# Stratonovich solution for the wave equation

Raluca M. Balan111University of Ottawa, Department of Mathematics and
Statistics, 150 Louis Pasteur Private, Ottawa, Ontario, K1G 0P8, Canada.
E-mail address<EMAIL_ADDRESS>supported by a grant from the
Natural Sciences and Engineering Research Council of Canada.

(May 15, 2021)

###### Abstract

In this article, we construct a Stratonovich solution for the stochastic wave
equation in spatial dimension $d\leq 2$, with time-independent noise and
linear term $\sigma(u)=u$ multiplying the noise. The noise is spatially
homogeneous and its spectral measure satisfies an integrability condition
which is stronger than Dalang’s condition. We give a probabilistic
representation for this solution, similar to the Feynman-Kac-type formula
given in [5] for the solution of the stochastic wave equation with spatially
homogeneous Gaussian noise, that is white in time. We also give the chaos
expansion of the Stratonovich solution and we compare it with the chaos
expansion of the Skorohod solution from [1].

MSC 2020: Primary 60H15; Secondary 60H07

Keywords: stochastic wave equation, Stratonovich solution, Malliavin calculus

###### Contents

  1. 1 Introduction
  2. 2 Equation with mollified noise
    1. 2.1 Existence of solution
    2. 2.2 Feynman-Kac-type formula
  3. 3 Proof of Theorem 1.2
    1. 3.1 Multiple Stratonovich integrals
    2. 3.2 Convergence as $\varepsilon\downarrow 0$
    3. 3.3 Convergence as $n\to\infty$
    4. 3.4 Final step: $v$ is Stratonovich solution
  4. 4 Feynman-Kac-type formula
  5. 5 Smooth noise case
  6. 6 Comparison with the Skorohod solution
    1. 6.1 Chaos expansions
    2. 6.2 Approximation for the Skorohod solution
  7. A Parseval-type identities
  8. B Products of Wiener integrals
  9. C Semigroup-type property of $G$

##  1 Introduction

In this article, we study the stochastic wave equation with time-independent
noise:

$\displaystyle\begin{cases}\dfrac{\partial^{2}v}{\partial t^{2}}(t,x)=\Delta
v(t,x)+v(t,x)\dot{W}(x)\quad\quad t>0,x\in\mathbb{R}^{d}\quad(d\leq 2)\\\
v(0,x)=1,\quad\dfrac{\partial v}{\partial t}(0,x)=0\quad\quad\quad\quad\quad\
x\in\mathbb{R}^{d}\end{cases}$ (1)

The noise is given by a zero-mean Gaussian process
$\\{W(\varphi);\varphi\in\mathcal{D}(\mathbb{R}^{d})\\}$ defined on a complete
probability space $(\Omega,\mathcal{F},P)$, with covariance

$\mathbb{E}[W(\varphi)W(\psi)]=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\varphi(x)\psi(y)\gamma(x-y)dxdy:=\langle\varphi,\psi\rangle_{\mathcal{H}},$

where $\mathcal{D}(\mathbb{R}^{d})$ is the space of infinitely differentiable
functions on $\mathbb{R}^{d}$ with compact support and
$\gamma:\mathbb{R}^{d}\to[0,\infty]$ is a non-negative-definite function. By
the Bochner-Schwartz theorem, there exists a tempered measure $\mathbb{R}^{d}$
such that $\gamma=\mathcal{F}\mu$ in the sense of distributions, and hence

$\langle\varphi,\psi\rangle_{\mathcal{H}}=\int_{\mathbb{R}^{d}}\mathcal{F}\varphi(\xi)\overline{\mathcal{F}\psi(\xi)}\mu(d\xi)\quad\mbox{for
any}\quad\varphi,\psi\in\mathcal{D}(\mathbb{R}^{d}).$

Let $\mathcal{H}$ be the completion of $\mathcal{D}(\mathbb{R}^{d})$ with
respect to $\langle\cdot,\cdot\rangle_{\mathcal{H}}$. Then
$W=\\{W(\varphi);\varphi\in\mathcal{H}\\}$ is an isonormal Gaussian process,
and we can use Malliavin calculus with respect to $W$ to analyze the solution
of (1). We recall briefly some basic facts from Malliavin calculus which are
needed in this paper. We refer the reader to [11] for more details. For any
$n\geq 1$, we denote by $\mathcal{H}_{n}$ the $n$-th Wiener chaos space
corresponding to $W$, i.e. the closed linear subspace of $L^{2}(\Omega)$
generated by
$\\{H_{n}(W(\varphi));\varphi\in\mathcal{H},\|\varphi\|_{\mathcal{H}}=1\\}$,
where $H_{n}(x)$ is the Hermite polynomial of order $n$. Any random variable
$F\in L^{2}(\Omega)$, which is measurable with respect to $W$, has the Wiener
chaos expansion:

$F=E(F)+\sum_{n\geq 1}I_{n}(f_{n})$

where $I_{n}:\mathcal{H}^{\otimes n}\to\mathcal{H}_{n}$ is the multiple Wiener
integral of order $n$. We let $\mathcal{H}_{0}=\mathbb{R}$ and
$I_{0}:\mathbb{R}\to\mathbb{R}$ be the identity map. Moreover,

$\mathbb{E}|F|^{2}=\big{(}\mathbb{E}(F)\big{)}^{2}+\sum_{n\geq
1}n!\|\widetilde{f}_{n}\|_{\mathcal{H}^{\otimes n}}^{2},$

where $\widetilde{f}_{n}$ is the symmetrization of $f_{n}$. Moreover,
$\|\widetilde{f}\|_{\mathcal{H}^{\otimes n}}\leq\|f\|_{\mathcal{H}^{\otimes
n}}$ for any $f\in\mathcal{H}^{\otimes n}$.

We impose the following assumption, which is needed for Lemmas A.1 and A.2
below:

Assumption A. (a) $\gamma$ is continuous, symmetric, and $\gamma(x)<\infty$ if
and only if $x\not=0$;
(b) $\mu$ is absolutely continuous with respect to the Lebesgue measure.

A basic example is the Riesz kernel $\gamma(x)=|x|^{-\alpha}$ with
$\alpha\in(0,d)$, in which case
$\mu(d\xi)=C_{d,\alpha}|\xi|^{-(d-\alpha)}d\xi$ and $C_{d,\alpha}>0$ is a
constant depending on $d$ and $\alpha$.

We are interested in the Stratonovich solution of (1), whose definition is
given below.

######  Definition 1.1.

The random field $\\{v(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ is a (mild)
Stratonovich solution of equation (1) if for any $t>0$ and
$x\in\mathbb{R}^{d}$, with probability 1,

$v(t,x)=1+\int_{0}^{t}\left(\int_{\mathbb{R}^{d}}G(t-s,x-y)v(s,y)W^{\circ}(dy)\right)ds,$

where the $W^{\circ}(dy)$ denotes the Stratonovich integral.

We recall that the Stratonovich integral
$\int_{\mathbb{R}^{d}}\Phi(x)W^{\circ}(dx)$ of the random field
$\\{\Phi(x)\\}_{x\in\mathbb{R}^{d}}$ is defined as the following limit in
probability, if it exists:

$\int_{\mathbb{R}^{d}}\Phi(x)W^{\circ}(dx)=\lim_{\varepsilon\downarrow
0}\int_{\mathbb{R}^{d}}\Phi(x)\dot{W}^{\varepsilon}(x)dx,$

where $\dot{W}^{\varepsilon}(x)=W(p_{\varepsilon}(x-\cdot))$ is a
mollification of $W$ and
$p_{\varepsilon}(x)=(2\pi\varepsilon)^{-d/2}e^{-|x^{2}|/(2\varepsilon)}$.

We denote by $G$ the fundamental solution of the wave equation on
$\mathbb{R}_{+}\times\mathbb{R}^{d}$:

$\displaystyle
G(t,x)=\begin{cases}\displaystyle\frac{1}{2}1_{\\{|x|<t\\}}&\text{if
$d=1$},\\\\[10.00002pt]
\displaystyle\frac{1}{2\pi}\frac{1}{\sqrt{t^{2}-|x|^{2}}}1_{\\{|x|<t\\}}&\text{if
$d=2$},\\\ \end{cases}$ (2)

Note that $G(t,\cdot)$ is integrable and $\int_{\mathbb{R}^{d}}G(t,x)dx=t$. We
denote by $\mathcal{F}\varphi(\xi)=\int_{\mathbb{R}^{d}}e^{-i\xi\cdot
x}\varphi(x)dx$ the Fourier transform of a function $\varphi\in
L^{1}(\mathbb{R}^{d})$. Then

$\mathcal{F}G(t,\cdot)(\xi)=\frac{\sin(t|\xi|)}{|\xi|}\quad\mbox{for all}\
\xi\in\mathbb{R}^{d}.$

The following inequality will play an important role in the present article:

$|\mathcal{F}G(t,\cdot)(\xi)|\leq
C_{t}\left(\frac{1}{1+|\xi|^{2}}\right)^{1/2},$ (3)

with $C_{t}=2\sqrt{2}(t\vee 1)$. To see this, use
$\frac{\sin(t|\xi|)}{|\xi|}\leq 2(t\vee 1)\frac{1}{1+|\xi|}$ and
$\left(\frac{1}{1+|\xi|}\right)^{2}\leq\frac{2}{1+|\xi|^{2}}$.

The first goal of this article is to prove the existence of a Stratonovich
solution to equation (1). This will be achieved under the following condition:

$\displaystyle
K_{\mu}:=\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|^{2}}\right)^{1/2}\mu(d\xi)<\infty,$
(C)

which, due to (3), implies that

$\int_{\mathbb{R}^{d}}|\mathcal{F}G(t,\cdot)(\xi)|\mu(d\xi)\leq C_{t}K_{\mu}.$
(4)

The Stratonovich solution $v$ is different than the Skorohod solution $u$ of
equation (1), which satisfies (by definition) the following integral equation:

$u(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)u(s,y)W(\delta y)ds,$ (5)

where $W(\delta y)$ denotes the Skorohod integral (see Section 6 below). In
[1], it was shown that under Dalang’s condition:

$\displaystyle\int_{\mathbb{R}^{d}}\frac{1}{1+|\xi|^{2}}\mu(d\xi)<\infty,$ (D)

the Skorohod solution exists and has the chaos expansion $u(t,x)=1+\sum_{n\geq
1}I_{n}\big{(}f_{n}(\cdot,x;t)\big{)}$, where

$f_{n}(x_{1},\ldots,x_{n},x;t)=\int_{T_{n}(t)}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})dt_{1}\ldots
dt_{n},$ (6)

and $T_{n}(t)=\\{(t_{1},\ldots,t_{n})\in[0,t]^{n};t_{1}<\ldots<t_{n}\\}$. Here
$t_{n+1}=t$ and $x_{n+1}=x$.

If $\gamma$ satisfies the scaling property $\gamma(cx)=c^{-\alpha}\gamma(x)$
for all $c>0$ and $x\in\mathbb{R}^{d}$, for some $\alpha>0$, then
$\mu(cA)=c^{\alpha}\mu(A)$ for any $c>0$ and $A\in{\cal B}(\mathbb{R}^{d})$,
and (C) is equivalent to $\alpha\in(0,1)$ while (D) is equivalent to
$\alpha\in(0,2)$.

In the case of the heat equation:

$\begin{cases}\dfrac{\partial u}{\partial t}(t,x)=\frac{1}{2}\Delta
u(t,x)+u(t,x)\dot{W}(x),\quad t\geq 0,x\in\mathbb{R}^{d}\ (d\geq 1)\\\ \quad
u(0,x)=1\qquad\qquad\qquad\qquad\qquad\quad x\in\mathbb{R}^{d}\end{cases}$ (7)

it was proved in [6] that the Skorohod solution $u_{h}$ and the Stratonovich
solution $v_{h}$ exist under Dalang’s condition, and they admit the following
Feynman-Kac representations:

$u_{h}(t,x)=\mathbb{E}^{B}[e^{V(t,x)-\frac{1}{2}M(t)}]\quad\mbox{and}\quad
v_{h}(t,x)=\mathbb{E}^{B}[e^{V(t,x)}].$

Here $B=(B_{t})_{t\geq 0}$ is a $d$-dimensional Brownian motion independent of
$W$ and $B_{t}^{x}=B_{t}+x$. The functional $V(t,x)$, defined formally by

$V(t,x):=\int_{0}^{t}\int_{\mathbb{R}^{d}}\delta_{0}(B_{r}^{x}-y)W(dy)dr,$

is the $L^{2}(\Omega)$-limit of
$V^{\varepsilon}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}p_{\varepsilon}(B_{r}^{x}-y)W(dy)dr$
as $\varepsilon\downarrow 0$, and the correction term

$M(t)=\int_{0}^{t}\int_{0}^{t}\gamma(B_{s}-B_{r})dsdr$

is the conditional variance of $V(t,x)$ given $B$. If in addition,

$\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|\xi|^{2}}\right)^{1-\delta}\mu(d\xi)<\infty\quad\mbox{for
some $\delta\in(0,1)$},$

then, by Proposition 5.28 of [6], $W$ has a version with values in a weighted
Besov space, and equation (7) has a pathwise solution which coincides with the
Stratonovich solution. The construction of the Stratonovich solution of
equation (7) (given by Theorem 5.7 of [6]) relies heavily on the Feynman-Kac
representation of this solution, and cannot be extended to the wave equation.

In this article, we develop a method for constructing a Stratonovich solution
based on chaos expansions, which seems to be new in the literature. We
implement this method for the wave equation, but we believe that the method is
so robust that can be applied to a large class of SPDEs with time-dependent
noise and linear term $\sigma(u)=u$ multiplying the noise. (We postpone the
study of the wave equation with time-dependent noise for future work.) We
believe that this method can also be applied to show the existence of a
Stratonovich solution for the heat equation under condition (D), using
inequality:

$\int_{0}^{T}\mathcal{F}G_{h}(t,\cdot)(\xi)dt\leq 4(T\vee
1)\frac{1}{1+|\xi|^{2}},$

instead of (3), where $\mathcal{F}G_{h}(t,\cdot)(\xi)=e^{-t|\xi|^{2}/2}$ is
Fourier transform in the space variable of the fundamental solution
$G_{h}(t,x)=p_{t}(x)$ of the heat equation on
$\mathbb{R}_{+}\times\mathbb{R}^{d}$.

The idea is to use the classical method of Picard’s iterations for solving the
equation with mollified noise $W^{\varepsilon}$:

$\displaystyle\begin{cases}\dfrac{\partial^{2}v^{\varepsilon}}{\partial
t^{2}}(t,x)=\Delta
v^{\varepsilon}(t,x)+v^{\varepsilon}(t,x)\dot{W}^{\varepsilon}(x),\quad\quad
t>0,x\in\mathbb{R}^{d}\quad(d\leq 2)\\\
v^{\varepsilon}(0,x)=1,\quad\dfrac{\partial v^{\varepsilon}}{\partial
t}(0,x)=0\end{cases}$ (8)

and then apply the product formula from Malliavin calculus for writing the
$n$-th Picard approximation $v_{n}^{\varepsilon}(t,x)$ of
$v^{\varepsilon}(t,x)$ as a finite sum of multiple Wiener integrals. The
delicate part is to let $\varepsilon\downarrow 0$. Finally, we let
$n\to\infty$. This method involves some non-trivial combinatorial arguments,
which appeared for the first time in the earlier works [7, 8] of Hu and Meyer
on multiple Stratonovich integrals with respect to Brownian motion.

We are now ready to state the first main result of this paper.

######  Theorem 1.2.

If Assumption A holds and $\mu$ satisfies condition (C), then there exists a
process $v=\\{v(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ such that for any $T>0$
and $p\geq 2$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|v^{\varepsilon}(t,x)-v(t,x)|^{p}\to
0\quad\mbox{as $\varepsilon\downarrow 0$},$ (9)

where $v^{\varepsilon}$ is the solution of (8). The random variable $v(t,x)$
can be represented as

$v(t,x)=1+\sum_{n\geq 1}H_{n}(t,x),$ (10)

where $H(t,x)=I_{n}^{\circ}\big{(}f_{n}(\cdot,x;t)\big{)}$ is the multiple
Stratonovich integral of $f_{n}(\cdot,x;t)$, and is given by relation (32)
below. Moreover, $v$ is a Stratonovich solution of equation (1).

The series (10) can be used to derive the Wiener chaos expansion of $v(t,x)$,
but the explicit expression of the projection on each Wiener chaos space is
quite involved, since $H_{n}(t,x)$ is a sum of multiple Wiener integrals of
order $n-2k$, for $k=0,1,\ldots,\lfloor n/2\rfloor$. Note that the term of
this sum which corresponds to $k=0$ is the $n$-th chaos term appearing in the
Wiener chaos expansion of the Skorohod solution $u(t,x)$ (see Section 6
below).

The second goal of this article is to give a Feynman-Kac-type formula for the
Stratonovich solution of equation (1), similar to the one given by Dalang,
Mueller and Tribe in [5] for Itô-Skorohod solution of the wave equation with
spatially-homogeneous Gaussian noise that is white in time. Unlike the
Feynman-Kac formula for the solution of the heat equation (which uses the
paths of Brownian motion), this formula is based on a process whose paths are
obtained by linear interpolation between the jump times of a Poisson process.

More precisely, let $N=(N_{t})_{t\geq 0}$ be a Poisson process of rate $1$,
with jump times $(\tau_{i})_{i\geq 1}$. Denote $\tau_{0}=0$. Let
$(U_{i})_{i\geq 1}$ be i.i.d. random variables with values in
$\mathbb{R}^{d}$, with density $G(1,\cdot)$. Note that if $d=1$, $U_{1}$ has a
uniform distribution on $(-1,1)$. Assume that $N$, $(U_{i})_{i\geq 1}$ and $W$
are independent.

