscr {L} ^{\gamma , \alpha }_t} \Vert g (t) \Vert _{\mathscr {C} ^{\beta }}, \qquad t > 0. \end{aligned}$$(77)
Proofs can be found in [7].
1.3 A.3 Estimation of finite-chaos diagrams
In this section we give some estimations of Feynman-like diagrams that appear in both the \(\Phi ^4_3\) model and in Sect. 3. Such diagrams were already estimated in [8, Chap.10] and [18].
Let \(K_{q, x} (y) = 2^{3 q} K (2^q (x - y))\) be the kernel associated to the q-th Littlewood–Paley block \(\Delta _q\) on \(\mathbb {R}^3\). For a function f defined on the torus \(\mathbb {T}^3\) we still write \(\Delta _q f (x) = \int _{\mathbb {T}^3} K_{q, x} (y) f (y) \mathrm {d}y\) where with an abuse of notation \(K_{q, x}\) stands for the kernel on \(\mathbb {T}^3\), which is \(\sum _{j \in \mathbb {Z}^3} K_{q, x} (y + 2 \pi j)\).Let P be the heat kernel on \(\mathbb {R} \times \mathbb {R}^3\), i.e.
and call P(t, x) also its periodized version (with another abuse of notation) which for \((t, x) \in \mathbb {R} \times \mathbb {T}^3\) can be obtained as \(\sum _{j \in \mathbb {Z}^3} P (t, x + 2 \pi j)\). Let \(Y_{\varepsilon }\) be the stationary Gaussian field which solves
where \(\eta _{\varepsilon }\) is a centered Gaussian field on \(\mathbb {R} \times \mathbb {T}^3\) with stationary covariance
such that \(\forall \varepsilon \in (0, 1]\)
where \(\Sigma \) is a smooth even function compactly supported in \([- 1, 1] \times B_{\mathbb {R}^3} (0, 1)\). We introduce the modified heat kernel
and take
We call \(C_{\varepsilon }\) the covariance of \(Y_{\varepsilon }\), i.e.
One can see easily that
with time-space convolutions in \(\mathbb {R} \times \mathbb {T}^3\) and the kernel \(\check{P} (- \cdot )\) that has reversed time. In the following we use the notations \(| k |_{\mathfrak {s}} = 2 k_1 + \sum _{j = 2}^4 k_j\) and \(\Vert \zeta \Vert _{\mathfrak {s}} = | t |^{1 / 2} + \sum _{j = 1}^3 | x_j |\) for \(k \in \mathbb {N}^4\), \(\zeta = (t, x) \in \mathbb {R} \times \mathbb {T}^3\).
Lemma A.8
Let P(t, x) be defined in (78). Then \(\forall (t, x) \in \mathbb {R} \times \mathbb {R}^3\)
Moreover, for every multi-index k we have:
Proof
and \(\exists C > 0\) such that \((1 + | \alpha |^3) e^{- \frac{| \alpha |}{4}} \leqslant C\) for every \(\alpha \in \mathbb {R}\). Calling \(k_t = k_1\) and \(k_x = (k_2, \ldots , k_4)\) one can see directly by taking derivatives of P(t, x) that
and then \(| \partial ^{k_t} \partial ^{k_x} P (t, x) | (| t |^{1 / 2} + | x |)^{3 + | k_x | + 2 | k_t |} \leqslant C\). \(\square \)
Remark A.9
It is immediate to note that the estimation of Lemma A.8 holds as well for the kernel P(t, x) on the torus \(\mathbb {R} \times \mathbb {T}^3\) and the stationary kernel \(\check{P} (t, x)\).
We recall [8, Lemma 10.14] in a restricted formulation that is enough for our purposes.
Lemma A.10
Let \(K_1, K_2 : \mathbb {R}^4 {\setminus } \{0\} \rightarrow \mathbb {R}\) smooth and such that \(\exists \alpha , \beta \in (- 5, 0)\), \(\exists m \in \mathbb {N}\) such that \(\forall | k |_{\mathfrak {s}} \leqslant m\), \(\forall \zeta \in \mathbb {R}^4 {\setminus } \{0\}\)
Let \(\gamma = \alpha + \beta + 5\). If \(\gamma < 0\) then
Remark A.11
It is clear that the argument of Lemma A.10 works as well for space-periodic kernels \(K_1, K_2 : \mathbb {R} \times \mathbb {T}^3 {\setminus } \{0\} \rightarrow \mathbb {R}\).