Based on $N$ and $(U_{i})_{i\geq 1}$, we define the linearly interpolated
process $(X_{t})_{t\geq 0}$ as follows:

$X_{t}=X_{\tau_{i}}+(t-\tau_{i})U_{i+1}\quad\mbox{for
$\tau_{i}<t\leq\tau_{i+1}$, $i\geq 0$,}$

with $X_{0}=0$. Let $X_{t}^{x}=x+X_{t}$.

The next theorem is the second main result of this paper, which gives the
Feynman-Kac-type representation for $v$.

######  Theorem 1.3.

If Assumption A holds, $\mu$ satisfies condition (C), and $v$ is the
Stratonovich solution of equation (1) given by Theorem 1.2, then for any
$t>0,x\in\mathbb{R}^{d}$, almost surely,

$\displaystyle v(t,x)$
$\displaystyle=e^{t}\mathbb{E}^{N,X}\Big{[}\prod_{i=1}^{N_{t}}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{N_{t}}\dot{W}(X_{\tau_{i}}^{x})\Big{]}$
(11) $\displaystyle:=e^{t}\sum_{n\geq
0}\mathbb{E}^{N,X}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{N_{t}}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{N_{t}}\dot{W}(X_{\tau_{i}}^{x})\Big{]},$

where the terms of the series are defined by relation (51) below if $n\geq 1$,
and we use the convention $\prod_{i=1}^{0}=1$ if $n=0$ (so that the first term
of the series is $e^{t}\mathbb{E}^{N,X}[1_{\\{N_{t}=0\\}}]=1$).

This article is organized as follows. In Section 2, we analyze equation (8)
with mollified noise. The proofs of Theorems 1.2 and 1.3 are given in Sections
3 and 4, respectively. In Section 5, we examine the particular case of a very
smooth noise $W$, which has finite spectral measure $\mu$. In Section 6, we
give the chaos expansion of the Stratonovich solution and compare it with the
chaos expansion of the Skorohod solution. Appendices A-C contain some
auxiliary results needed in the sequel.

##  2 Equation with mollified noise

In this section, we study equation (8) with mollified noise
$\dot{W}^{\varepsilon}$.

###  2.1 Existence of solution

The mollified noise $\\{\dot{W}^{\varepsilon}(x)\\}_{x\in\mathbb{R}^{d}}$ is a
zero-mean Gaussian process with stationary increments (in Yaglom sense) and
covariance function:

$\mathbb{E}|\dot{W}^{\varepsilon}(x)-\dot{W}^{\varepsilon}(y)|^{2}=\int_{\mathbb{R}^{d}}(1-e^{i\xi\cdot(x-y)})^{2}e^{-\varepsilon|\xi|^{2}}\mu(d\xi).$

Under condition (D),

$\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{2}=\|p_{\varepsilon}(x-\cdot)\|_{\mathcal{H}}^{2}=\int_{\mathbb{R}^{d}}e^{-\varepsilon|\xi|^{2}}\mu(d\xi)\leq\int_{\mathbb{R}^{d}}\frac{1}{1+\varepsilon|\xi|^{2}}\mu(d\xi)=:C_{\mu,\varepsilon}<\infty$

and

$\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{p}=z_{p}(\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{2})^{p/2}\leq
z_{p}C_{\mu,\varepsilon}^{p/2}\quad\mbox{for any $p>0$},$ (12)

where

$z_{p}=E|Z|^{p}=\frac{2^{p/2}}{\sqrt{\pi}}\Gamma\left(\frac{p+1}{2}\right)\quad\mbox{with
$Z\sim N(0,1)$}.$

######  Definition 2.1.

We say that the random field $v^{\varepsilon}=\\{v^{\varepsilon}(t,x);t\geq
0,x\in\mathbb{R}^{d}\\}$ is a solution of equation (8) if for any $t>0$ and
$x\in\mathbb{R}^{d}$, with probability $1$,

$v^{\varepsilon}(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v^{\varepsilon}(s,y)\dot{W}^{\varepsilon}(y)dyds.$
(13)

Intuitively, $v^{\varepsilon}(t,x)$ should be given by the series:

$v^{\varepsilon}(t,x)=1+\sum_{n\geq
1}\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})d{\bf
x}d{\bf t},$ (14)

where ${\bf x}=(x_{1},\ldots,x_{n})$, ${\bf t}=(t_{1},\ldots,t_{n})$ and we
let $t_{n+1}=t$ and $x_{n+1}=x$.

Note that (14) cannot be obtained by a pathwise application of Proposition 2.2
of [5], since the potential $\dot{W}^{\varepsilon}(x)$ is not bounded. We will
justify below relation (14) for fixed $\varepsilon>0$. The fact that the
series (14) converges uniformly in $\varepsilon$ will be proved in Section
3.3.

Let $v_{n}^{\varepsilon}(t,x)=1+\sum_{k=1}^{n}H_{k}^{\varepsilon}(t,x)$ be the
$n$-th Picard iteration, where

$H_{n}^{\varepsilon}(t,x)=\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})d{\bf
x}d{\bf t}.$ (15)

Note that $H_{n}^{\varepsilon}(t,x)$ is well-defined due to inequality (16)
below. Let $H_{0}^{\varepsilon}(t,x)=1$. We have the following recurrence
relation: for any $n\geq 0$,

$H_{n+1}^{\varepsilon}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)H_{n}^{\varepsilon}(s,y)\dot{W}^{\varepsilon}(y)dyds.$

The next result shows that the series (14) converges, for any $\varepsilon>0$
fixed.

######  Lemma 2.2.

Under condition (D), for any $\varepsilon>0$ fixed,
$\lim_{n\to\infty}v_{n}^{\varepsilon}(t,x)=v^{\varepsilon}(t,x)$ exists in
$L^{p}(\Omega)$, uniformly in $[0,T]\times\mathbb{R}^{d}$, for any $p\geq 1$
and $T>0$, i.e.

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|v_{n}^{\varepsilon}(t,x)-v^{\varepsilon}(t,x)|^{p}\to
0\quad\mbox{as $n\to\infty$}.$

Moreover, $v^{\varepsilon}$ is a solution of equation (8).

###### Proof.

Let $\|\cdot\|_{p}$ be the norm in $L^{p}(\Omega)$. Using Minkowski
inequality, the generalized Hölder inequality and inequality (12), we obtain
that for any $p\geq 1$,

$\displaystyle\|H_{n}^{\varepsilon}(t,x)\|_{p}$
$\displaystyle\leq\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\Big{\|}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})\Big{\|}_{p}d{\bf
x}d{\bf t}$
$\displaystyle\leq\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}\|\dot{W}^{\varepsilon}(x_{i})\|_{pn}d{\bf
x}d{\bf t}$ $\displaystyle\leq
z_{pn}^{1/p}C_{\varepsilon}^{n/2}\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})d{\bf
x}d{\bf t}$
$\displaystyle=z_{pn}^{1/p}C_{\mu,\varepsilon}^{n/2}\int_{T_{n}(t)}\prod_{i=1}^{n}(t_{i+1}-t_{i})d{\bf
t}$
$\displaystyle=\left\\{\frac{2^{pn/2}}{\sqrt{\pi}}\Gamma\left(\frac{pn+1}{2}\right)\right\\}^{1/p}C_{\mu,\varepsilon}^{n/2}\frac{t^{2n}}{(2n)!}.$

Since $\Gamma(\frac{pn+1}{2})\leq C_{p}^{n}(n!)^{p/2}$ and $(2n)!\geq
c_{0}^{n}(n!)^{2}$ (due to Stirling’s formula), we obtain:

$\|H_{n}^{\varepsilon}(t,x)\|_{p}\leq\frac{2^{n/2}}{\pi^{1/(2p)}}C_{p}^{n/p}C_{\mu,\varepsilon}^{n/2}c_{0}^{n}\frac{t^{2n}}{(n!)^{3/2}}.$
(16)

Hence,

$\sum_{n\geq
0}\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\|H_{n}^{\varepsilon}(t,x)\|_{p}<\infty$

and the sequence $\\{v_{n}^{\varepsilon}(t,x)\\}_{n}$ is Cauchy in
$L^{p}(\Omega)$, uniformly in $(t,x)\in[0,T]\times\mathbb{R}^{d}$.

To prove that $v^{\varepsilon}$ is a solution to (8), we let $n\to\infty$ in
the recurrence relation:

$v_{n+1}^{\varepsilon}(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v_{n}^{\varepsilon}(s,y)\dot{W}^{\varepsilon}(y)dyds.$
(17)

The left-hand side converges in $L^{p}(\Omega)$ to $v^{\varepsilon}(t,x)$. The
right-hand side converges in $L^{1}(\Omega)$ to
$\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v^{\varepsilon}(s,y)\dot{W}^{\varepsilon}(y)dyds$,
by the dominated convergence theorem. To justify the application of this
theorem, we use the fact that

$\mathbb{E}|\big{(}v_{n}^{\varepsilon}(s,y)-v^{\varepsilon}(s,y)\big{)}\dot{W}^{\varepsilon}(y)|\leq\left(\mathbb{E}|v_{n}^{\varepsilon}(s,y)-v^{\varepsilon}(s,y)|^{2}\right)^{1/2}\left(\mathbb{E}|\dot{W}^{\varepsilon}(y)|^{2}\right)^{1/2}.$

The last term is bounded by a constant (that depends on $\varepsilon$), due to
(12) and the uniform convergence of $(v_{n}^{\varepsilon})_{n}$. ∎

###  2.2 Feynman-Kac-type formula

In this section, we give a Feynman-Kac-type representation for
$v^{\varepsilon}$. Note that this cannot be obtained directly by a pathwise
application of Theorem 3.2 of [5] since the random potential $W^{\varepsilon}$
is not bounded.

Recall the definition of the linearly interpolated process $(X_{t})_{t\geq 0}$
given in the introduction. Note that
$X_{\tau_{i+1}}-X_{\tau_{i}}=(\tau_{i+1}-\tau_{i})U_{i}$. Hence
$(X_{\tau_{i+1}}-X_{\tau_{i}})_{i\geq 0}$ are independent given $N$. Since
$tU_{i}$ has density $G(t,\cdot)/t$ for any $t>0$,
$X_{\tau_{i+1}}-X_{\tau_{i}}$ has density
$G(\tau_{i+1}-\tau_{i},\cdot)/(\tau_{i+1}-\tau_{i})$ given $N$. It follows
that:

$(x_{1},\ldots,x_{n})\to\prod_{i=1}^{n}\frac{G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})}{\tau_{i}-\tau_{i-1}}\
\mbox{is the density of $(X_{\tau_{1}}^{x},\ldots,X_{\tau_{n}}^{x})$ given
$N$},$ (18)

with the convention $x_{0}=x$.

The key idea is the following identity, which gives a representation of an
integral over the simplex $T_{n}(t)$ using the points of the Poisson process
$N$: for any integrable function $h$ on $T_{n}(t)$,

$\int_{T_{n}(t)}h(t_{1},\ldots,t_{n})dt_{1}\ldots
dt_{n}=e^{t}\mathbb{E}[h(\tau_{1},\ldots,\tau_{n})1_{\\{N_{t}=n\\}}],$ (19)

This identity follows from the fact that $(\tau_{1},\ldots,\tau_{n})$ has a
uniform distribution over $T_{n}(t)$, given $N_{t}=n$.

We will use the following general facts. If $X:\Omega\to\mathcal{X}$ and
$Y:\Omega\to\mathcal{Y}$ are independent random variables, then for any non-
negative measurable functions $f,g,h$, we have

$\mathbb{E}^{X}[f(X,Y)]=\mathbb{E}[f(X,Y)|Y]$ (20)
$\mathbb{E}^{X}[g(X)h(Y)]=\mathbb{E}[g(X)]\,h(Y),$ (21)

where $\mathbb{E}^{X}$ denotes the expectation with respect to $X$. More
precisely, $\mathbb{E}^{X}[f(X,Y)]$ is the random variable given by
$\mathbb{E}^{X}[f(X,Y)]=\int_{\mathcal{X}}f(x,Y)\mu_{X}(dx)$, where $\mu_{X}$
is the law of $X$.

The next result gives the Feynman-Kac-type representation of
$v^{\varepsilon}$.

######  Lemma 2.3.

For any $\varepsilon>0$, $t>0$ and $x\in\mathbb{R}^{d}$, with probability $1$,

$v^{\varepsilon}(t,x)=e^{t}\mathbb{E}^{N,X}\Big{[}\prod_{i=1}^{N_{t}}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{N_{t}}\dot{W}^{\varepsilon}(X_{\tau_{i}}^{x})\Big{]}.$
(22)

###### Proof.

Fix $\varepsilon>0$, $t>0$ and $x\in\mathbb{R}^{d}$. By Lemma 2.2,
$v_{n}^{\varepsilon}(t,x)\to v^{\varepsilon}(t,x)$ a.s. along a subsequence.
So it is enough to prove that for any $n\geq 1$, with probability $1$,

$v_{n}^{\varepsilon}(t,x)=e^{t}\mathbb{E}^{N,X}\Big{[}1_{\\{N_{t}\leq
n\\}}\prod_{i=1}^{N_{t}}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{N_{t}}\dot{W}^{\varepsilon}(X_{\tau_{i}}^{x})\Big{]},$

and then let $n\to\infty$. For this, it suffices to prove that for any $n\geq
1$, with probability $1$,

$H_{n}^{\varepsilon}(t,x)=e^{t}\mathbb{E}^{N,X}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{n}\dot{W}^{\varepsilon}(X_{\tau_{i}}^{x})\Big{]}.$
(23)

In definition (15) of $H_{n}(t,x)$, we use the fact that $G(t,x)=G(t,-x)$ and
the change of variables $\bar{t}_{i}=t-t_{n+1-i}$ and $\bar{x}_{i}=x_{n+1-i}$
for all $i=1,\ldots,n$. Then

$H_{n}^{\varepsilon}(t,x)=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}G(t_{i}-t_{i-1},x_{i}-x_{i-1})\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})d{\bf
x}d{\bf t},$ (24)

with $t_{0}=0$ and $x_{0}=x$. We interchange the integrals. Using
representation (19) for the integral over the simplex, followed by (21) (since
$W$ and $N$ are independent), we obtain:

$\displaystyle H_{n}^{\varepsilon}(t,x)$
$\displaystyle=e^{t}\int_{(\mathbb{R}^{d})^{n}}\mathbb{E}\Big{[}\prod_{i=1}^{n}G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})1_{\\{N_{t}=n\\}}\Big{]}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})d{\bf
x}$
$\displaystyle=e^{t}\int_{(\mathbb{R}^{d})^{n}}\mathbb{E}^{N}\Big{[}\prod_{i=1}^{n}G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})\Big{]}d{\bf
x}$
$\displaystyle=e^{t}\mathbb{E}^{N}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1})\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}\frac{G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})}{\tau_{i}-\tau_{i-1}}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})d{\bf
x}\Big{]}.$

Relation (23) follows using Lemma 2.4 below with
$X=(X_{\tau_{1}}^{x},\ldots,X_{\tau_{n}}^{x})$, $Y=N$, $Z=W$,

$g(x_{1},\ldots,x_{n},W)=\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})\quad\mbox{and}\quad
h(N)=1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1}),$

noticing that

$f_{(X_{\tau_{1}}^{x},\ldots,X_{\tau_{n}}^{x})|N}(x_{1},\ldots,x_{n}|N)=\prod_{i=1}^{n}\frac{G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})}{\tau_{i}-\tau_{i-1}}$

(with $x_{0}=x$) is the conditional density of
$(X_{\tau_{1}}^{x},\ldots,X_{\tau_{n}}^{x})$ given $N$ (see (18)). ∎

######  Lemma 2.4.

If random variables $X:\Omega\to\mathcal{X}$, $Y:\Omega\to\mathcal{Y}$ and
$Z:\Omega\to\mathcal{Z}$ are such that $Z$ is independent of $(X,Y)$ and the
conditional density $f_{X|Y}(x|Y)$ of $X$ given $Y$ exists, then for any non-
negative measurable functions $g$ and $h$,

$\mathbb{E}^{Y}\Big{[}h(Y)\int_{\mathcal{X}}f_{X|Y}(x|Y)g(x,Z)dx\Big{]}=\mathbb{E}^{Y,X}\Big{[}h(Y)g(X,Z)\Big{]}.$

###### Proof.

We denote by $\mu_{Y}$ the law of $Y$ and $\mu_{Y,X}$ the law of $(Y,X)$. Then

$\mu_{Y,X}(B\times A)=\int_{B}P(X\in
A|Y=y)\mu_{Y}(dy)=\int_{B}\int_{A}f_{X|Y}(x|y)dx\mu_{Y}(dy).$

Since $Y$ and $Z$ are independent, using (20), we have:

$\mathbb{E}^{Y}\Big{[}h(Y)\int_{\mathbb{R}^{n}}f_{X|Y}(x|Y)g(x,Z)dx\Big{]}=\mathbb{E}\Big{[}h(Y)\int_{\mathbb{R}^{n}}f_{X|Y}(x|Y)g(x,Z)dx\Big{|}Z\Big{]}=g_{1}(Z),$

where

$\displaystyle g_{1}(z)$
$\displaystyle=\mathbb{E}\Big{[}h(Y)\int_{\mathcal{X}}f_{X|Y}(x|Y)g(x,z)dx\Big{]}=\int_{\mathcal{Y}}h(y)\int_{\mathbb{R}^{n}}f_{X|Y}(x|y)g(x,z)dx\mu_{Y}(dy)$
$\displaystyle=\int_{\mathcal{Y}\times\mathcal{X}}h(y)g(x,z)\mu_{Y,X}(dy,dx).$

Since $(X,Y)$ and $Z$ are independent, using (20), we have:

$\mathbb{E}^{Y,X}\Big{[}h(Y)g(X,Z)\Big{]}=\mathbb{E}[\Big{[}h(Y)g(X,Z)|Z\Big{]}=g_{2}(Z),$

where $g_{2}(z)=\mathbb{E}[h(Y)g(X,z)]$. The conclusion follows since
$g_{1}(z)=g_{2}(z)$ for any $z$. ∎

##  3 Proof of Theorem 1.2

In this section, we give the proof of Theorem 1.2, which contains also the
construction of the Stratonovich solution $v$.