Lemma A.12
([8], Lemma 10.17) Let \(K : \mathbb {R}^4 {\setminus } \{0\} \rightarrow \mathbb {R}\) smooth and such that \(\exists m \in \mathbb {N}\), \(\exists \alpha \in (- 5, 0)\) such that
Let \(\psi \in C^{\infty }_c (\mathbb {R}^4)\) with unit mass. Let \(\psi _{\varepsilon } (t, x) = \varepsilon ^{- 5} \psi (\varepsilon ^{- 2} t, \varepsilon ^{- 1} x)\) and define \(K_{\varepsilon } = K *\psi _{\varepsilon }\). Then
Proof
Let w.l.o.g. \(\psi _{\varepsilon }\) be supported on \(B_{\mathfrak {s}} (0, \varepsilon ) = \{ \zeta \in \mathbb {R}^4 | \Vert \zeta \Vert _{\mathfrak {s}} < \varepsilon \}\). For \(\Vert \zeta \Vert _{\mathfrak {s}} > 2 \varepsilon \) we bound \(\partial ^k K_{\varepsilon }\) as
since for \(\Vert \zeta ' \Vert < \varepsilon \) we have \(\Vert \zeta - \zeta ' \Vert \geqslant \Vert \zeta \Vert - \varepsilon \geqslant \Vert \zeta \Vert / 2\). For \(\Vert \zeta \Vert _{\mathfrak {s}} \leqslant 2 \varepsilon \) we bound
\(\square \)
Lemma A.13
The covariance \(C_{\varepsilon }\) on \(\mathbb {R} \times \mathbb {T}^3\) has the bound, for every multi-index \(k \in \mathbb {N}^4\):
Proof
Note that from Lemma A.8 it follows immediately that \(\check{P}\) satisfies the assumptions of Lemma A.10. We obtain then the estimation
for every multi-inded \(k \in \mathbb {N}^4\), and from Lemma A.12 we obtain the result. \(\square \)
Lemma A.14
We have for every \(\sigma \in [0, 1]\)
Proof
Since for every \(\varepsilon \in (0, 1]\)\(C_{\varepsilon }\) is smooth, the result is immediately obtained by Taylor expansion and interpolation from the bound of Lemma A.13. \(\square \)
Lemma A.15
For every \(t > 0\) we have
and for every \(n \geqslant 3\)
Proof
From the estimations of Lemmas A.8 and A.13 we have
In the same way for \(n \geqslant 3\)
\(\square \)
Lemma A.16
We have for every \(\alpha \in (0, 3)\)
with the integral over \(\mathbb {T}^3\) or \(\mathbb {R}^3\). Moreover for \(P : \mathbb {R}^4 \rightarrow \mathbb {R}\) as in (78) we have \(\forall \delta \in [0, 1]\)
Proof
We start with inequality (85), which can be obtained in a similar way as Lemma A.12. When \(| t |^{1 / 2} \geqslant 2^{- i} \vee | x |\) we have
When \(2^{- i} \geqslant | t |^{1 / 2} \vee | x |\) we estimate for \(\alpha \in (0, 3)\)
Finally, when \(| x | \geqslant 2^{- i} \vee | t |^{1 / 2}\) we split the domains \(| x | \geqslant 2^{- i + 1} | y |\) or \(| x | < 2^{- i + 1} | y |\). In the first region \(| x - 2^{- i} y | \gtrsim | x |\) so
while in the second region \(| y | \geqslant 2^i | x | / 2\), then \(| K (y) | \leqslant | K (y) |^{1 / 2} f (2^i | x | / 2)\) where f is another rapidly decreasing function which is defined on the torus as \(f (\cdot ) = \sum _{j \in \mathbb {Z}^3} f (\cdot + 2 \pi j)\) by an abuse of notation. Then for \(\alpha \in (0, 3)\)
concluding the argument. Taking the integral over \(\mathbb {R}^3\) in (85) does not change the estimations, and the second inequality (86) is obtained in the same way.
Let us show (87). Note that since \(\forall i \geqslant 0, \forall x \in \mathbb {R}^3\)\(\int K_{i, x} (y) \mathrm {d}y = 0\) (obvious from its Fourier transform) we have
where \(P'\) denotes the derivative of P with respect to the space variable and can be estimated with Lemma A.8. As before we can bound \((| t |^{1 / 2} + | x + \tau 2^{- i} y |)^{- 4}\) by considering three separate regions: when \(| t |^{1 / 2} \geqslant 2^{- i} \vee | x |\) we have
When \(2^{- i} \geqslant | t |^{1 / 2} \vee | x |\) we estimate simply
since \(P (2^{- 2 i} t, 2^i y) = 2^{- 3 i} P (t, y)\).
When \(| x | \geqslant 2^{- i} \vee | t |^{1 / 2}\) we have instead that either \(| x | \geqslant 2 \tau 2^{- i} | y |\) or \(| x | < 2 \tau 2^{- i} | y |\). In the first region \(| x + \tau 2^{- i} y | \gtrsim | x |\) so
In the region \(| y | > 2^i | x | / (2 \tau )\) we have \(| y K (y) | \lesssim (2^i | x | / (2 \tau )) K (2^i | x | / (2 \tau ))\) and then the integral can be estimated as follows:
This concludes the proof for (87). \(\square \)
Lemma A.17
We have for every \(\alpha \in (0, 3)\)
Proof
From (85) and (87) we deduce that
Bounding the sum over i with an integral, we conclude that
\(\square \)
Lemma A.18
For a fixed \(\bar{\zeta } = (t, \bar{x}) \in \mathbb {R} \times \mathbb {T}^3\) and \(\forall q \in \mathbb {Z}, q \geqslant - 1\) define the measures
Let \(C_{\varepsilon } (\zeta )\) for \(\zeta \in \mathbb {R} \times \mathbb {R}^3\) be (\(\mathbb {T}^3\)-periodic) covariance of \(Y_{\varepsilon }\). Then for every \(\alpha < 3\)
For every \(\beta \in (3, 5) \backslash \{ 4 \}\)
Proof
We have
by Lemma A.13. By (86) we have
and using again (86) we obtain the result. For the second estimation we obtain from (87) that
and from Lemma A.8
By the estimation on convolutions of Lemma A.10 we obtain then
by (86), and this concludes the proof. \(\square \)
Lemma A.19
For a fixed \(\bar{\zeta } = (t, \bar{x}) \in \mathbb {R} \times \mathbb {T}^3\) and \(\forall q \in \mathbb {Z}, q \geqslant - 1\) define the measure
with the notation \(\zeta _i = (s_i, x_i) \in \mathbb {R} \times \mathbb {T}^d\) for \(i = 1, 2\). For \(k, \ell \in [0, 2)\) and \(m \in (0, 5)\), \(n \in (0, 3)\) define
If \(k + m - 1 \in (0, 5)\) or \(\ell + m - 1 \in (0, 5)\), \(k + \ell + m - 4 \in (- 2, 5)\) we have the bound
Proof
From the Fourier support properties of the kernel K we have
By Cauchy–Schwarz inequality it suffices then to bound the terms \(\bar{I}_{k, \ell , m, n}\) and \(\tilde{I}_{k, \ell , m, n}\) where \(\mu _{q, \zeta _1, \zeta _2}\) in (88) is replaced respectively by \(\bar{\mu }_{q, \zeta _1, \zeta _2}\) and \(\tilde{\mu }_{q, \zeta _1, \zeta _2}\).
The first term can be estimated by repeated change of variables, using the fact that \(P (2^{- 2 i} s, 2^{- i} y) = 2^{- 3 i} P (s, y)\) and \(| C_{\varepsilon } (2^{- 2 i} s, 2^{- i} y) | \lesssim 2^i (| s |^{1 / 2} + | y |)^{- 1}\).