As mentioned in the introduction, the proof is based on chaos expansions.
Recall that $v_{n}^{\varepsilon}$ is the $n$-th Picard approximations of
$v^{\varepsilon}$. The proof of (9) is summarized by the diagram below, in
which all arrows indicate convergence in $L^{p}(\Omega)$, uniform in
$(t,x)\in[0,T]\times\mathbb{R}^{d}$:

$\begin{array}[]{ccccc}&v_{n}^{\varepsilon}(t,x)&\longrightarrow_{\varepsilon}&v_{n}(t,x)&\mbox{for
any $n$}\\\ \mbox{(uniform in
$\varepsilon$)}&\downarrow_{n}&&\downarrow_{n}&\\\
&v^{\varepsilon}(t,x)&\---\to_{\varepsilon}&v(t,x)&\end{array}$

From this diagram, we see that there are are three facts that need to be
proved:
(i) the convergence of $v_{n}^{\varepsilon}(t,x)$ to some $v_{n}(t,x)$ (to be
determined), as $\varepsilon\downarrow 0$;
(ii) the convergence of $v_{n}(t,x)$ to some $v(t,x)$ (to be determined), as
$n\to\infty$;
(iii) the convergence of $v_{n}^{\varepsilon}(t,x)$ to $v^{\varepsilon}(t,x)$
as $n\to\infty$, uniform in $\varepsilon$.

In Section 3.1, we examine briefly the multiple Stratonovich integrals. We
will prove (i) in Section 3.2, and (ii)-(iii) in Section 3.3. The fact that
$v$ is a Stratonovich solution will be proved in Section 3.4.

###  3.1 Multiple Stratonovich integrals

In this section, we examine the integral $I_{n}^{\circ}(f)$. First, note

$\langle
p_{\varepsilon}(x_{1}-\cdot),p_{\varepsilon}(x_{2}-\cdot)\rangle_{\mathcal{H}}=(p_{2\varepsilon}*\gamma)(x_{1}-x_{2}),$
(25)

and $p_{2\varepsilon}*\gamma=\mathcal{F}\mu_{\varepsilon}$ in the sense of
distributions, where

$\mu_{\varepsilon}(d\xi)=e^{-\varepsilon|\xi|^{2}}\mu(d\xi).$ (26)

We need to introduce some notation. We let $[n]=\\{1,\ldots,n\\}$. For any set
$J\subset[n]$, we let $|J|$ be the cardinality of $J$, and $J^{c}$ be the
complement of $J$ in $[n]$. We denote $\lfloor x\rfloor=k$ if $k\in\mathbb{Z}$
and $k\leq x<k+1$.

######  Definition 3.1.

The multiple Stratonovich integral of order $n$ of a measurable function
$f:\mathbb{R}^{d}\to\mathbb{R}$ is defined as the following limit in
probability, if it exists:

$I_{n}^{\circ}(f):=\int_{(\mathbb{R}^{d})^{n}}f(x_{1},\ldots,x_{n})\prod_{i=1}^{n}W^{\circ}(dx_{i})=\lim_{\varepsilon\downarrow
0}\int_{(\mathbb{R}^{d})^{n}}f(x_{1},\ldots,x_{n})\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})dx_{1}\ldots
dx_{n}.$

Using Theorem B.1 for expressing the product
$\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})$ and (25), we obtain that:

$\displaystyle\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})$
$\displaystyle=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k}\big{(}\bigotimes_{j\in
J}p_{\varepsilon}(x_{j}-\cdot)\big{)}\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}}).$
(27)

This shows that $I_{n}^{\circ}(f)$ exists if
$I_{n-2k}(f_{k,J,I_{1},\ldots,I_{k}}^{\varepsilon})\stackrel{{\scriptstyle
P}}{{\to}}I_{n-2k}(f_{k,J,I_{1},\ldots,I_{k}})$ as $\varepsilon\downarrow 0$,
for any $k=0,1,\ldots,\lfloor n/2\rfloor$, for any set $J\subset[n]$ and for
any partition $\\{I_{1},\ldots,I_{k}\\}$ as above, where

$\displaystyle f_{k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(y_{j})_{j\in
J}\big{)}$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n}}f(x_{1},\ldots,x_{n})\prod_{j\in
J}p_{\varepsilon}(x_{j}-y_{j})\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}})dx_{1}\ldots
dx_{n}$ $\displaystyle f_{k,J,I_{1},\ldots,I_{k}}\big{(}(x_{j})_{j\in
J}\big{)}$
$\displaystyle=\int_{(\mathbb{R}^{d})^{2k}}f(x_{1},\ldots,x_{n})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}.$

Moreover, in this case,

$I_{n}^{\circ}(f)=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k}(f_{k,J,I_{1},\ldots,I_{k}}).$

For $n=1$, $I_{1}^{\circ}(f)=\int_{\mathbb{R}^{d}}f(x)W(dx)$ if
$f\in\mathcal{H}$, and hence $W^{\circ}(dx)=W(dx)$. For $n=2$,

$\displaystyle I_{2}^{\circ}(f)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{2}}f(x_{1},x_{2})W(dx_{1})W(dx_{2})+\int_{(\mathbb{R}^{d})^{2}}f(x_{1},x_{2})\gamma(x_{1}-x_{2})dx_{1}dx_{2},$

provided that both integrals above are well-defined, and hence

$W^{\circ}(dx_{1})W^{\circ}(dx_{2})=W(dx_{1})W(dx_{2})+\gamma(x_{1}-x_{2})dx_{1}dx_{2}.$

For $n=3$, we have:

$\displaystyle W^{\circ}(dx_{1})W^{\circ}(dx_{2})W^{\circ}(dx_{3})$
$\displaystyle=W(dx_{1})W(dx_{2})W(dx_{3})+W(dx_{1})\gamma(x_{2}-x_{3})dx_{2}dx_{3}+$
$\displaystyle\quad
W(dx_{2})\gamma(x_{1}-x_{3})dx_{1}dx_{3}+W(dx_{3})\gamma(x_{1}-x_{2})dx_{1}dx_{2}.$

In general,

$\displaystyle\prod_{i=1}^{n}W^{\circ}(dx_{i})=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\prod_{j\in
J}W(dx_{j})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}.$ (28)

Formally, we can say that

$\prod_{i=1}^{n}W^{\circ}(dx_{i})=\lim_{\varepsilon\downarrow
0}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x_{i})dx_{1}\ldots
dx_{n}=\prod_{i=1}^{n}\dot{W}(x_{i})dx_{1}\ldots dx_{n}.$ (29)

###  3.2 Convergence as $\varepsilon\downarrow 0$

In this section, we identify a random variable $v_{n}(t,x)$ such that
$v_{n}^{\varepsilon}(t,x)\to v_{n}(t,x)$ in $L^{p}(\Omega)$, as
$\varepsilon\downarrow 0$, uniformly in $(t,x)\in[0,T]\times\mathbb{R}^{d}$,
for any $p\geq 2$.

The next result shows that the multiple Stratonovich integral of the kernel
$f_{n}(\cdot,x;t)$ exists. For any ${\bf t}=(t_{1},\ldots,t_{n})\in T_{n}(t)$
and ${\bf x}=(x_{1},\ldots,x_{n})\in(\mathbb{R}^{d})^{n}$, we define:

$h_{{\bf t},(x_{j})_{j\in J^{c}}}^{(t,x)}\big{(}(x_{j})_{j\in
J}\big{)}:=\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j}),$ (30)

with $t_{n+1}=t$ and $x_{n+1}=x$.

######  Theorem 3.2.

If Assumption A and condition (C) hold, then for any $n\geq 1$, $p\geq 2$ and
$T>0$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|H_{n}^{\varepsilon}(t,x)-H_{n}(t,x)|^{p}\to
0\quad\mbox{as $\varepsilon\downarrow 0$,}$ (31)

where

$H_{n}(t,x)=I_{n}^{\circ}\big{(}f_{n}(\cdot,x;t)\big{)}=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}H_{n,k,J,I_{1},\ldots,I_{k}}(t,x)$ (32)

and the variable

$\displaystyle H_{n,k,J,I_{1},\ldots,I_{k}}(t,x)$
$\displaystyle:=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}\left(\int_{(\mathbb{R}^{d})^{n-2k}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})\prod_{j\in
J}W(dx_{j})\right)$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}dt_{1}\ldots dt_{n}$

is well-defined. Here we let $t_{n+1}=t$ and $x_{n+1}=x$.

Consequently, if $v_{n}(t,x)=1+\sum_{k=1}^{n}H_{k}(t,x)$, then for any $n\geq
1$, $p\geq 2$ and $T>0$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|v_{n}^{\varepsilon}(t,x)-v_{n}(t,x)|^{p}\to
0\quad\mbox{as $\varepsilon\downarrow 0$}.$

###### Proof.

Step 1. We consider first the case $n=1$. By the stochastic Fubini’s theorem,

$\displaystyle H_{1}^{\varepsilon}(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-t_{1},x-x_{1})\dot{W}^{\varepsilon}(x_{1})dx_{1}dt_{1}$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-t_{1},x-x_{1})p_{\varepsilon}(x_{1}-y_{1})dx_{1}dt_{1}\right)W(dy_{1}).$

Letting

$H_{1}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-t_{1},x-x_{1})W(dx_{1})dt_{1},$

it is not difficult to see that for any $p\geq 2$ and $T>0$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|H_{1}^{\varepsilon}(t,x)-H_{1}(t,x)|^{p}\to
0\quad\mbox{as $\varepsilon\downarrow 0$.}$ (33)

Step 2. We consider now the case $n\geq 2$. In definition (15) of
$H_{n}^{\varepsilon}(t,x)$, we use (27) for computing the product of the $n$
Wiener integrals. We obtain the representation:

$H_{n}^{\varepsilon}(t,x)=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}H_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x),$
(34)

where

$\displaystyle H_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})I_{n-2k}\big{(}\bigotimes_{j\in
J}p_{\varepsilon}(x_{j}-\cdot)\big{)}$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}})d{\bf
x}d{\bf t}$

with ${\bf t}=(t_{1},\ldots,t_{n})$ and ${\bf x}=(x_{1},\ldots,x_{n})$. Using
stochastic Fubini’s theorem, we see that

$\displaystyle H_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=I_{n-2k}\big{(}A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\big{)}$
$\displaystyle H_{n,k,J,I_{1},\ldots,I_{k}}(t,x)$
$\displaystyle=I_{n-2k}\big{(}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{)},$

where

$\displaystyle A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(y_{j})_{j\in
J},t,x\big{)}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})\prod_{j\in
J}p_{\varepsilon}(x_{j}-y_{j})$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}})d{\bf
x}d{\bf t}$ $\displaystyle A_{n,k,J,I_{1},\ldots,I_{k}}\big{(}(x_{j})_{j\in
J},t,x\big{)}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d{\bf t}.$

We will prove that

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\|A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)-A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\to
0\quad\mbox{as $\varepsilon\downarrow 0$}.$ (35)

Relation (31) will follow, since by Minkowksi’s inequality and
hypercontractivity,

$\displaystyle\|H_{n}^{\varepsilon}(t,x)-H_{n}(t,x)\|_{p}\leq\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}(p-1)^{\frac{n-2k}{2}}\Big{(}(n-2k)!\Big{)}^{1/2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\|A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)-A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}},$

where $\|\cdot\|_{p}$ is the norm in $L^{p}(\Omega)$.

Step 3. We prove that $H_{n,k,J,I_{1},\ldots,I_{k}}(t,x)$ is well-defined,
i.e. $A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\in\mathcal{H}^{\otimes(n-2k)}$.

Step 3.(a) We first prove that $A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\in
L^{1}((\mathbb{R}^{d})^{n-2k})$, i.e.

${\cal
I}:=\int_{(\mathbb{R}^{d})^{n}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d{\bf
x}d{\bf t}<\infty.$

We denote by $j_{1}<\ldots<j_{2k}$ the elements of
$J^{c}=\cup_{i=1}^{k}I_{i}$. By applying Lemma C.2 successively, we can skip
the indices $j\in J$ when estimating ${\cal I}$, so that ${\cal
I}\leq(C_{t}^{(1)})^{n-2k}{\cal I}^{\prime}$, where

$\displaystyle{\cal I}^{\prime}$
$\displaystyle=\int_{0<t_{j_{1}}<\ldots<t_{j_{2k}}<t}\int_{(\mathbb{R}^{d})^{2k}}\prod_{s=1}^{2k}G(t_{j_{s+1}}-t_{j_{s}},x_{j_{s+1}}-x_{j_{s}})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d\big{(}(t_{j})_{j\in J^{c}}\big{)}$
$\displaystyle=\int_{0<t_{j_{1}}<\ldots<t_{j_{2k}}<t}\int_{(\mathbb{R}^{d})^{k}}\prod_{s=1}^{2k}\mathcal{F}G(t_{j_{s+1}}-t_{j_{s}},\cdot)(\xi_{j_{1}}+\ldots+\xi_{j_{s}})\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})d\big{(}(t_{j})_{j\in
J^{c}}\big{)},$

where we used Lemma A.2 for the second line. Note that if
$\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}$ for all $i=1,\ldots,k$, then
$\xi_{j_{1}}+\ldots+\xi_{j_{2k}}=0$.

Suppose that $\ell_{i}<m_{i}$ for all $i=1,\ldots,k$ and
$\ell_{1}<\ldots<\ell_{k}$. Say $\ell_{i}=j_{s_{i}}$ and $m_{i}=j_{r_{i}}$ for
all $i=1,\ldots,k$ with $s_{1}<\ldots<s_{k}$. Then
$\\{s_{1},\ldots,s_{k}\\}\cup\\{r_{1},\ldots,r_{k}\\}=\\{1,\ldots,2k\\}$. We
assume that $\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}$ for all $i=1,\ldots,k$.
Pick $s\in\\{1,\ldots,2k\\}$. If $s=s_{i}$ for some $i=1,\ldots,k$ then
$\xi_{j_{s}}=\xi_{\ell_{i}}=\eta_{i}$ and we use inequality (3):

$|\mathcal{F}G(t_{j_{s+1}}-t_{j_{s}},\cdot)(\xi_{j_{1}}+\ldots+\xi_{j_{s}})|\leq
C_{t}\left(\frac{1}{1+|\xi_{j_{1}}+\ldots+\xi_{j_{s}}|^{2}}\right)^{1/2}$

If $s=r_{i}$ for some $i=1,\ldots,k$ then $\xi_{j_{s}}=\xi_{m_{i}}=-\eta_{i}$
and we use the inequality

$|\mathcal{F}G(t_{j_{s+1}}-t_{j_{s}},\cdot)(\xi_{j_{1}}+\ldots+\xi_{j_{s}})|\leq
t_{j_{s+1}}-t_{j_{s}}\leq t.$

Applying Lemma 4.1 of [2], we obtain:

$\displaystyle{\cal
I}^{\prime}\leq(tC_{t})^{k}\frac{t^{2k}}{(2k)!}\prod_{i=1}^{k}\left(\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|z+\eta_{i}|^{2}}\right)^{1/2}\mu(d\eta_{i})\right)=C_{t}^{k}\frac{t^{3k}}{(2k)!}K_{\mu}^{k}<\infty.$

Step 3.(b) We compute the Fourier transform of
$A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)$. Recall definition (30) of $h_{{\bf
t},(x_{j})_{j\in J^{c}}}$. By Fubini’s theorem,

$\displaystyle\mathcal{F}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}=\int_{(\mathbb{R}^{d})^{n-2k}}e^{-i\sum_{j\in J}\xi_{j}\cdot
x_{j}}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(x_{j})_{j\in
J},t,x\big{)}d\big{(}(x_{j})_{j\in J}\big{)}$
$\displaystyle=\quad\int_{(\mathbb{R}^{d})^{n}}e^{-i\sum_{j\in J}\xi_{j}\cdot
x_{j}}\int_{T_{n}(t)}h_{{\bf t},(x_{j})_{j\in J^{c}}}\big{(}(x_{j})_{j\in
J}\big{)}\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d{\bf t}d{\bf x}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}\mathcal{F}h_{{\bf
t},(x_{j})_{j\in J^{c}}}\big{(}(\xi_{j})_{j\in
J}\big{)}\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d{\bf t}.$

We apply Lemma A.2 to the function $\varphi_{{\bf t},(\xi_{j})(j\in
J)}((x_{j})_{j\in J^{c}}):=\mathcal{F}h_{{\bf t},(x_{j})_{j\in
J^{c}}}\big{(}(\xi_{j})_{j\in J}\big{)}$. Then,

$\displaystyle\mathcal{F}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}\varphi_{{\bf
t},(\xi_{j})(j\in J)}((x_{j})_{j\in
J^{c}})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d{\bf t}$
$\displaystyle=\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}\varphi_{{\bf
t},(\xi_{j})(j\in J)}((\xi_{j})_{j\in
J^{c}})\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t}.$

Note that

$\displaystyle\mathcal{F}\varphi_{{\bf t},(\xi_{j})(j\in J)}((\xi_{j})_{j\in
J^{c}})$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n}}e^{-i\sum_{j=1}^{n}\xi_{j}\cdot
x_{j}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})d{\bf x}$
$\displaystyle=e^{-i(\sum_{j=1}^{n}\xi_{j})\cdot
x}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)(\xi_{1}+\ldots+\xi_{j})$

and $\sum_{j=1}^{n}\xi_{j}=\sum_{j\in J}\xi_{j}$ if
$\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}$ for all $i=1,\ldots,k$. Hence,

$\displaystyle\mathcal{F}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}$ $\displaystyle=e^{-i(\sum_{j\in J}\xi_{j})\cdot
x}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{k}}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)(\xi_{1}+\ldots+\xi_{j})$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t}.$ (36)

Step 3.(c) We show that the $\mathcal{H}^{\otimes(n-2k)}$-norm of
$A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)$ is finite. This argument plays an
important role in this paper. By definition,

$\|A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}=\int_{(\mathbb{R}^{d})^{n-2k}}|\mathcal{F}A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}|^{2}\prod_{j\in J}\mu(d\xi_{j}).$

In relation (36), we split the product over $j$ into 3 products: for $j\in J$,
$j\in\\{m_{1},\ldots,m_{k}\\}$ and $j\in\\{\ell_{1},\ldots,\ell_{k}\\}$. For
the second product, we use the inequality

$\prod_{i=1}^{k}|\mathcal{F}G(t_{m_{i}+1}-t_{m_{i}},\cdot)(\xi_{1}+\ldots+\xi_{m_{i}})|\leq\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}}).$
(37)

Hence,

$\displaystyle\|A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\int_{(\mathbb{R}^{d})^{n-2k}}\left(\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j})(\xi_{1}+\ldots+\xi_{j})|\right.$
$\displaystyle\left.\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\xi_{1}+\ldots+\xi_{\ell_{i}})|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t}\right)^{2}\prod_{j\in J}\mu(d\xi_{j}).$

For each $(\xi_{j})_{j\in J}$ fixed, we will apply Hölder’s inequality to the
measure

$\nu_{(\xi_{j})_{j\in J}}\big{(}d{\bf
t},d((\eta_{i})_{i=1,\ldots,k})\big{)}:=\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\xi_{1}+\ldots+\xi_{\ell_{i}})|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t}.$

For this, we need to show that $\nu_{(\xi_{j})_{j\in J}}$ is a finite measure
on $T_{n}(t)\times(\mathbb{R}^{d})^{k}$. By (3),

$\displaystyle\nu_{(\xi_{j})_{j\in
J}}\big{(}T_{n}(t)\times(\mathbb{R}^{d})^{k}\big{)}\leq\int_{T_{n}(t)}\prod_{i=1}^{k}\left(\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(z+\eta_{i})|\mu(d\eta_{i})\right)d{\bf
t}$
$\displaystyle\quad\quad\quad\leq\frac{t^{n}}{n!}\left(C_{t}\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|z+\eta|^{2}}\right)^{1/2}\mu(d\eta)\right)^{k}=\frac{t^{n}}{n!}C_{t}^{k}K_{\mu}^{k}.$

Hence,

$\displaystyle\|A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}C_{t}^{k}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\quad\int_{(\mathbb{R}^{d})^{n-2k}}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j})(\xi_{1}+\ldots+\xi_{j})|^{2}$
$\displaystyle\quad\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\xi_{1}+\ldots+\xi_{\ell_{i}})|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}.$

We have arrived now at the most delicate part of the argument, which requires
a careful analysis of the positions of the indices $\ell_{i}$ and $m_{i}$.
Recall that $\ell_{i}<m_{i}$ for all $i=1,\ldots,k$ and
$\ell_{1}<\ldots\ell_{k}$. We calculate the sums $\xi_{1}+\ldots+\xi_{j}$
successively one after the other and we arrange them in a row-wise manner, one
per row. When we encounter an index
$j\not\in\\{\ell_{1},\ldots,\ell_{k}\\}\cup\\{m_{1},\ldots,m_{k}\\}$, we add
$\xi_{j}$ to the previous sum. When we encounter $j=\ell_{i}$ for some
$i\in\\{1,\ldots,k\\}$, we add $\eta_{i}$. When we encounter $j=m_{i}$ for
some $i\in\\{1,\ldots,k\\}$, we subtract $\eta_{i}$ and delete this row. Here
is an illustration for $n=11$, $k=4$, $J=\\{1,4,6\\}$, $I_{1}=\\{2,8\\}$,
$I_{2}=\\{3,5\\}$, $I_{3}=\\{7,10\\}$, $I_{4}=\\{9,11\\}$:

|  | $\xi_{1}$ |  |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| $\ell_{1}=2$ | $\xi_{1}$ | $+\eta_{1}$ |  |  |  |  |  |  |  |  |  |
| $\ell_{2}=3$ | $\xi_{1}$ | $+\eta_{1}$ | $+\eta_{2}$ |  |  |  |  |  |  |  |  |
|  | $\xi_{1}$ | $+\eta_{1}$ | $+\eta_{2}$ | $+\xi_{4}$ |  |  |  |  |  |  |  |
delete | $m_{2}=5$ | $\xi_{1}$ | $+\eta_{1}$ | $+\eta_{2}$ | $+\xi_{4}$ | $-\eta_{2}$ |  |  |  |  |  |  |
|  | $\xi_{1}$ | $+\eta_{1}$ |  | $+\xi_{4}$ |  | $+\xi_{6}$ |  |  |  |  |  |
| $\ell_{3}=7$ | $\xi_{1}$ | $+\eta_{1}$ |  | $+\xi_{4}$ |  | $+\xi_{6}$ | $+\eta_{3}$ |  |  |  |  |
delete | $m_{1}=8$ | $\xi_{1}$ | $+\eta_{1}$ |  | $+\xi_{4}$ |  | $+\xi_{6}$ | $+\eta_{3}$ |  | $-\eta_{1}$ |  |  |
| $\ell_{4}=9$ | $\xi_{1}$ |  |  | $+\xi_{4}$ |  | $+\xi_{6}$ | $+\eta_{3}$ |  |  | $+\eta_{4}$ |  |
delete | $m_{3}=10$ | $\xi_{1}$ |  |  | $+\xi_{4}$ |  | $+\xi_{6}$ | $+\eta_{3}$ |  |  | $+\eta_{4}$ | $-\eta_{3}$ |
delete | $m_{4}=11$ | $\xi_{1}$ |  |  | $+\xi_{4}$ |  | $+\xi_{6}$ |  |  |  | $+\eta_{4}$ |  | $-\eta_{4}$

When we move from one row to the next one, we will always have just one new
term added to the sum, and this term can be either some $\eta_{i}$ with
$i\in\\{1,\ldots,k\\}$ which did not appear in the rows above, or some
$\xi_{j}$ with $j\in J$ which did not appear in the rows above. The
problematic rows corresponding to $j\in\\{m_{1},\ldots,m_{k}\\}$ have been
deleted, due to our use of inequality (37).

We decompose $\\{1,\ldots,\ell_{i}\\}$ into the blocks
$B_{s}=\\{j;\ell_{s-1}+1\leq j\leq\ell_{s}\\}$ for $s=1,\ldots,k$, where
$\ell_{0}=0$. Then we write

$\sum_{j=1}^{\ell_{i}}\xi_{j}=\sum_{s=1}^{i}(\xi_{\ell_{s-1}+1}\ldots+\xi_{\ell_{s}-1}+\eta_{s}).$

The sum $\sum_{j=1}^{\ell_{i}}\xi_{j}$ will always contain the term
$\eta_{i}$. (This sum may or may not contain $\eta_{s}$ for $s<i$, depending
on whether $m_{s}>\ell_{i}$ or $m_{s}<\ell_{i}$. If $m_{s}<\ell_{i}$, then one
of the blocks $B_{s+1},\ldots,B_{i}$ will contain $m_{s}$, and the term
$-\eta_{s}$ added at the position corresponding to $m_{s}$ will cancel out
with the term $\eta_{s}$ at the end of block $B_{s}$.)

Based on this procedure and using inequality (3) and Lemma 4.1 of [2], we
obtain:

$\displaystyle\|A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}K_{\mu}^{k}C_{t}^{k}\int_{T_{n}(t)}d{\bf
t}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\prod_{j\in
J}\left(\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|\mathcal{F}G(t_{j+1}-t_{j},\cdot)(z+\xi_{j})|^{2}\mu(d\xi_{j})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\prod_{i=1}^{k}\left(\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(z+\eta_{i})|\mu(d\eta_{i})\right)$
$\displaystyle\leq\frac{t^{n}}{n!}K_{\mu}^{k}C_{t}^{k}\left(\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}d{\bf
t}\right)\left(C_{t}^{2}\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\frac{1}{1+|z+\xi|^{2}}\mu(d\xi)\right)^{n-2k}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\left(C_{t}\sup_{z\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\frac{1}{1+|z+\eta|^{2}}\right)^{1/2}\mu(d\eta)\right)^{k}$
$\displaystyle=\frac{t^{n}}{n!}K_{\mu}^{n}C_{t}^{2(n-k)}\frac{2^{k}t^{n+2k}}{(n+2k)!}.$
(38)

Step 4. We prove (35). For this, we will use the decomposition:

$A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)-A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)=T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)+T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x),$
(39)

where

$\displaystyle T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(y_{j})_{j\in
J},t,x\big{)}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}\big{(}h_{{\bf
t},(x_{j})_{j\in J^{c}}}^{(t,x)}-h_{{\bf t},(x_{j})_{j\in
J^{c}}}^{(t,x)}*p_{\varepsilon}^{\otimes(n-2k)}\big{)}\big{(}(y_{j})_{j\in
J}\big{)}$ (40)
$\displaystyle\quad\quad\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}})d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d{\bf t}$ $\displaystyle
T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}\big{(}(y_{j})_{j\in
J},t,x\big{)}$
$\displaystyle=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{2k}}h_{{\bf
t},(x_{j})_{j\in J^{c}}}^{(t,x)}\big{(}(y_{j})_{j\in J}\big{)}$ (41)
$\displaystyle\quad\quad\Big{(}\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})-\prod_{i=1}^{k}(p_{2\varepsilon}*\gamma)(x_{\ell_{i}}-x_{m_{i}})\Big{)}d\big{(}(x_{j})_{j\in
J^{c}}\big{)}d{\bf t}.$

We treat separately the two terms. Recall that
$p_{2\varepsilon}*\gamma=\mathcal{F}\mu_{\varepsilon}$ where
$\mu_{\varepsilon}$ is given by (26). Similarly to Step 3, it can be proved
that the functions $T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)$ and
$T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)$ belong to
$L^{1}((\mathbb{R}^{d})^{n-2k})$ and their Fourier transforms are given by:

$\displaystyle\mathcal{F}T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}=e^{-i(\sum_{j\in J}\xi_{j})\cdot
x}\big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in J}|\xi_{j}|^{2}}\big{)}$
$\displaystyle\quad\quad\quad\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{k}}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\mu_{\varepsilon}(d\eta_{i})d{\bf
t}$ (42)
$\displaystyle\mathcal{F}T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J}\big{)}=e^{-i(\sum_{j\in J}\xi_{j})\cdot
x}\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{k}}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\Big{(}1-e^{-\varepsilon\sum_{i=1}^{k}|\eta_{i}|^{2}}\Big{)}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t}.$ (43)

(In these calculations, we have applied Lemma A.2 to the function
$p_{2\varepsilon}*\gamma=\mathcal{F}\mu_{\varepsilon}$.)

By applying Hölder’s inequality and (3) (as in Step 3), we obtain:

$\displaystyle\|T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}C_{t}^{k}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\quad\quad\int_{(\mathbb{R}^{d})^{n-2k}}\big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\big{)}^{2}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j})(\sum_{s=1}^{j}\xi_{s})|^{2}$
$\displaystyle\quad\quad\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\sum_{s=1}^{\ell_{i}}\xi_{s})|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}$
$\displaystyle\leq\frac{t^{n}}{n!}C_{t}^{2(n-k)}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\quad\quad\int_{(\mathbb{R}^{d})^{n-2k}}\big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\big{)}^{2}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}\frac{1}{1+|\sum_{s=1}^{j}\xi_{s}|^{2}}$
$\displaystyle\quad\quad\prod_{i=1}^{k}\left(\frac{1}{1+|\sum_{s=1}^{\ell_{i}}\xi_{s}|^{2}}\right)^{1/2}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}$ (44)

and

$\displaystyle\|T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}C_{t}^{k}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\quad\quad\int_{(\mathbb{R}^{d})^{n-2k}}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j})(\sum_{s=1}^{j}\xi_{s})|^{2}\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\sum_{s=1}^{\ell_{i}}\xi_{s})|$
$\displaystyle\quad\quad\Big{(}1-e^{-\varepsilon\sum_{i=1}^{k}|\eta_{i}|^{2}}\Big{)}^{2}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}$
$\displaystyle\leq\frac{t^{n}}{n!}C_{t}^{2(n-k)}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}$
$\displaystyle\quad\quad\int_{(\mathbb{R}^{d})^{n-2k}}\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}\frac{1}{1+|\sum_{s=1}^{j}\xi_{s}|^{2}}\prod_{i=1}^{k}\left(\frac{1}{1+|\sum_{s=1}^{\ell_{i}}\xi_{s}|^{2}}\right)^{1/2}$
$\displaystyle\quad\quad\Big{(}1-e^{-\varepsilon\sum_{i=1}^{k}|\eta_{i}|^{2}}\Big{)}^{2}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}.$

By the dominated convergence theorem, both
$\|T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}$
and
$\|T_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}$
converge to 0 as $\varepsilon\downarrow 0$, uniformly in
$(t,x)\in[0,T]\times\mathbb{R}^{d}$. The application of this theorem is
justified by (38).

This proves (35) and concludes the proof of the theorem. ∎

###  3.3 Convergence as $n\to\infty$

In this section, we examine the two vertical sides of the diagram above,
involving the convergence in $n$.

######  Theorem 3.3.

If Assumption A and condition (C) hold, then for any $p\geq 2$ and $T>0$, the
following limit

$v(t,x):=\lim_{n\to\infty}v_{n}(t,x)\quad\mbox{exists in $L^{p}(\Omega)$,}$

uniformly in $[0,T]\times\mathbb{R}^{d}$, where $v_{n}(t,x)$ is defined in
Theorem 3.2.

###### Proof.

We will prove that $\\{v_{n}(t,x)\\}_{n\geq 0}$ is a Cauchy sequence in
$L^{p}(\Omega)$, uniformly in $(t,x)\in[0,T]\times\mathbb{R}^{d}$. For this,
it is enough to prove that

$\sum_{n\geq
0}\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\|H_{n}(t,x)\|_{p}<\infty.$ (45)

Recall definition (32) of $H_{n}(t,x)$. By Minkowski’s inequality and
hypercontractivity,

$\displaystyle\|H_{n}(t,x)\|_{p}\leq$ $\displaystyle\quad\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}(p-1)^{\frac{n-2k}{2}}\big{(}(n-2k)!\big{)}^{1/2}\|A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}.$

We now use (38). Since there are $\binom{n}{2k}$ sets $J\subset[n]$ with
$|J|=n-2k$, and for each set $J$ there are $\frac{(2k)!}{2^{k}k!}$ (unordered)
partitions $\\{I_{1},\ldots,I_{k}\\}$ of $J^{c}$ with $|I_{i}|=2$ for all
$i=1,\ldots,k$,

$\displaystyle\|H_{n}(t,x)\|_{p}$ $\displaystyle\leq\sum_{k=0}^{\lfloor
n/2\rfloor}\frac{n!}{(n-2k)!\,2^{k}k!}(p-1)^{\frac{n-2k}{2}}\big{(}(n-2k)!\big{)}^{1/2}\left[\frac{t^{n}}{n!}K_{\mu}^{n}C_{t}^{2(n-k)}\frac{2^{k}t^{n+2k}}{(n+2k)!}\right]^{1/2}$
$\displaystyle\leq C_{p,\mu,t}^{n/2}\sum_{k=0}^{\lfloor
n/2\rfloor}\frac{(2n)!}{(n-2k)!\,k!\,(n+k)!}\cdot\frac{n!}{(2n)!}\cdot\left(\frac{(n-2k)!}{n!}\right)^{1/2}\big{(}(n+k)!\big{)}^{1/2},$

where $C_{p,\mu,t}=(p-1)K_{\mu}C_{t}^{2}t(t\vee 1)^{3/2}$ and we used the fact
that $(n+2k)!\geq(n+k)!$. By Stirling’s formula, $(2n)!\geq
c_{1}^{-n}(n!)^{2}$ and $(n+k)!=\Gamma(n+k+1)\leq\Gamma(\frac{3}{2}n+1)\leq
c_{2}^{n}(n!)^{3/2}$, where $c_{1}>0$ and $c_{2}>0$ are positive constants.
Since $(n-2k)!\leq n!$, we obtain:

$\displaystyle\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\|H_{n}(t,x)\|_{p}$
$\displaystyle\leq C_{p,\mu,T}^{n/2}\sum_{k=0}^{\lfloor
n/2\rfloor}\binom{2n}{n-2k,k,n+k}\cdot\frac{c_{1}^{n}}{n!}\cdot
c_{2}^{n/2}(n!)^{3/4}$ $\displaystyle\leq
C_{p,\mu,T}^{n/2}\,c_{1}^{n}c_{2}^{n/2}\frac{1}{(n!)^{1/4}}3^{2n}.$

This proves (45). ∎

We now prove the uniform convergence in $\varepsilon$ which was mentioned in
Section 2.

######  Theorem 3.4.

If Assumption A and condition (C) holds, then for any $p\geq 2$ and $T>0$,

$\lim_{n\to\infty}v_{n}^{\varepsilon}(t,x)=v^{\varepsilon}(t,x)\ \mbox{in
$L^{p}(\Omega)$},$

uniformly in $[0,T]\times\mathbb{R}^{d}$ and $\varepsilon>0$, where
$v^{\varepsilon}$ is the solution of equation (8).