To bound the term \(\tilde{\mu }_{q, \zeta _1, \zeta _2}\) we use the estimation (87) choosing \(\delta \in [0, 1]\) such that the hypotheses of Lemma A.10 are satisfied. A repeated application of these convolution estimations yields the result. \(\square \)
Appendix B: Some Malliavin calculus
We recall here some tools from Malliavin calculus that are widely used in the rest of the paper. An introduction to Malliavin calculus and the proofs of some results of this Appendix can be found in [20, 21, 25]. Lemma B.7 was inspired by the calculations of [19].
1.1 B.1 Notation
Let \(\{ W (h) \}_{h \in H}\) be an isonormal Gaussian process indexed by a real separable Hilbert space H. Let \((\Omega , \mathcal {F}, \mathbb {P})\) a probability space with \(\mathcal {F}\) generated by the isonormal Gaussian process W, we note \(L^2 (\Omega ) = \bigoplus _{n \geqslant 0} \mathcal {H}_n\) the well-known Wiener chaos decomposition of \(L^2 (\Omega )\). For any real separable Hilbert space V and \(k \in \mathbb {N}\), let \(\mathrm {D}^k : L^2 (\Omega ; V) \rightarrow L^2 (\Omega ; H^{\odot k} \otimes V)\) be the Malliavin derivative and \(\delta ^k : L^2 (\Omega ; H^{\otimes k} \otimes V) \rightarrow L^2 (\Omega ; V)\) the divergence operator (also called Skorohod integral) defined as the adjoint of \(\mathrm {D}^k\). For \(p \geqslant 1\) we will write \(\mathbb {D}^{k, p} (V) \subset L^p (\Omega ; V)\) for the closure of smooth random variables with respect to the norm
with the notation \(\mathbb {D}^{k, p} :=\mathbb {D}^{k, p} (\mathbb {R})\). Let \(\{ P_t \}_{t \in \mathbb {R}^+}\) the Ornstein–Uhlenbeck semigroup and \(L : L^2 (\Omega ) \rightarrow L^2 (\Omega )\) its generator (i.e. \(e^{tL} = P_t\)). Following [25] we introduce the Green operator
with the notation
so that \(G_{[j]}^{[j]} = G_j\). To avoid confusion, it is worth stressing that \(G_{[j]}^{[m]}\) is not the m-th power of the operator \(G_j\) but just a shortcut for \(\prod _{k = j}^m G_k \).
1.2 B.2 Partial chaos expansion
Let \(\Psi \in L^2 (\Omega )\) which has the Wiener chaos decomposition \(\Psi = \sum _{n \geqslant 0} J_n \Psi \). Then by Proposition 1.2.2 of [21] \(\mathrm {D}J_n \Psi = J_{n - 1} \mathrm {D}\Psi \), and knowing that \(LJ_n \Psi = - nJ_n \Psi \) we obtain the commutation property
for every \(\alpha > 0\), \(j > 0\). The above formula holds also for \(j = 0\) if \(\mathbb {E} (\Psi ) = 0\).
The results we have recalled so far let us write an nth-order Wiener chaos expansion for a random variable in \(\mathbb {D}^{n, 2}\):
Lemma B.1
Let \(\Psi \in \mathbb {D}^{n, 2}\) and \(G_{[j]}^{[m]}\) as in (89). Then for every \(n \in \mathbb {N}\backslash \{ 0 \}\)\(G_{[1]}^{[n]} \mathrm {D}^n \Psi \in {\text {Dom}} \delta ^n\), \(J_0 \mathrm {D}^k \Psi \in {\text {Dom}} \delta ^k\)\(\forall 0 \leqslant k < n\) and
Proof
We have for any \(\Psi \in L^2 (\Omega )\), since \(L = - \delta \mathrm {D}\) ([21], Proposition 1.4.3):
where we used (90), and the fact that \((1 - L)^{- 1} \mathrm {D}\Psi \in {\text {Dom}} \delta \) is obvious by construction. This yields the first order expansion \(\Psi =\mathbb {E} (\Psi ) + \delta (1 - L)^{- 1} \mathrm {D}\Psi \). Iterating the expansion up to order n we obtain (91). It is clear that \(J_0 \mathrm {D}^k \Psi \in {\text {Dom}} \delta ^k\) since \(J_0 \mathrm {D}^k \Psi \) is constant with values in \(H^{\otimes k}\). The second equality comes from the fact that \(\delta ^k J_0 \mathrm {D}^k \Psi \in \mathcal {H}_k\)\(\forall k \in \mathbb {N}\), indeed \(\forall \Psi ' \in \mathbb {D}^{k, 2}\):
\(\square \)
In order to obtain \(L^p\) estimations of the remainder term \(\delta ^n G_{[1]}^{[n]} \mathrm {D}^n \Psi \) generated by expansion (91), we used the following lemmas:
Lemma B.2
([21], Prop. 1.5.7) Let V be a real separable Hilbert space. For every \(p > 1\) and every \(q \in \mathbb {N}, k \geqslant q\) and every \(\Psi \in \mathbb {D}^{k, p} (H^q \otimes V)\) we have
Remark B.3
Let V be a real separable Hilbert space. For every \(v \in V\) and every \(\Psi \in \mathbb {D}^{q, 2} (H^{\otimes q})\) with \(q \in \mathbb {N}\) we have \(\Psi \otimes v \in {\text {Dom}} \delta ^q\) and
Indeed, notice that for every smooth \(\Psi ' \in \mathbb {D}^{q, 2} (V)\) and every smooth \(\Psi \in \mathbb {D}^{q, 2} (H^{\otimes q})\) we have
Now since \(\mathrm {D}^q (\Psi \otimes v) = \mathrm {D}^q \Psi \otimes v\) and \(\Psi \in \mathbb {D}^{q, 2} (H^{\otimes q})\), we have \(\Psi \otimes v \in \mathbb {D}^{q, 2} (H^{\otimes q} \otimes V)\). Lemma B.2 yields the bound \(\Vert \delta ^q (\Psi \otimes v) \Vert _{L^2 (V)} \lesssim \Vert \Psi \otimes v \Vert _{\mathbb {D}^{q, 2} (H^{\otimes q} \otimes V)}\) which allows to pass to the limit for \(\Psi '\) and \(\delta ^q (\Psi \otimes v)\) in \(L^2 (V)\).
Lemma B.4
([25], Prop. 4.3) For every \(j > 0\) the operator \((j - L)^{- 1 / 2}\) is bounded in \(L^p\) for every \(1 \leqslant p < \infty \).
Lemma B.5
Let \(j \in \mathbb {N}\backslash \{ 0 \}\) and V a real separable Hilbert space. There exists a finite constant \(c_p\) such that for every \(\Psi \in L^p (\Omega , V)\):
(where the operator \(\mathrm {D}(j - L)^{- 1 / 2}\) is defined on every \(\Psi \) which is polynomial in \(W (h_1), \ldots , W (h_n)\) and can be extended by density on \(L^p\)).
Proof
First notice that we can suppose w.l.o.g. \(\mathbb {E} (\Psi ) = 0\) thanks to (90). Therefore we can write \(\mathrm {D}(j - L)^{- \frac{1}{2}}\) as
with \(C = - \sqrt{- L}\). We decompose the second part as \(- C (j - L)^{- 1 / 2} \Psi = \sum _{n = 1}^{\infty } \left( \frac{n}{j + n} \right) ^{1 / 2} J_n \Psi = T_{\phi } \Psi \), with \(T_{\phi } \Psi :=\sum _{n = 0}^{\infty } \phi (n) J_n \Psi \). We apply Theorem 1.4.2 of [21] to show that \(T_{\phi }\) is bounded in \(L^p\), indeed \(\phi (n) = h (1 / n)\) and \(h (x) = (jx + 1)^{- 1 / 2}\) which is analytic in a neighbourhood of 0. Finally, we can apply Proposition 1.5.2 of [21] to show that \(\mathrm {D}C^{- 1}\) is bounded in \(L^p\), thus concluding the proof. \(\square \)
The two lemmas above give the following immediate corollary:
Corollary B.6
For every \(1 \leqslant m \leqslant n\) the operator \(G_{[m]}^{[n]} :=\prod _{j = m}^n (j - L)^{- 1}\) is bounded in \(L^p\) for every \(1 \leqslant p < \infty \).