###### Proof.

As in Theorem 3.3, it is enough to prove that

$\sum_{n\geq
0}\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\sup_{\varepsilon>0}\|H_{n}^{\varepsilon}(t,x)\|_{p}<\infty.$

Recall decomposition (34) of $H_{n}^{\varepsilon}(t,x)$ and the fact that

$H_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)=I_{n-2k}\big{(}A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\big{)}.$

Note that $A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)$ has the same
expression as $T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)$ (defined
by (40)), but with $h_{{\bf t},(x_{j})_{j\in J^{c}}}^{(t,x)}-h_{{\bf
t},(x_{j})_{j\in J^{c}}}^{(t,x)}*p_{\varepsilon}^{\otimes(n-2k)}$ replaced by
$h_{{\bf t},(x_{j})_{j\in J^{c}}}^{(t,x)}*p_{\varepsilon}^{\otimes(n-2k)}$.
Similarly to (44), we have:

$\displaystyle\|A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}C_{t}^{2(n-k)}K_{\mu}^{k}\int_{T_{n}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}\int_{(\mathbb{R}^{d})^{n-2k}}e^{-\varepsilon\sum_{j\in
J}|\xi_{j}|^{2}}$ $\displaystyle\quad\int_{(\mathbb{R}^{d})^{k}}\prod_{j\in
J}\frac{1}{1+|\sum_{s=1}^{j}\xi_{s}|^{2}}\prod_{i=1}^{k}\left(\frac{1}{1+|\sum_{s=1}^{\ell_{i}}\xi_{s}|^{2}}\right)^{1/2}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu_{\varepsilon}(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}.$

Using the bound $e^{-x}\leq 1$ for $x\geq 0$, and proceeding as for (38), we
obtain:

$\|A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k)}}^{2}\leq\frac{t^{n}}{n!}K_{\mu}^{n}C_{t}^{2(n-k)}\frac{2^{k}t^{n+2k}}{(n+2k)!}.$

The rest of the proof is the same as for Theorem 3.3. ∎

###  3.4 Final step: $v$ is Stratonovich solution

In this section, we give the proof of the last statement in Theorem 1.2.

######  Theorem 3.5.

If Assumption A and condition (C) hold, then $v$ is a Stratonovich solution to
equation (1), i.e. for any $t>0$ and $x\in\mathbb{R}^{d}$,

$v(t,x)\stackrel{{\scriptstyle P}}{{=}}\lim_{\varepsilon\downarrow
0}\left(1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v(s,y)\dot{W}^{\varepsilon}(y)dyds\right).$

###### Proof.

Step 1. Since $v^{\varepsilon}$ satisfies (13), we have the decomposition

$T^{\varepsilon}(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v(s,y)\dot{W}^{\varepsilon}(y)dyds=v^{\varepsilon}(t,x)+L^{\varepsilon}(t,x),$
(46)

where

$L^{\varepsilon}(t,x)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\big{(}v(s,y)-v^{\varepsilon}(s,y)\big{)}\dot{W}^{\varepsilon}(y)dyds.$

Since $v^{\varepsilon}(t,x)\to v(t,x)$ in $L^{p}(\Omega)$ as
$\varepsilon\downarrow 0$, it suffices to prove that
$L^{\varepsilon}(t,x)\stackrel{{\scriptstyle P}}{{\to}}0$ as
$\varepsilon\downarrow 0$. In fact, we will show that:

$\mathbb{E}|L^{\varepsilon}(t,x)|\to 0\quad{\mbox{a}s}\ \varepsilon\downarrow
0.$ (47)

Step 2. We calculate $L^{\varepsilon}(t,x)$. Note that

$v(s,y)-v^{\varepsilon}(s,y)=\sum_{n\geq
1}\big{(}H_{n}(s,y)-H_{n}^{\varepsilon}(s,y)\big{)}.$

We use relations (32) and (34) for expressing $H_{n}(s,y)$ and
$H_{n}^{\varepsilon}(s,y)$. We obtain:

$\displaystyle v(s,y)-v^{\varepsilon}(s,y)=\sum_{n\geq 1}\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k}\big{(}B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\big{)},$

where
$B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)=A_{n,k,J,I_{1},\ldots,I_{k}}(\cdot,s,y)-A_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)$.

We multiply the last equation on display by
$\dot{W}^{\varepsilon}(y)=I_{1}\big{(}p_{\varepsilon}(y-\cdot)\big{)}$. We
obtain:

$\displaystyle\big{(}v(s,y)-v^{\varepsilon}(s,y)\big{)}\dot{W}^{\varepsilon}(y)=$
$\displaystyle\quad\sum_{n\geq 1}\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k}\big{(}B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\big{)}I_{1}\big{(}p_{\varepsilon}(y-\cdot)\big{)}.$
(48)

We apply the product formula (60) for computing the product of the two Wiener
integrals:

$\displaystyle
I_{n-2k}\big{(}B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\big{)}I_{1}\big{(}p_{\varepsilon}(y-\cdot)\big{)}$
$\displaystyle=I_{n-2k+1}\big{(}\widetilde{B}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\otimes
p_{\varepsilon}(y-\cdot)\big{)}$
$\displaystyle+(n-2k)I_{n-2k-1}\big{(}\widetilde{B}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\otimes_{1}p_{\varepsilon}(y-\cdot)\big{)},$

where $\widetilde{B}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)$ is
the symmetrization of $B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)$.

Note that for any functions $f\in\mathcal{H}^{\otimes m}$ and
$g\in\mathcal{H}$,

$I_{m}(\widetilde{f}\otimes g)=I_{m}(f\otimes g),$

since the symmetrization of $\widetilde{f}\otimes g$ coincides with the
symmetrization of $f\otimes g$, and

$mI_{m-1}(\widetilde{f}\otimes_{1}g)=\sum_{\ell=1}^{m}\int_{(\mathbb{R}^{d})^{m-1}}\langle
f(x_{1},\ldots,x_{\ell},\cdot,x_{\ell+1},\ldots,x_{m}),g\rangle_{\mathcal{H}}\prod_{\begin{subarray}{c}j=1\\\
j\not=\ell\end{subarray}}^{m}W(dx_{j}).$

Hence,

$\displaystyle
I_{n-2k}\big{(}B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\big{)}I_{1}\big{(}p_{\varepsilon}(y-\cdot)\big{)}=I_{n-2k+1}\big{(}B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\otimes
p_{\varepsilon}(y-\cdot)\big{)}+$ $\displaystyle\quad\quad\quad\sum_{\ell\in
J}\int_{(\mathbb{R}^{d})^{n-2k-1}}\langle
B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}((x_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}},\cdot,s,y),p_{\varepsilon}(y-\cdot)\rangle_{\mathcal{H}}\prod_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}^{m}W(dx_{j}),$ (49)

where
$B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}((x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}},\cdot,s,y)$ denotes the function
$x_{\ell}\mapsto B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}((x_{j})_{j\in
J},s,y)$.

We introduce expression (49) into (48). Then we multiply by $G(t-s,x-y)$ and
we integrate $dsdy$ on $[0,t]\times\mathbb{R}^{d}$. Using stochastic Fubini’s
theorem for interchanging the multiple Wiener integral with the $dsdy$
integral, we obtain that:

$\displaystyle L^{\varepsilon}(t,x)$ $\displaystyle=\sum_{n\geq
1}\sum_{k=0}^{\lfloor n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\Big{(}\mathcal{I}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)+\mathcal{J}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)\Big{)},$

where

$\displaystyle\mathcal{I}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n-2k+1}}F_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},z,t,x\big{)}\prod_{j\in J}W(dx_{j})W(dz)$ $\displaystyle
F_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},z,t,x\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}((x_{j})_{j\in
J},s,y)p_{\varepsilon}(y-z)dyds$

and

$\displaystyle\mathcal{J}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=\sum_{\ell\in
J}\mathcal{J}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)$
$\displaystyle\mathcal{J}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n-2k-1}}Q_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}},t,x\big{)}\prod_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}W(dx_{j})$ $\displaystyle
Q_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}},t,x\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\langle
B_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}},\cdot,s,y\big{)},p_{\varepsilon}(y-\cdot)\rangle_{\mathcal{H}}dyds.$

Since $B^{\varepsilon}=T^{\varepsilon}+T^{\prime\varepsilon}$ (see (39)), we
write $F^{\varepsilon}=G^{\varepsilon}+G^{\prime\varepsilon}$ and
$Q^{\varepsilon,\ell}=R^{\varepsilon,\ell}+R^{\prime\varepsilon,\ell}$, where

$\displaystyle G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},z,t,x\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},s,y\big{)}p_{\varepsilon}(y-z)dyds$ $\displaystyle
R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}},t,x\big{)}$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\langle
T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}},\cdot,s,y\big{)},p_{\varepsilon}(y-\cdot)\rangle_{\mathcal{H}}dyds,$

and $G^{\prime\varepsilon},R^{\prime\varepsilon,\ell}$ have the same form as
$G^{\varepsilon}$, respectively $R^{\varepsilon}$, but with $T^{\varepsilon}$
replaced by $T^{\prime\varepsilon}$.

We obtain the decomposition:

$\displaystyle L^{\varepsilon}(t,x)$ $\displaystyle=\sum_{n\geq
1}\sum_{k=0}^{\lfloor n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\Big{(}\mathcal{G}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)+\mathcal{G}_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(t,x)+$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)+\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(t,x)\Big{)}$
$\displaystyle=:L_{1}^{\varepsilon}(t,x)+L_{2}^{\varepsilon}(t,x)+L_{3}^{\varepsilon}(t,x)+L_{4}^{\varepsilon}(t,x),$

where

$\displaystyle\mathcal{G}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n-2k+1}}G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},z,t,x\big{)}\prod_{j\in J}W(dx_{j})W(dz)$
$\displaystyle\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)$
$\displaystyle=\sum_{\ell\in
J}\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)$
$\displaystyle\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)$
$\displaystyle=\int_{(\mathbb{R}^{d})^{n-2k-1}}R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}},t,x\big{)}\prod_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}W(dx_{j}),$

$\mathcal{G}^{\prime\varepsilon}$ has the same form as
$\mathcal{G}^{\varepsilon}$ but with $G^{\varepsilon}$ replaced by
$G^{\prime\varepsilon}$, $\mathcal{R}^{\prime\varepsilon}=\sum_{\ell\in
J}\mathcal{R}^{\prime\varepsilon,\ell}$ and
$\mathcal{R}^{\prime\varepsilon,\ell}$ has the same form as
$\mathcal{R}^{\varepsilon,\ell}$ but with $R^{\varepsilon,\ell}$ replaced by
$R^{\prime\varepsilon,\ell}$.

We will prove that

$\mathbb{E}|L_{i}^{\varepsilon}(t,x)|\to 0\quad\mbox{as}\
\varepsilon\downarrow 0,\quad\mbox{for}\ i=1,2,3,4.$ (50)

Step 3. We treat $L_{1}^{\varepsilon}(t,x)$. We proceed similarly to Step 3 of
the proof of Theorem 3.2. Note that the function
$G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\in
L^{1}((\mathbb{R}^{d})^{n-2k+1})$. Letting $t_{n+1}=s$ and ${\bf
t}=(t_{1},\ldots,t_{n+1})$, it can be proved that

$\displaystyle\mathcal{F}G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J},\eta\big{)}=e^{-i(\sum_{j\in J}\xi_{j})\cdot x}e^{-i\eta\cdot
x}e^{-\frac{\varepsilon}{2}|\eta|^{2}}\Big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\Big{)}$
$\displaystyle\quad\int_{T_{n+1}(t)}\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{j\in
J}\xi_{j}+\eta\big{)}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}$
$\displaystyle\qquad\qquad\qquad\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu_{\varepsilon}(d\eta_{i})d{\bf
t}.$

Applying Hölder’s inequality as above, and using the estimate (37) for the
terms corresponding to $m_{i}$, we obtain that:

$\displaystyle\|G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k+1)}}^{2}=\int_{(\mathbb{R}^{d})^{n-2k+1}}|\mathcal{F}G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J},\eta\big{)}|^{2}\prod_{j\in J}\mu(d\xi_{j})\mu(d\eta)$
$\displaystyle\quad\leq
C_{t}^{k}K_{\mu}^{k}\frac{t^{n+1}}{(n+1)!}\int_{T_{n+1}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}\int_{(\mathbb{R}^{d})^{n-2k+1}}e^{-\varepsilon|\eta|^{2}}\Big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\Big{)}^{2}$
$\displaystyle\quad\quad\quad\int_{(\mathbb{R}^{d})^{k}}|\mathcal{F}G(t-t_{n+1},\cdot)(\sum_{j\in
J}\xi_{j}+\eta)|^{2}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j},\cdot)(\sum_{s=1}^{j}\xi_{s})|^{2}$
$\displaystyle\qquad\qquad\quad\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)(\sum_{s=1}^{\ell_{i}}\xi_{s})|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\prod_{j\in
J}\mu(d\xi_{j})d{\bf t}.$

The last integral converges to $0$ as $\varepsilon\downarrow 0$, by the
dominated convergence theorem. Hence,

$\displaystyle\mathbb{E}|L_{1}^{\varepsilon}(t,x)|\leq\sum_{n\geq
1}\sum_{k=0}^{\lfloor n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\Big{(}\mathbb{E}|\mathcal{G}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)|^{2}\Big{)}^{1/2}$
$\displaystyle\quad\leq\sum_{n\geq 1}\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\big{(}(n-2k+1)!\big{)}^{1/2}\|G_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(t,x)\|_{\mathcal{H}^{\otimes(n-2k+1)}}.$

The last series converges to $0$ as $\varepsilon\downarrow 0$, again by the
dominated convergence theorem. The application of this theorem is justified as
in the proof of Theorem 3.3.

Step 4. We treat $L_{2}^{\varepsilon}(t,x)$. It can be proved that the
function $G_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)$ belongs
to $L^{1}((\mathbb{R}^{d})^{n-2k+1})$ and its Fourier transform is

$\displaystyle\mathcal{F}G_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon}(\cdot,t,x)\big{(}(\xi_{j})_{j\in
J},\eta\big{)}=e^{-i(\sum_{j\in J}\xi_{j})\cdot x}e^{-i\eta\cdot
x}e^{-\frac{\varepsilon}{2}|\eta|^{2}}$
$\displaystyle\quad\int_{T_{n+1}(t)}\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{j\in
J}\xi_{j}+\eta\big{)}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)(\sum_{s=1}^{j}\xi_{s})$
$\displaystyle\qquad\qquad\qquad\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\Big{(}1-e^{-\varepsilon\sum_{i=1}^{k}|\eta_{i}|^{2}}\Big{)}\prod_{i=1}^{k}\mu(d\eta_{i})d{\bf
t},$

where ${\bf t}=(t_{1},\ldots,t_{n+1})$. As in Step 4, one can show that
$\mathbb{E}|L_{2}^{\varepsilon}(t,x)|\to 0$ as $\varepsilon\downarrow 0$.

Step 5. We treat $L_{3}^{\varepsilon}(t,x)$. Expressing the
$\mathcal{H}$-inner product in terms of Fourier transforms, we see that

$\displaystyle
R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}},t,x\big{)}=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\int_{\mathbb{R}^{d}}e^{i\xi_{\ell}\cdot
y}e^{-\frac{\varepsilon}{2}|\xi_{\ell}|^{2}}$
$\displaystyle\qquad\qquad\qquad\left(\int_{\mathbb{R}^{d}}e^{-i\xi_{\ell}\cdot
x_{l}}T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}\big{(}(x_{j})_{j\in
J},s,y\big{)}dx_{\ell}\right)\mu(d\xi_{\ell})dyds.$

Therefore,

$\displaystyle\mathcal{F}R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}}\big{)}=\int_{(\mathbb{R}^{d})^{n-2k-1}}e^{-i\sum_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\xi_{j}\cdot
x_{j}}R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}},t,x\big{)}d\big{(}(x_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\big{)}$
$\displaystyle\quad=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)\int_{\mathbb{R}^{d}}\mathcal{F}T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)\big{(}(\xi_{j})_{j\in
J}\big{)}e^{i\xi_{\ell}\cdot
y}e^{-\frac{\varepsilon}{2}|\xi_{\ell}|^{2}}\mu(d\xi_{\ell})dyds.$

We now insert formula (42) for
$\mathcal{F}T_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon}(\cdot,s,y)$. We use the
fact that

$\int_{\mathbb{R}^{d}}e^{-i\big{(}\sum_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\xi_{j}\big{)}\cdot
y}G(t-s,x-y)dy=e^{-i\big{(}\sum_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\xi_{j}\big{)}\cdot
x}\mathcal{F}G(t-s,\cdot)\big{(}\sum_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\xi_{j}\big{)}.$

Changing the notation $t_{n+1}=s$ and denoting ${\bf
t}=(t_{1},\ldots,t_{n+1})$, we obtain that

$\displaystyle\mathcal{F}R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}}\big{)}=e^{-i\big{(}\sum_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\xi_{j}\big{)}\cdot
x}\int_{T_{n+1}(t)}\int_{(\mathbb{R}^{d})^{k+1}}e^{-\frac{\varepsilon}{2}|\xi_{\ell}|^{2}}\Big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\Big{)}$ $\displaystyle\
\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\xi_{j}\big{)}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu_{\varepsilon}(d\eta_{i})\mu(d\xi_{\ell})d{\bf
t}.$

By applying Hölder’s inequality, (37) and inequality $e^{-x}\leq 1$ for $x\geq
0$, it can be proved that

$\displaystyle\|R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(\cdot,t,x)\|_{\mathcal{H}^{\otimes(n-2k-1)}}^{2}=\int_{(\mathbb{R}^{d})^{n-2k-1}}|\mathcal{F}R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\big{)}|^{2}\prod_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\mu(d\xi_{j})$ $\displaystyle\quad\leq
C_{t}^{k+1}K_{\mu}^{k+1}\frac{t^{n+1}}{(n+1)!}\int_{T_{n+1}(t)}\prod_{i=1}^{k}(t_{m_{i}+1}-t_{m_{i}})^{2}\int_{(\mathbb{R}^{d})^{n-2k-1}}\int_{(\mathbb{R}^{d})^{k+1}}e^{-\varepsilon|\xi_{\ell}|^{2}}$
$\displaystyle\quad\quad\quad\Big{(}1-e^{-\frac{\varepsilon}{2}\sum_{j\in
J}|\xi_{j}|^{2}}\Big{)}^{2}|\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\xi_{j}\big{)}|^{2}\prod_{j\in
J}|\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}|^{2}$
$\displaystyle\quad\quad\quad\prod_{i=1}^{k}|\mathcal{F}G(t_{\ell_{i}+1}-t_{\ell_{i}},\cdot)\big{(}\sum_{s=1}^{\ell_{i}}\xi_{s}\big{)}|\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\mu(d\xi_{\ell})\prod_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\mu(d\xi_{j})d{\bf t}.$

The last integral converges to $0$ by the dominated convergence theorem.
Hence,

$\displaystyle\mathbb{E}|L_{3}^{\varepsilon}(t,x)|\leq\sum_{n\geq
1}\sum_{k=0}^{\lfloor n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\sum_{\ell\in
J}\Big{(}\mathbb{E}|\mathcal{R}_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)|^{2}\Big{)}^{1/2}$
$\displaystyle\quad\leq\sum_{n\geq 1}\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}\sum_{\ell\in
J}\big{(}(n-2k-1)!\big{)}^{1/2}\|R_{n,k,J,I_{1},\ldots,I_{k}}^{\varepsilon,\ell}(t,x)\|_{\mathcal{H}^{\otimes(n-2k+1)}}.$

The last series converges to $0$ as $\varepsilon\downarrow 0$, again by the
dominated convergence theorem.

Step 6. We treat $L_{4}^{\varepsilon}(t,x)$. Similarly to Step 5, it can be
proved that

$\displaystyle\mathcal{F}R_{n,k,J,I_{1},\ldots,I_{k}}^{\prime\varepsilon,\ell}(\cdot,t,x)\big{(}(\xi_{j})_{\begin{subarray}{c}j\in
J\\\
j\not=\ell\end{subarray}}\big{)}=e^{-i\big{(}\sum_{\begin{subarray}{c}j\in
J\\\ j\not=\ell\end{subarray}}\xi_{j}\big{)}\cdot
x}\int_{T_{n+1}(t)}\int_{(\mathbb{R}^{d})^{k+1}}e^{-\frac{\varepsilon}{2}|\xi_{\ell}|^{2}}\Big{(}1-e^{-\varepsilon\sum_{i=1}^{k}|\eta_{i}|^{2}}\Big{)}$
$\displaystyle\
\mathcal{F}G(t-t_{n+1},\cdot)\big{(}\sum_{\begin{subarray}{c}j\in J\\\
j\not=\ell\end{subarray}}\xi_{j}\big{)}\prod_{j=1}^{n}\mathcal{F}G(t_{j+1}-t_{j},\cdot)\big{(}\sum_{s=1}^{j}\xi_{s}\big{)}\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i})\mu(d\xi_{\ell})d{\bf
t}$

and $\mathbb{E}|L_{4}^{\varepsilon}(t,x)|\to 0$ as $\varepsilon\downarrow 0$.

This concludes the proof of (50) and the proof of the theorem. ∎

##  4 Feynman-Kac-type formula

In this section, we give the proof of Theorem 1.3. First, we introduce the
notation:

$\displaystyle\mathbb{E}^{N,X}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{n}\dot{W}(X_{\tau_{i}}^{x})\Big{]}:=$
(51)
$\displaystyle\quad\quad\mathbb{E}^{N}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1})\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}\frac{G(\tau_{i}-\tau_{i-1},x_{i}-x_{i-1})}{\tau_{i}-\tau_{i-1}}\prod_{i=1}^{n}W^{\circ}(dx_{i})\Big{]},$

where $\tau_{0}=0$, $x_{0}=x$. This notation is motivated by the formal
relation
$\prod_{i=1}^{n}W^{\circ}(dx_{i})=\prod_{i=1}^{n}\dot{W}(x_{i})dx_{1}\ldots
dx_{n}$ (see (29)) and a formal application of Lemma 2.4 (similar to the proof
of Lemma 2.3), with $X=(X_{\tau_{1}}^{x},\ldots,X_{\tau_{n}}^{x})$, $Y=N$,
$Z=W$,

$g(x_{1},\ldots,x_{n},W)=\prod_{i=1}^{n}\dot{W}(x_{i})\quad\mbox{and}\quad
h(N)=1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1}),$

We need an auxiliary result.

######  Lemma 4.1.

For any random variable $Y:\Omega\to\mathcal{Y}$ and a non-negative measurable
function $f$ on $(\mathbb{R}^{d})^{n}\times\mathcal{Y}$, such that
$f(\cdot,y)\in\mathcal{H}^{\otimes n}$ for any $y\in\mathcal{Y}$ and
$\mathbb{E}[f(\cdot,Y)]\in\mathcal{H}^{\otimes n}$, we have:

$\int_{\mathbb{R}^{2d}}\mathbb{E}[f(x_{1},\ldots,y_{n},Y)]\prod_{i=1}^{n}W^{\circ}(dx_{i})=\mathbb{E}^{Y}\left[\int_{\mathbb{R}^{2d}}f(x_{1},\ldots,x_{n},Y)\prod_{i=1}^{n}W^{\circ}(dx_{i})\right].$
(52)

###### Proof.

We first examine the case $n=1$. Recall that (for deterministic integrands),
$W^{\circ}(dx)=W(dx)$. Let $\mu_{Y}$ be the law of $Y$. By the stochastic
Fubini theorem,

$\displaystyle\mathbb{E}^{Y}\left[\int_{\mathbb{R}^{d}}f(x,Y)W(dx)\right]$
$\displaystyle=\int_{\mathcal{Y}}\left(\int_{\mathbb{R}^{d}}f(x,y)W(dx)\right)\mu_{Y}(dy)$
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\int_{\mathcal{Y}}f(x,y)\mu_{Y}(dy)\right)W(dx)=\int_{\mathbb{R}^{d}}\mathbb{E}[f(x,Y)]W(dx).$
(53)

The general case is proved in the same way, using linearity. For instance, for
$n=2$, we use the fact that
$W^{\circ}(dx_{1})W^{\circ}(dx_{2})=W(dx_{1})W(dx_{2})+\gamma(x_{1}-x_{2})dx_{1}dx_{2}$.
∎

The proof of Theorem 1.3 follows from the following result.

######  Proposition 4.2.

If Assumption A and condition (C) hold, for any $n\geq 1$, $t>0$,
$x\in\mathbb{R}^{d}$, with probability 1,

$H_{n}(t,x)=\mathbb{E}^{N,X}\Big{[}1_{\\{N_{t}=n\\}}\prod_{i=1}^{n}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{n}\dot{W}(X_{\tau_{i}}^{x})\Big{]},$

where the right-hand-side is defined by (51).

###### Proof.

We treat only the cases $n=1$ and $n=2$. The general case is similar.

Let $n=1$. Using the Poisson representation (19) and relation (53) (with
$Y=N$),

$\displaystyle H_{1}(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-t_{1},x-x_{1})W(dx_{1})dt_{1}=\int_{\mathbb{R}^{d}}\left(\int_{0}^{t}G(s_{1},x_{1}-x)ds_{1}\right)W(dx_{1})$
$\displaystyle=e^{t}\int_{\mathbb{R}^{d}}\mathbb{E}\big{[}G(\tau_{1},x_{1}-x)1_{\\{N_{t}=1\\}}\big{]}W(dx_{1})=e^{t}\mathbb{E}^{N}\left[\int_{\mathbb{R}^{d}}G(\tau_{1},x_{1}-x)1_{\\{N_{t}=1\\}}W(dx_{1})\right]$
$\displaystyle=e^{t}\mathbb{E}^{N}\left[1_{\\{N_{t}=1\\}}\tau_{1}\int_{\mathbb{R}^{d}}\frac{G(\tau_{1},x_{1}-x)}{\tau_{1}}W(dx_{1})\right]$

If $n=2$, we use the change of variables $s_{1}=t-t_{2},s_{2}=t-t_{1}$,
$x_{1}^{\prime}=x_{2}$, $x_{2}^{\prime}=x_{1}$, followed by Poisson
representation (19) and relation (52) (with $Y=N$):

$\displaystyle H_{2}(t,x)$
$\displaystyle=\int_{T_{2}(t)}\int_{\mathbb{R}^{2d}}G(t-t_{2},x-x_{2})G(t_{2}-t_{1},x_{2}-x_{1})W^{\circ}(dx_{1})W^{\circ}(dx_{2})d{\bf
t}$
$\displaystyle=\int_{\mathbb{R}^{2d}}\left(\int_{T_{2}(t)}G(s_{1},x_{1}^{\prime}-x)G(s_{2}-s_{1},x_{2}^{\prime}-x_{1}^{\prime})d{\bf
s}\right)W^{\circ}(dx_{2}^{\prime})W^{\circ}(dx_{1}^{\prime})$
$\displaystyle=e^{t}\int_{\mathbb{R}^{2d}}\mathbb{E}\big{[}G(\tau_{1},x_{1}-x)G(\tau_{2}-\tau_{1},x_{2}-x_{1})1_{\\{N_{t}=2\\}}\big{]}W^{\circ}(dx_{1})W^{\circ}(dx_{2})$
$\displaystyle=e^{t}\mathbb{E}^{N}\left[\int_{\mathbb{R}^{2d}}G(\tau_{1},x_{1}-x)G(\tau_{2}-\tau_{1},x_{2}-x_{1})1_{\\{N_{t}=2\\}}W^{\circ}(dx_{1})W^{\circ}(dx_{2})\right]$
$\displaystyle=e^{t}\mathbb{E}^{N}\left[1_{\\{N_{t}=2\\}}\tau_{1}(\tau_{2}-\tau_{1})\int_{\mathbb{R}^{2d}}\frac{G(\tau_{1},x_{1}-x)}{\tau_{1}}\frac{G(\tau_{2}-\tau_{1},x_{2}-x_{1})}{\tau_{2}-\tau_{1}}W^{\circ}(dx_{1})W^{\circ}(dx_{2})\right].$

∎

##  5 Smooth noise case

In this section, we consider the case of a smooth noise $W$ with a finite
spectral measure $\mu$. In this case,
$\gamma(x)=\int_{\mathbb{R}^{d}}e^{-i\xi\cdot x}\mu(d\xi)$ for any
$x\in\mathbb{R}^{d}$, and we do not need Assumption A.

Since $\mu(\mathbb{R}^{d})<\infty$, the sequence
$\\{p_{\varepsilon}(x-\cdot)\\}_{\varepsilon>0}$ is Cauchy in $\mathcal{H}$,
uniformly in $x\in\mathbb{R}^{d}$:

$\|p_{\varepsilon}(x-\cdot)-p_{\varepsilon^{\prime}}(x-\cdot)\|_{\mathcal{H}}^{2}=\int_{\mathbb{R}^{2}}\left(e^{-\varepsilon|\xi|^{2}/2}-e^{-\varepsilon^{\prime}|\xi|^{2}/2}\right)^{2}\mu(d\xi)\to
0\quad\mbox{as $\varepsilon,\varepsilon^{\prime}\to 0$}.$

We denote its limit by $A(x)$. Then $\lim_{\varepsilon\downarrow
0}\|p_{\varepsilon}(x-\cdot)-A(x)\|_{\mathcal{H}}=0$ uniformly in
$x\in\mathbb{R}^{d}$, and

$\|A(x)\|_{\mathcal{H}}^{2}=\lim_{\varepsilon\downarrow
0}\|p_{\varepsilon}(x-\cdot)\|_{\mathcal{H}}^{2}=\lim_{\varepsilon\downarrow
0}\int_{\mathbb{R}^{d}}e^{-\varepsilon|\xi|^{2}}\mu(d\xi)=\mu(\mathbb{R}^{d}).$

The random variable

$\dot{W}(x):=W(A(x))$

is well-defined and $\dot{W}^{\varepsilon}(x)\to\dot{W}(x)$ in
$L^{2}(\Omega)$, as $\varepsilon\downarrow 0$, uniformly in $x$. For any
$p>0$,

$\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{p}=z_{p}(\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{2})^{p/2}\leq
z_{p}(\mu(\mathbb{R}^{d}))^{p/2}$ (54)
$\mathbb{E}|\dot{W}(x)|^{p}=z_{p}(\mathbb{E}|\dot{W}(x)|^{2})^{p/2}=z_{p}(\mu(\mathbb{R}^{d}))^{p/2}$

######  Remark 5.1.

The Gaussian process $\\{\dot{W}(x)\\}_{x\in\mathbb{R}^{d}}$ has stationary
increments (in Yaglom sense) and covariance function:

$\mathbb{E}|\dot{W}(x)-\dot{W}(y)|^{2}=\int_{\mathbb{R}^{d}}(1-e^{i\xi\cdot(x-y)})^{2}\mu(d\xi).$

This process is not the same as the mean-square derivative process
$W^{\prime}$ of the random field $W(x)=W(1_{[0,x]})$, which is defined as the
limit $W^{\prime}(x)=\lim_{h\to 0}\frac{W(x+h)-W(x)}{h}$ in $L^{2}(\Omega)$.
(This limit exists if $\mu(\mathbb{R}^{d})<\infty$.) To see this, it suffices
to note that $\\{W^{\prime}(x)\\}_{x\in\mathbb{R}^{d}}$ is a stationary
Gaussian process with zero-mean and covariance

$\mathbb{E}[W^{\prime}(x)W^{\prime}(y)]=\int_{\mathbb{R}^{d}}e^{-i\xi\cdot(x-y)}\mu(d\xi).$

Let $\overline{v}_{n}(t,x)=1+\sum_{k=1}^{n}\overline{H}_{k}(t,x)$, where

$\overline{H}_{n}(t,x)=\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}\dot{W}(x_{i})d{\bf
x}d{\bf t}.$

Since $\mathbb{E}|\dot{W}^{\varepsilon}(x)|^{p}$ is bounded by a constant that
does not depend on $\varepsilon$, similarly to (16), we have:

$\|\overline{H}_{n}(t,x)\|_{p}\leq\frac{2^{n/2}}{\pi^{1/(2p)}}C_{p}^{n/p}\big{(}\mu(\mathbb{R}^{d})\big{)}^{n/2}c_{0}^{n}\frac{t^{2n}}{(n!)^{3/2}}.$

and the convergence in Lemma 2.2 is uniform in $\varepsilon$. Moreover, the
proof of Lemma 2.2 can be repeated with $\dot{W}^{\varepsilon}$ replaced by
$\dot{W}$, to infer that $\\{\overline{v}_{n}(t,x)\\}_{n}$ is a Cauchy
sequence in $L^{p}(\Omega)$, uniformly in $(t,x)\in[0,T]\times\mathbb{R}^{d}$.

The next lemma deals with the issue of convergence when $\varepsilon\downarrow
0$.

######  Lemma 5.2.

If $\mu(\mathbb{R}^{d})<\infty$, then for any $p\geq 1$, $T>0$ and $n\geq 1$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|H_{n}^{\varepsilon}(t,x)-\overline{H}_{n}(t,x)|^{p}\to
0\quad\mbox{as $\varepsilon\downarrow 0$}.$ (55)

Consequently, for any $t>0,x\in\mathbb{R}^{d}$,
$H_{n}(t,x)=\overline{H}_{n}(t,x)$ a.s. and $v_{n}(t,x)=\overline{v}_{n}(t,x)$
a.s.