Moreover, Let \(j \in \mathbb {N}\backslash \{ 0 \}\) and V a real separable Hilbert space. Then for every \(\Psi \in L^p (\Omega , V)\) we have:
Moreover, for every \(0 \leqslant k \leqslant 2 m\), \(i \geqslant 0\) we have
The next lemma is one of the most useful tools of this paper. It allows us to write products of decompositions of the type (91) as sums of iterated Skorohod integrals. From now on we will note \(\langle \cdot , \cdot \rangle _{H^{\otimes r}}\) the r-th contraction, which to avoid inconsistency has to be taken between symmetric tensors. We also note \(h^{\odot n}_{v_1, \ldots , v_n} :=h_{v_1} \odot \cdots \odot h_{v_n}\) for \(h_{v_1}, \ldots , h_{v_n} \in H\).
Lemma B.7
Let \(u = f (W (h_u)) h^{\otimes m}_u\) and \(v = Fh^{\odot n}_{v_1, \ldots , v_n}\) with \(f \in C^{m + n} (\mathbb {R})\) and \(F \in \mathbb {D}^{m + n, 2}\). Then
with \(C_{m, n, q, r, i} :=\left( {\begin{array}{c}m\\ q\end{array}}\right) \left( {\begin{array}{c}n\\ r\end{array}}\right) \left( {\begin{array}{c}q\\ i\end{array}}\right) \left( {\begin{array}{c}r\\ i\end{array}}\right) i!\) and \(I_{m, n} :=\{ (q, r, i) \in \mathbb {N}^3 : 0 \leqslant q \leqslant m, 0 \leqslant r \leqslant n, 0 \leqslant i \leqslant q \wedge r \}\).
A trivial change of variables gives also:
with \(I'_{m, n} :=\{ (i, q, r) \in \mathbb {N}^3 : 0 \leqslant i \leqslant m \wedge n, 0 \leqslant q \leqslant m - i, 0 \leqslant r \leqslant n - i \}\).
Remark B.8
In the special case \(v = g (W (h_v)) h^{\otimes n}_v\) Eq. (92) takes the form
which is just a generalization to Skorohod integrals of the multiplication formula for multiple Wiener integrals ([24, 26]). We can write the above formula more explicitly as
Remark B.9
Note that one can assume w.l.o.g. the argument of \(\delta ^{m + n - q - r}\) in (92) to be symmetric, and this would allow to iterate Lemma B.7.
Remark B.10
We can give the following intuition for the second formula in Lemma B.7. The random variables u and v have an infinite chaos decomposition, and following the tree-like notation of [6] or [8] they could be thought of as having an infinite number of leaves which need to be contracted with each other.
It is apparent that the index i in the second equation denotes contractions between the already existing leaves of the trees u, v. The indexes r and q count new leaves in each vertex that are created by the Malliavin derivatives, which are then contracted with other leaves from the other tree. There are then \(m + n - r - q - 2 i\) overall unmatched leaves which are arguments to the iterated Skorokhod integral.
The more intuitive interpretation of the second equation in Lemma B.7 is the reason why we gave two distinct expression for the same quantity. Nevertheless, the formula (92) is more practical in the calculations and is more widely used throughout the paper.
Proof
[Lemma B.7] Using Cauchy–Schwarz inequality and Lemma B.2 we can show that \(\langle D^r \delta ^n (v), \delta ^j (u) \rangle _{H^{\otimes r}} \in L^2 (\Omega , H^{\otimes m - j - r})\) for every \(0 \leqslant r + j \leqslant m\). Then we apply Lemma B.11 to get:
Using the commutation formula (95) we rewrite the r.h.s. as
We obtain
and using again Lemma B.11 we obtain
where we used \(\delta ^k (\Psi ) h^{\otimes n - r} = \delta ^k (\Psi \otimes h^{\otimes n - r})\) for \(\Psi \in {\text {Dom}} \delta ^k\), as seen in Remark B.3. Substituting this expression into \(\delta ^m (u) \delta ^n (v)\) we get
where we set \(A_{m, n, r, i, \ell } :=\left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}n\\ r\end{array}}\right) \left( {\begin{array}{c}m - i\\ \ell \end{array}}\right) \left( {\begin{array}{c}r\\ i\end{array}}\right) i!\) and \(J_{m, n} :=\{ (r, i, \ell ) \in \mathbb {N}^3 : 0 \leqslant r \leqslant n, 0 \leqslant i \leqslant r \wedge n, 0 \leqslant \ell \leqslant m - i \}\).