###### Proof.

By Minkowski’s inequality,

$\displaystyle\|H_{n}^{\varepsilon}(t,x)-\overline{H}_{n}(t,x)\|_{p}\leq$
$\displaystyle\quad\quad\int_{T_{n}(t)}\int_{\mathbb{R}^{nd}}\prod_{i=1}^{n}G(s_{i}-s_{i-1},y_{i}-y_{i-1})\Big{\|}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x+y_{i})-\prod_{i=1}^{n}\dot{W}(x+y_{i})\Big{\|}_{p}d{\bf
y}d{\bf s},$

where $s_{0}=0$ and $y_{0}=0$. Using the inequality

$\Big{|}\prod_{i=1}^{n}a_{i}-\prod_{i=1}^{n}b_{i}\Big{|}\leq\sum_{i=1}^{n}|a_{1}\ldots
a_{i-1}b_{i+1}\ldots b_{n}||a_{i}-b_{i}|.$

for $a_{1},b_{1},\ldots,a_{n},b_{n}\in\mathbb{R}$, followed by the generalized
Hölder inequality, we obtain:

$\displaystyle\mathbb{E}\Big{|}\prod_{i=1}^{n}\dot{W}^{\varepsilon}(x+y_{i})-\prod_{i=1}^{n}\dot{W}(x+y_{i})\Big{|}^{p}\leq$
$\displaystyle\quad
n^{p-1}\sum_{i=1}^{n}(\mathbb{E}|\dot{W}^{\varepsilon}(x+y_{1})|^{pn})^{1/n}\ldots(\mathbb{E}|\dot{W}^{\varepsilon}(x+y_{i-1})|^{pn})^{1/n}(\mathbb{E}|\dot{W}(x+y_{i+1})|^{pn})^{1/n}\ldots$
$\displaystyle\quad\quad\quad(\mathbb{E}|\dot{W}(x+y_{n})|^{pn})^{1/n}(\mathbb{E}|\dot{W}^{\varepsilon}(x+y_{i})-\dot{W}(x+y_{i})|^{pn})^{1/n}\leq$
$\displaystyle\quad
n^{p-1}z_{pn}\Big{(}\mu(\mathbb{R}^{d})\Big{)}^{\frac{p(n-1)}{2}}\sum_{i=1}^{n}(\mathbb{E}|\dot{W}^{\varepsilon}(x+y_{i})-\dot{W}(x+y_{i})|^{2})^{p/2}.$

Since the right-hand-side above converges to $0$ when $\varepsilon\downarrow
0$, uniformly in $x\in\mathbb{R}^{d}$, the conclusion follows by the dominated
convergence theorem. ∎

######  Remark 5.3.

If $\mu(\mathbb{R}^{d})<\infty$, it is much easier to prove that $v$ is a
Stratonovich solution. To see this, recall decomposition (46). Then
$\mathbb{E}|L^{\varepsilon}(t,x)|\to 0$ as $\varepsilon\downarrow 0$, by the
dominated convergence theorem. To justify the application of this theorem, we
simply use the bound

$\mathbb{E}|\big{(}v^{\varepsilon}(s,y)-v(s,y)\big{)}\dot{W}^{\varepsilon}(y)|\leq\Big{(}\mathbb{E}|v^{\varepsilon}(s,y)-v(s,y)|^{2}\Big{)}^{1/2}\Big{(}\mathbb{E}|\dot{W}^{\varepsilon}(y)|^{2}\Big{)}^{1/2}.$

The last term is bounded by a constant (which does not depend on
$\varepsilon$) due to (54) and the uniform convergence in (9).

######  Remark 5.4.

If $\mu(\mathbb{R}^{d})<\infty$, $v$ satisfies also the equation:

$v(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v(s,y)\dot{W}(y)dyds.$

To see this, consider the recurrence relation (17). Taking the limit as
$\varepsilon\downarrow 0$ in $L^{1}(\Omega)$, we obtain:

$v_{n+1}(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)v_{n}(s,y)\dot{W}(y)dyds.$

Now take the limit as $n\to\infty$ in $L^{1}(\Omega)$.

######  Remark 5.5.

If $\mu(\mathbb{R}^{d})<\infty$, we have the following Feynman-Kac
representation of $v$:

$v(t,x)=e^{t}\mathbb{E}^{N,X}\Big{[}\prod_{i=1}^{N_{t}}(\tau_{i}-\tau_{i-1})\prod_{i=1}^{N_{t}}\dot{W}(X_{\tau_{i}}^{x})\Big{]}.$

which is proved similarly to (22), replacing $\dot{W}^{\varepsilon}$ by
$\dot{W}$.

##  6 Comparison with the Skorohod solution

In this section, we compare the Stratonovich solution $v$ with the Skorohod
solution $u$.

###  6.1 Chaos expansions

In this section, we give the chaos expansions of $v(t,x)$ and $u(t,x)$.

Recall that $u$ satisfies equation (5). By Theorem 2.2 of [1], we know that
under condition (D), $u$ exists and has the chaos expansion
$u(t,x)=1+\sum_{n\geq 1}J_{n}(t,x)$, where
$J_{n}(t,x)=I_{n}(f_{n}(\cdot,x;t))$ and $f_{n}(\cdot,x;t)$ is given by (6).
An important observation is that

$J_{n}(t,x)=H_{n,0,[n]}(t,x)=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{j=1}^{n}G(t_{j+1}-t_{j},x_{j+1}-x_{j})\prod_{j=1}^{n}W(dx_{j})d{\bf
t}.$

Recalling definition (32) of $H_{n}(t,x)$, we write
$H_{n}(t,x)=J_{n}(t,x)+M_{n}(t,x)$, where

$M_{n}(t,x)=\sum_{k=1}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}H_{n,k,J,I_{1},\ldots,I_{k}}(t,x).$

Note that $M_{1}(t,x)=0$ since $H_{1}(t,x)=J_{1}(t,x)$. We have:

$v(t,x)=1+\sum_{n\geq 1}J_{n}(t,x)+\sum_{n\geq 1}M_{n}(t,x)=u(t,x)+\sum_{n\geq
1}M_{n}(t,x).$

The following table shows shows the composition of $M_{n}(t,x)$ in comparison
with $J_{n}(t,x)$:

$n$ | $J_{n}(t,x)$ | $M_{n}(t,x)$ | $k$ | $n-2k$
---|---|---|---|---
1 | $\in\mathcal{H}_{1}$ | 0 |  |
2 | $\in\mathcal{H}_{2}$ | 1 term in $\mathcal{H}_{0}$ | 1 | 0
3 | $\in\mathcal{H}_{3}$ | 3 terms in $\mathcal{H}_{1}$ | 1 | 1
4 | $\in\mathcal{H}_{4}$ | 6 terms in $\mathcal{H}_{2}$ | 1 | 2
|  | 3 terms in $\mathcal{H}_{0}$ | 2 | 0
5 | $\in\mathcal{H}_{5}$ | 10 terms in $\mathcal{H}_{3}$ | 1 | 3
|  | 15 terms in $\mathcal{H}_{1}$ | 2 | 1
6 | $\in\mathcal{H}_{6}$ | 15 terms in $\mathcal{H}_{4}$ | 1 | 4
|  | 45 terms in $\mathcal{H}_{2}$ | 2 | 2
|  | 15 terms in $\mathcal{H}_{0}$ | 3 | 0

In general, $M_{n}(t,x)$ contains $\binom{n}{2k}\cdot\frac{(2k)!}{2^{k}k!}$
terms in $\mathcal{H}_{n-2k}$, for any $k=1,2,\ldots,\lfloor n/2\rfloor$.

The projection of $v(t,x)$ on $\mathcal{H}_{m}$ is equal to $J_{m}(t,x)$ plus
the sum of all terms in $\mathcal{H}_{m}$ which appear in the 3rd column of
the table above. This shows that the chaos expansion

$v(t,x)=\sum_{m\geq 0}K_{m}(t,x)\quad\mbox{with}\quad
K_{m}(t,x)\in\mathcal{H}_{m}$

of $v(t,x)$ is much more complicated than that of $u(t,x)$. More precisely,

$\displaystyle K_{0}(t,x)$ $\displaystyle=1+\sum_{\begin{subarray}{c}n\geq 1\
{\rm even}\\\
n/2=:k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ [n]\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}H_{n,k,\emptyset,I_{1},\ldots,I_{k}}(t,x)=\mathbb{E}\big{(}v(t,x)\big{)}$
$\displaystyle K_{m}(t,x)$ $\displaystyle=\sum_{\begin{subarray}{c}n\geq 1,\
n-m\ {\rm even}\\\
(n-m)/2=:k\end{subarray}}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=m\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}H_{n,k,J,I_{1},\ldots,I_{k}}(t,x)\quad\mbox{for
$m\geq 1$}.$

The term of the sum which corresponds to $n=m$ is
$H_{m,0,[m]}(t,x)=J_{m}(t,x)$, for $m\geq 1$.

###  6.2 Approximation for the Skorohod solution

In this section, we present an approximation procedure for $u(t,x)$, which
turns out to be quite different than the approximation of $v(t,x)$. This
section is included only for the sake of comparison, and it is not needed for
the results in this paper.

We consider the equation with mollified noise $\dot{W}^{\varepsilon}$ and Wick
product $\diamond$:

$\displaystyle\begin{cases}\dfrac{\partial^{2}u^{\varepsilon}}{\partial
t^{2}}(t,x)=\Delta
u^{\varepsilon}(t,x)+u^{\varepsilon}(t,x)\diamond\dot{W}^{\varepsilon}(x),\quad\quad
t>0,x\in\mathbb{R}^{d}\quad(d\leq 2)\\\
u^{\varepsilon}(0,x)=1,\quad\dfrac{\partial u^{\varepsilon}}{\partial
t}(0,x)=0\end{cases}$ (56)

Recall that $F\diamond W(h)=\delta(Fh)$ where $\delta$ is the divergence
operator and $F$ is a random variable in $L^{2}(\Omega)$ which is Malliavin
differentiable with respect to $W$ (see [11]).

######  Definition 6.1.

A process $u^{\varepsilon}=\\{u_{\varepsilon}(t,x);t\geq
0,x\in\mathbb{R}^{d}\\}$ is a solution to (56) if it satisfies

$u^{\varepsilon}(t,x)=1+\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)u^{\varepsilon}(s,y)\diamond\dot{W}^{\varepsilon}(y)dyds.$
(57)

By the definition of the Wick product and stochastic Fubini theorem, (57) is
equivalent to

$u^{\varepsilon}(t,x)=1+\int_{\mathbb{R}^{d}}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)p_{\varepsilon}(y-z)u^{\varepsilon}(s,y)dydz\right)W(\delta
z).$

We define the kernel

$f_{n}^{\varepsilon}(y_{1},\ldots,y_{n},x;t)=\int_{T_{n}(t)}\int_{(\mathbb{R}^{d})^{n}}\prod_{i=1}^{n}G(t_{i+1}-t_{i},x_{i+1}-x_{i})\prod_{i=1}^{n}p_{\varepsilon}(x_{i}-y_{i})d{\bf
x}d{\bf t}.$

It is not difficult to prove that
$f_{n}^{\varepsilon}(\cdot,x;t)\in\mathcal{H}^{\otimes n}$, using the fact
that $f_{n}(\cdot,x;t)\in\mathcal{H}^{\otimes n}$ and

$\mathcal{F}f_{n}^{\varepsilon}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n})=\exp\left(-\frac{\varepsilon}{2}\sum_{j=1}^{n}|\xi_{j}|^{2}\right)\mathcal{F}f_{n}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n}).$

######  Lemma 6.2.

Under condition (D), for any $T>0$ and $p\geq 2$, the series

$u^{\varepsilon}(t,x):=1+\sum_{n\geq
1}I_{n}(f_{n}^{\varepsilon}(\cdot,x;t))\quad\mbox{converges in
$L^{p}(\Omega)$}$ (58)

uniformly in $(t,x)\in[0,T]\times\mathbb{R}^{d}$ and $\varepsilon>0$.
Moreover, $\\{u^{\varepsilon}(t,x);t\geq 0,x\in\mathbb{R}^{d}\\}$ is the
unique solution of equation (56).

###### Proof.

Note that

$\mathcal{F}\widetilde{f}_{n}^{\varepsilon}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n})=\exp\left(-\frac{\varepsilon}{2}\sum_{j=1}^{n}|\xi_{j}|^{2}\right)\mathcal{F}\widetilde{f}_{n}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n}).$

Hence $\|\widetilde{f}_{n,\varepsilon}(\cdot,x;t)\|_{\mathcal{H}^{\otimes
n}}\leq\|\widetilde{f}_{n}(\cdot,x;t)\|_{\mathcal{H}^{\otimes n}}$ and the
$L^{2}(\Omega)$-convergence of the series (58) (uniform in $(t,x)$) follows
from the uniform convergence of $\sum_{n\geq
0}n!\|\widetilde{f}_{n}(\cdot,x;t)\|_{\mathcal{H}^{\otimes n}}^{2}$. The
$L^{p}(\Omega)$-convergence follows by hypercontractivity.

Next, we prove that $u^{\varepsilon}$ is a solution of (56). We have to show
that $u^{\varepsilon,t,x}\in{\rm Dom}\,\delta$ and
$\delta(u^{\varepsilon,t,x})=u^{\varepsilon}(t,x)-1$, where

$u^{\varepsilon,t,x}(z)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)p_{\varepsilon}(y-z)u^{\varepsilon}(s,y)dyds.$

First, we deduce the chaos expansion of $u^{\varepsilon,t,x}(z)$:

$\displaystyle u^{\varepsilon,t,x}(z)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)p_{\varepsilon}(y-z)\Big{(}\sum_{n\geq
0}I_{n}(f_{n}^{\varepsilon}(\cdot,y;s))\Big{)}dyds$ $\displaystyle=\sum_{n\geq
0}I_{n}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}G(t-s,x-y)p_{\varepsilon}(y-z)f_{n}^{\varepsilon}(\cdot,y;s)dyds\right)$
$\displaystyle=\sum_{n\geq
0}I_{n}\big{(}f_{n+1}^{\varepsilon}(\cdot,z,x;t)\big{)}.$

Using an analogue of Proposition 1.3.7 of [11], we infer that
$u^{\varepsilon,t,x}\in{\rm Dom}\,\delta$ and

$\delta(u^{\varepsilon,t,x})=\sum_{n\geq
0}I_{n+1}\big{(}f_{n+1}^{\varepsilon}(\cdot,x;t)\big{)}=u^{\varepsilon}(t,x)-1.$

Uniqueness follows by classical arguments. ∎

######  Lemma 6.3.

Under condition (D), for any $T>0$ and $p\geq 2$,

$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|u^{\varepsilon}(t,x)-u(t,x)|^{p}\to
0\quad\mbox{as}\quad\varepsilon\downarrow 0.$

###### Proof.

By hypercontractivity, it suffices to consider the case $p=2$. Note that
$u(t,x)-u^{\varepsilon}(t,x)=\sum_{n\geq
1}I_{n}(h_{n}^{\varepsilon}(\cdot,x;t))$, where
$h_{n}^{\varepsilon}(\cdot,x;t)=f_{n}(\cdot,x;t)-f_{n}^{\varepsilon}(\cdot,x;t)$,
and

$\mathcal{F}\widetilde{h}_{n}^{\varepsilon}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n})=\left[1-\exp\left(-\frac{\varepsilon}{2}\sum_{j=1}^{n}|\xi_{j}|^{2}\right)\right]\mathcal{F}\widetilde{f}_{n}(\cdot,x;t)(\xi_{1},\ldots\xi_{n}).$

By the dominated convergence theorem,

$\|\widetilde{h}_{n}^{\varepsilon}(\cdot,x;t)\|_{\mathcal{H}^{\otimes
n}}^{2}=\int_{\mathbb{R}^{d}}|\mathcal{F}\widetilde{h}_{n}^{\varepsilon}(\cdot,x;t)(\xi_{1},\ldots,\xi_{n})|^{2}\mu(d\xi_{1})\ldots\mu(d\xi_{n})\to
0\ \mbox{as}\ \varepsilon\to 0.$

Moreover,
$\|\widetilde{h}_{n}^{\varepsilon}(\cdot,x;t)\|_{\mathcal{H}^{\otimes
n}}\leq\|\widetilde{f}_{n}(\cdot,x;t)\|_{\mathcal{H}^{\otimes n}}$. Note that

$\displaystyle\mathbb{E}|u(t,x)-u^{\varepsilon}(t,x)|^{2}$
$\displaystyle=\sum_{n\geq
1}n!\|\widetilde{h}_{n}^{\varepsilon}(\cdot,x;t)\|_{\mathcal{H}^{\otimes
n}}^{2}.$

and the series converges to $0$ as $\varepsilon\downarrow 0$, uniformly in
$(t,x)\in[0,T]\times\mathbb{R}^{d}$, using the dominated convergence theorem
and the fact that
$\sup_{(t,x)\in[0,T]\times\mathbb{R}^{d}}\mathbb{E}|u(t,x)|^{2}<\infty$. ∎

##  Appendix A Parseval-type identities

In this section, we give two Parseval-type identities which are used in this
paper.

######  Lemma A.1.

Let $d\geq 1$ be arbitrary and $\gamma:\mathbb{R}^{d}\to[0,\infty]$ be non-
negative-definite function satisfying Assumption A. Let $\mu$ be the tempered
measure on $\mathbb{R}^{d}$ such that $\gamma=\mathcal{F}\mu$. If $\varphi$ is
a non-negative integrable function on $\mathbb{R}^{d}$ such that
$\int_{\mathbb{R}^{d}}|\mathcal{F}\varphi(\xi)|^{2}\mu(d\xi)<\infty$, then

$\int_{\mathbb{R}^{d}}\varphi(x)\gamma(x)dx=\int_{\mathbb{R}^{d}}\mathcal{F}\varphi(\xi)\mu(d\xi).$

###### Proof.

By relation (5.37) of [9],

$\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\gamma(x-y)\nu_{1}(dx)\nu_{2}(dy)=\int_{\mathbb{R}^{d}}\mathcal{F}\nu_{1}(\xi)\overline{\mathcal{F}\nu_{2}(\xi)}\mu(d\xi),$

for any finite measures $\nu_{1},\nu_{2}$ on $\mathbb{R}^{d}$ such that
$\int_{\mathbb{R}^{d}}|\mathcal{F}\nu_{i}(\xi)|^{2}\mu(d\xi)<\infty$ for
$i=1,2$. The conclusion follows taking $\nu_{1}(dx)=\varphi(x)dx$ and
$\nu_{2}=\delta_{0}$. ∎

######  Lemma A.2.

Let $\gamma$ and $\mu$ be as in Lemma A.1. If $\varphi$ is a non-negative
integrable function on $(\mathbb{R}^{d})^{2k}$ and $\\{I_{1},\ldots,I_{k}\\}$
is a partition of $\\{1,\ldots,2k\\}$ with $I_{i}=\\{\ell_{i},m_{i}\\}$ for
$i=1,\ldots,k$, then

$\displaystyle\int_{(\mathbb{R}^{d})^{2k}}\varphi(x_{1},\ldots,x_{2k})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})dx_{1}\ldots
dx_{2k}=$
$\displaystyle\quad\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}\varphi(\xi_{1},\ldots,\xi_{2k})\prod_{i=1}^{k}1_{\\{\xi_{\ell_{i}}=-\xi_{m_{i}}=\eta_{i}\\}}\mu(d\eta_{1})\ldots\mu(d\eta_{k}),$

provided that the integral on the right-hand-side is finite.

###### Proof.

Without loss of generality, we will assume that $I_{i}=\\{2i-1,2i\\}$ for all
$i=1,\ldots,k$. (If not, we use the change of variables
$y_{1}=x_{\ell_{1}},y_{2}=x_{m_{1}},\ldots,y_{2k-1}=x_{\ell_{k}},y_{2k}=x_{m_{k}}$
and we let $\rho$ be the permutation of $\\{1,\ldots,2k\\}$ given by
$\ell_{1}=\rho(1),m_{1}=\rho(2),\ldots,\ell_{k}=\rho(2k-1),m_{k}=\rho(2k)$.
Then $y_{i}=x_{\rho(i)}$ for $i=1,\ldots,2k$, and

$\int_{(\mathbb{R}^{d})^{2k}}\varphi(x_{1},\ldots,x_{2k})\prod_{i=1}^{k}\gamma(x_{\ell_{i}}-x_{m_{i}})d{\bf
x}=\int_{(\mathbb{R}^{d})^{2k}}\varphi(y_{\rho^{-1}(1)},\ldots
y_{\rho^{-1}(2k)})\prod_{i=1}^{k}\gamma(y_{2i-1}-y_{2i})d{\bf y},$

where ${\bf x}=(x_{1},\ldots,x_{2k})$ and ${\bf y}=(y_{1},\ldots,y_{2k})$.) We
have to prove that

$\int_{(\mathbb{R}^{d})^{2k}}\varphi(x_{1},\ldots,x_{2k})\prod_{i=1}^{k}\gamma(x_{2i-1}-x_{2i})d{\bf
x}=\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}\varphi(\xi_{1},\ldots,\xi_{2k})\prod_{i=1}^{k}1_{\\{\xi_{2i-1}=-\xi_{2i}=\eta_{i}\\}}\prod_{i=1}^{k}\mu(d\eta_{i}),$

which we will re-write as:

$\int_{(\mathbb{R}^{d})^{k}}\varphi(x_{1},x_{1}^{\prime},\ldots,x_{k},x_{k}^{\prime})\prod_{i=1}^{k}\gamma(x_{i}-x_{i}^{\prime})d{\bf
x}d{\bf
x^{\prime}}=\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}\varphi(\xi_{1},-\xi_{1},\ldots,\xi_{k},-\xi_{k})\prod_{i=1}^{k}\mu(d\xi_{i}),$
(59)

with ${\bf x}=(x_{1},\ldots,x_{k})$ and ${\bf
x}^{\prime}=(x_{1}^{\prime},\ldots,x_{k}^{\prime})$.

We prove (59) by induction on $k\geq 1$. If $k=1$, then by Lemma A.1, we have:

$\displaystyle\int_{(\mathbb{R}^{d})^{2}}\varphi(x_{1},x_{1}^{\prime})\gamma(x_{1}-x_{1}^{\prime})dx_{1}dx_{1}^{\prime}=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\varphi(x_{1},x_{1}+y_{1})\gamma(y_{1})dy_{1}\right)dx_{1}$
$\displaystyle\quad\quad\quad=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\mathcal{F}\varphi(x_{1},x_{1}+\cdot)(\xi_{1})\mu(d\xi_{1})\right)dx_{1}$
$\displaystyle\quad\quad\quad=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}e^{-i\xi_{1}\cdot(x_{1}^{\prime}-x_{1})}\varphi(x_{1},x_{1}^{\prime})dx_{1}^{\prime}\right)\mu(d\xi_{1})dx_{1}$
$\displaystyle\quad\quad\quad=\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}e^{-i\xi_{1}\cdot(x_{1}-x_{1}^{\prime})}\varphi(x_{1},x_{1}^{\prime})dx_{1}dx_{1}^{\prime}\right)\mu(d\xi_{1})$
$\displaystyle\quad\quad\quad=\int_{\mathbb{R}^{d}}\mathcal{F}\varphi(\xi_{1},-\xi_{1})\mu(d\xi_{1}),$

where for the second last line we used the fact that $\mu$ is symmetric.
Suppose now that the statement holds for $k$. By applying the result for
$k=1$, we obtain:

$\displaystyle
I:=\int_{(\mathbb{R}^{d})^{k+1}}\varphi(x_{1},x_{1}^{\prime},\ldots,x_{k+1},x_{k+1}^{\prime})\prod_{i=1}^{k+1}\gamma(x_{i}-x_{i}^{\prime})dx_{1}\ldots
dx_{k+1}dx_{1}^{\prime}\ldots dx_{k+1}^{\prime}=$
$\displaystyle\int_{(\mathbb{R}^{d})^{2k}}\left(\int_{\mathbb{R}^{d}}\psi_{\xi_{k+1}}(x_{1},x_{1}^{\prime},\ldots,x_{k},x_{k}^{\prime})\mu(d\xi_{k+1})\right)\prod_{i=1}^{k}\gamma(x_{i}-x_{i}^{\prime})dx_{1}\ldots
dx_{k}dx_{1}^{\prime}\ldots dx_{k}^{\prime},$

where
$\psi_{\xi_{k+1}}(x_{1},x_{1}^{\prime},\ldots,x_{k},x_{k}^{\prime}):=\mathcal{F}\varphi(x_{1},x_{1}^{\prime},\ldots,x_{k},x_{k}^{\prime},\cdot,\cdot)(\xi_{k+1},-\xi_{k+1})$.
We apply Fubini’s theorem and the induction hypothesis for the inner integral
on $(\mathbb{R}^{d})^{2k}$. We obtain:

$I=\int_{\mathbb{R}^{d}}\left(\int_{(\mathbb{R}^{d})^{k}}\mathcal{F}\psi_{\xi_{k+1}}(\xi_{1},-\xi_{1},\ldots,\xi_{k},-\xi_{k})\mu(d\xi_{1})\ldots\mu(d\xi_{k})\right)\mu(d\xi_{k+1}).$

The conclusion follows since
$\mathcal{F}\psi_{\xi_{k+1}}(\xi_{1},-\xi_{1},\ldots,\xi_{k},-\xi_{k})=\mathcal{F}\varphi(\xi_{1},-\xi_{1},\ldots,\xi_{k+1},-\xi_{k+1})$.
∎

##  Appendix B Products of Wiener integrals

In this section, we develop a formula for the product of $n$ Wiener integrals
with respect to the noise $W$, which plays a crucial role in the present
paper. This formula may be known but we could not find a reference for it.

We recall the product formula from Malliavin calculus: for any integer $p\geq
1$, for any symmetric function $f\in\mathcal{H}^{\otimes p}$ and for any
$g\in\mathcal{H}$,

$I_{p}(f)I_{1}(g)=I_{p+1}(f\otimes g)+pI_{p-1}(f\otimes_{1}g),$ (60)

where $f\otimes_{1}g$ is the first contraction of $f$ and $g$, defined by:

$(f\otimes_{1}g)(y_{1},\ldots,y_{p-1})=\langle
f(\cdot,y_{1},\ldots,y_{p-1}),g\rangle_{\mathcal{H}}$

(see e.g. Theorem 2.7.10 of [10], or Proposition 1.1.3 of [11]).

We will use the fact that $I_{p}(f)=I_{p}(\widetilde{f})$ for any
$f\in\mathcal{H}^{\otimes p}$, where $\widetilde{f}$ is the symmetrization of
$f$. We let $f\otimes_{0}g=f\otimes g$ and we denote by
$f\widetilde{\otimes}_{r}g$ the symmetrization of $f\otimes_{r}g$.

######  Theorem B.1.

For any $n\geq 2$ and for any functions $f_{1},\ldots,f_{n}\in\mathcal{H}$,

$\prod_{j=1}^{n}I_{1}(f_{j})=\sum_{k=0}^{\lfloor
n/2\rfloor}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ J^{c}\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k}(\bigotimes_{j\in
J}f_{j})\prod_{i=1}^{k}\langle f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}},$
(61)

with the convention that $\otimes_{j\in\emptyset}f_{j}=1$ and $I_{0}(1)=1$.

###### Proof.

We use induction on $n$. The case $n=2$ is clear.

Suppose that the result is true for $n$. Assume that $n$ is even, $n=2K$ for
$K\in\mathbb{Z}_{+}$. The case when $n$ is odd is similar and will be omitted.

We multiply relation (61) by $I_{n+1}(f_{n+1})$. To evaluate the product
$I_{n-2k}(\bigotimes_{j\in J}f_{j})I_{1}(f_{n+1})$, we dstinguish 2 cases:
(a) if $k=K$, then $J=\emptyset$ and $I_{n-2k}(\bigotimes_{j\in
J}f_{j})I_{1}(f_{n+1})=I_{1}(f_{n+1})$;
(b) if $k\leq K-1$, then by (60),

$I_{n-2k}(\bigotimes_{j\in
J}f_{j})I_{1}(f_{n+1})=I_{n+1-2k}\Big{(}\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes
f_{n+1}\Big{)}+(n-2k)I_{n-2k-1}\Big{(}\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes_{1}f_{n+1}\Big{)}.$

Note that $I_{n+1-2k}\Big{(}\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes f_{n+1}\Big{)}=I_{n+1-2k}\Big{(}\bigotimes_{j\in
J}f_{j}\otimes f_{n+1}\Big{)}$ since $\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes f_{n+1}$ and $\bigotimes_{j\in J}f_{j}\otimes f_{n+1}$
have the same symmetrizations. Moreover, $\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes_{1}f_{n+1}=\frac{1}{n-2k}\sum_{j\in J}\langle
f_{j},f_{n+1}\rangle_{\mathcal{H}}\widetilde{\bigotimes}_{\ell\in J\verb
2\2\\{j\\}}f_{\ell}$ and hence,

$I_{n-2k-1}\Big{(}\big{(}\widetilde{\bigotimes}_{j\in
J}f_{j}\big{)}\otimes_{1}f_{n+1}\Big{)}=\frac{1}{n-2k}\sum_{j\in J}\langle
f_{j},f_{n+1}\rangle_{\mathcal{H}}I_{n-2k-1}\Big{(}\bigotimes_{\ell\in
J-\\{j\\}}f_{\ell}\Big{)}.$

It follows that

$\displaystyle A$
$\displaystyle:=\prod_{i=1}^{n+1}I_{1}(f_{i})=\sum_{k=0}^{K-1}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ [n]-J\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n+1-2k}\Big{(}\bigotimes_{j\in J}f_{j}\otimes
f_{n+1}\Big{)}\prod_{i=1}^{k}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}$
$\displaystyle+\sum_{k=0}^{K-1}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{j\in
J}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm partition}\ {\rm
of}\ [n]-J\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k-1}\Big{(}\bigotimes_{\ell\in
J-\\{j\\}}f_{j}\Big{)}\prod_{i=1}^{k}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}\langle
f_{j},f_{n+1}\rangle_{\mathcal{H}}$
$\displaystyle+\sum_{\begin{subarray}{c}(\\{I_{1},\ldots,I_{k}\\}\ {\rm
partition}\ {\rm of}\ [n]-J\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{1}(f_{n+1})\prod_{i=1}^{K}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}=:A_{1}+A_{2}+A_{3}.$

We have to prove that $A=B$, where

$\displaystyle
B=\sum_{k=0}^{K}\sum_{\begin{subarray}{c}J^{\prime}\subset[n+1]\\\
|J^{\prime}|=n+1-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\
{\rm partition}\ {\rm of}\ [n+1]-J^{\prime}\\\
I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n+1-2k}(\bigotimes_{j\in
J^{\prime}}f_{j})\prod_{i=1}^{k}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}.$

We split $B$ using $k\leq K-1$ and $k=K$: (if $k=K$ then
$|J^{\prime}|=n+1-2K=1$)

$\displaystyle B$
$\displaystyle=\sum_{k=0}^{K-1}\sum_{\begin{subarray}{c}J^{\prime}\subset[n+1]\\\
|J^{\prime}|=n+1-2k\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\
{\rm partition}\ {\rm of}\ [n+1]-J^{\prime}\\\
I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n+1-2k}(\bigotimes_{j\in
J^{\prime}}f_{j})\prod_{i=1}^{k}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}$
$\displaystyle+\sum_{j=1}^{n}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{K}\\}\
{\rm partition}\ {\rm of}\ [n+1]-J^{\prime}\\\
I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,K\end{subarray}}I_{1}(f_{j})\prod_{i=1}^{K}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}=:B_{1}+B_{2}.$

We split $A_{2}$ using $k\leq K-2$ and $k=K-1$: (if $k=K-1$ then $n-2k-1=1$)

$\displaystyle A_{2}$
$\displaystyle=\sum_{k=0}^{K-2}\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=n-2k\end{subarray}}\sum_{j\in
J}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{k}\\}\ {\rm partition}\ {\rm
of}\ [n]-J\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,k\end{subarray}}I_{n-2k-1}\Big{(}\bigotimes_{\ell\in
J-\\{j\\}}f_{j}\Big{)}\prod_{i=1}^{k}\langle
f_{\ell_{i}},f_{m_{i}}\rangle_{\mathcal{H}}\langle
f_{j},f_{n+1}\rangle_{\mathcal{H}}$
$\displaystyle+\sum_{\begin{subarray}{c}J\subset[n]\\\
|J|=2\end{subarray}}\sum_{\begin{subarray}{c}\\{I_{1},\ldots,I_{K-1}\\}\ {\rm
partition}\ {\rm of}\ [n]-J\\\ I_{i}=\\{\ell_{i},m_{i}\\}\forall
i=1,\ldots,K-1\end{subarray}}\sum_{j\in J}I_{1}\big{(}\bigotimes_{\ell\in
J-\\{j\\}}f_{\ell}\big{)}\langle
f_{j},f_{n+1}\rangle_{\mathcal{H}}\prod_{i=1}^{K-1}\langle
f_{\ell_{i}},f_{m_{1}}\rangle_{\mathcal{H}}:=A_{2}^{\prime}+A_{2}^{\prime\prime}.$

We notice that:

$(a)\ A_{1}+A_{2}^{\prime}=B_{1}\quad\mbox{and}\quad(b)\
A_{2}^{\prime\prime}+A_{3}=B_{2}.$

(a) We use the decomposition $B_{1}=B_{1}^{\prime}+B_{1}^{\prime\prime}$,
where $B_{1}^{\prime}$ and $B_{1}^{\prime\prime}$ correspond to the cases
$n+1\in J^{\prime}$, respectively $n+1\not\in J^{\prime}$. We observe that
$A_{1}=B_{1}^{\prime\prime}$ and $A_{2}^{\prime}=B_{1}^{\prime}$.

(b) We use the decomposition $B_{2}=B_{2}^{\prime}+B_{2}^{\prime\prime}$,
where $B_{2}^{\prime}$ and $B_{2}^{\prime\prime}$ correspond to the cases
$j=n+1$, respectively $j\leq n$. We observe that $A_{3}=B_{2}^{\prime}$ and
$A_{2}^{\prime\prime}=B_{2}^{\prime\prime}$.

∎

##  Appendix C Semigroup-type property of $G$

In this section, we prove a property of $G$ which was used in Step 3.(a) of
the proof of Theorem 3.2. We denote $G_{t}(x)=G(t,x)$.

######  Lemma C.1 (Lemma 4.3 of [3]).

Suppose that $d=2$. Let $q\in(\frac{1}{2},1)$ be arbitrary. For any $0<r<t$
and $x,z\in\mathbb{R}^{d}$

$\int_{r}^{t}\big{(}G_{t-s}^{2q}*G_{s-r}^{2q}\big{)}^{1/q}(x-z)ds\leq
A_{q}(t-r)^{\frac{1}{q}-1}G_{t-r}^{2-\frac{1}{q}}(x-z),$

where $A_{q}>0$ is a constant depending on $q$.

As an immediate consequence of the previous lemma, we obtain the following
semigroup-type property of $G$.

######  Lemma C.2.

For any $0<r<t$ and $x,z\in\mathbb{R}^{d}$,

$\int_{r}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)G_{s-r}(y-z)dyds\leq
C_{t}^{(1)}G_{t-r}(x-z),$

where $C_{t}^{(1)}=Ct^{2}$ and $C$ is a positive constant.

###### Proof.

If $d=1$, this follows by relation (2.6) of [12].

Suppose that $d=2$. Let $q\in(\frac{1}{2},1)$ be arbitrary. Note that
$G_{t}(x)\leq C_{t,q}G_{t}^{2q}(x)$, where $C_{t,q}=(2\pi t)^{2q-1}$. By
Hölder’s inequality and Lemma C.1,

$\displaystyle\int_{r}^{t}\int_{\mathbb{R}^{d}}G_{t-s}(x-y)G_{s-r}(y-z)dydr\leq
C_{t,q}^{2}\int_{r}^{t}\big{(}G_{t}^{2q}*G_{s}^{2q}\big{)}(x-z)dr\leq$
$\displaystyle\qquad
C_{t,q}^{2}(t-r)^{1-q}\left[\int_{r}^{t}\big{(}G_{t}^{2q}*G_{s}^{2q}\big{)}^{1/q}(x-z)ds\right]^{q}\leq
C_{t,q}^{2}(t-r)^{2(1-q)}A_{q}^{q}G_{t-r}^{2q-1}(x-z).$

The conclusion follows using the fact that $G_{t-r}^{2q-1}(x-z)\leq(2\pi
t)^{2-2q}G_{t-r}(x-z)$. ∎

Acknowledgement. The author is grateful to Guangqu Zheng for sharing the proof
of Theorem B.1 in the case $n=3$, and to Samy Tindel for pointing out
references [7] and [8].

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