In order to complete the proof we just have to perform some basic changes of indexes. Taking \(q = \ell + i\) and noting that \(\left( {\begin{array}{c}m\\ i\end{array}}\right) \left( {\begin{array}{c}m - i\\ q - i\end{array}}\right) = \left( {\begin{array}{c}q\\ i\end{array}}\right) \left( {\begin{array}{c}m\\ q\end{array}}\right) \) we have \(A_{m, n, r, i, \ell } = \left( {\begin{array}{c}m\\ q\end{array}}\right) \left( {\begin{array}{c}q\\ i\end{array}}\right) \left( {\begin{array}{c}n\\ r\end{array}}\right) \left( {\begin{array}{c}r\\ i\end{array}}\right) i!\) and this yields (92).
Finally, we perform the change of variables \(q - i \rightarrow q\), \(r - i \rightarrow r\) to get the second formula. \(\square \)
We give below the results we used to prove Lemma B.7.
Lemma B.11
([19], Lemma 2.1) Let \(q \in \mathbb {N}\backslash \{ 0 \}\), \(\Psi \in \mathbb {D}^{q, 2}\), \(u \in {\text {Dom}} \delta ^q\) and symmetric. Assume also that \(\forall 0 \leqslant r + j \leqslant q\)\(\langle D^r \Psi , \delta ^j (u) \rangle _{H^{\otimes r}} \in L^2 (\Omega , H^{\otimes q - r - j})\). Then \(\forall 0 \leqslant r < q\)\(\langle D^r \Psi , u \rangle _r \in {\text {Dom}} \delta ^{q - r}\) and
Remark B.12
Note that
where \(\llbracket \cdot \rrbracket \) stands for the Wick product. Indeed \(\forall \Psi \in \mathbb {D}^{1, 2}\) we know that \(\mathbb {E} [\delta (h^{\otimes n}) \Psi ] =\mathbb {E} [W (h) h^{\otimes n - 1} \Psi ]\) using the definition of \(\delta \), and then \(\delta ^n (h^{\otimes n}) = \delta ^{n - 1} (W (h) h^{\otimes n - 1})\). From Lemma B.11 we have, since \(\mathrm {D}W (h) = h\):
which gives by induction \(\delta ^n (h^{\otimes n}) = \llbracket W^n (h) \rrbracket \).
Lemma B.13
Let \(j, k \in \mathbb {N}\), \(u \in \mathbb {D}^{j + k, 2} (H^{\otimes j})\) symmetric and such that all its derivatives are symmetric. We have
Proof
If \(j = 0\), \(k = 1\) or \(k = 0, j = 1\) Eq. (95) is trivial. Let \(j = k = 1\) and \(u \in \mathbb {D}^{2, 2} (H) \subset \mathbb {D}^{1, 2} (H)\). We can apply Proposition 1.3.2 of [21] to obtain \(\langle \mathrm {D}\delta (u), h \rangle = \langle u, h \rangle + \delta (\langle \mathrm {D}u, h \rangle )\)\(\forall h \in H\). Since by hypothesis \(\mathrm {D}u\) is symmetric we have \(\delta (\langle \mathrm {D}u, h \rangle ) = \langle \delta \mathrm {D}u, h \rangle \), and then \(\mathrm {D}\delta (u) = u + \delta \mathrm {D}u\). The proof by induction is easy noticing that \(\mathrm {D}\delta ^j = \delta ^j \mathrm {D}+ j \delta ^{j - 1}\). \(\square \)
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Furlan, M., Gubinelli, M. Weak universality for a class of 3d stochastic reaction–diffusion models. Probab. Theory Relat. Fields 173, 1099–1164 (2019). https://doi.org/10.1007/s00440-018-0849-6
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DOI: https://doi.org/10.1007/s00440-018-0849-6
Keywords
- Weak universality
- Paracontrolled distributions
- Stochastic quantisation equation
- Malliavin calculus
- Partial chaos expansion
Mathematics Subject Classification
- 60H15
- 60H07