# Path-by-path uniqueness of multidimensional SDE’s on the plane with
nondecreasing coefficients

Antoine-Marie Bogso  University of Yaounde I
Faculty of Sciences, Department of Mathematics
P.O. Box 812, Yaounde, Cameroon
and African institute for Mathematical Sciences Ghana, P.O. Box LGDTD 20046,
Summerhill Estates, East Legon Hills, Santoe, Acrra, Ghana
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Moustapha Dieye
École polytechnique de Thiès, Département tronc commun, BP A10, Thiès, Sénégal
<EMAIL_ADDRESS>and  Olivier Menoukeu Pamen  Institute for Financial and
Actuarial Mathematics (IFAM)
Department of Mathematical Sciences, University of Liverpool
Liverpool L69 7ZL, UK
and African institute for Mathematical Sciences Ghana, P.O. Box LGDTD 20046,
Summerhill Estates, East Legon Hills, Santoe, Acrra, Ghana
<EMAIL_ADDRESS>
###### Abstract.

In this paper we study path-by-path uniqueness for multidimensional stochastic
differential equations driven by the Brownian sheet. We assume that the drift
coefficient is unbounded, verifies a spatial linear growth condition and is
componentwise nondeacreasing. Our approach consists of showing the result for
bounded and componentwise nondecreasing drift using both a local time-space
representation and a law of iterated logarithm for Brownian sheets. The
desired result follows using a Gronwall type lemma on the plane. As a by
product, we obtain the existence of a unique strong solution of
multidimensional SDEs driven by the Brownian sheet when the drift is non-
decreasing and satisfies a spatial linear growth condition.

###### Key words and phrases:

Brownian sheet, SDEs on the plane, path-by-path uniqueness, stochastic wave
equations

###### 1991 Mathematics Subject Classification:

Primary 60H05, 60H15.

The project on which this publication is based has been carried out with
funding provided by the Alexander von Humboldt Foundation, under the programme
financed by the German Federal Ministry of Education and Research entitled
German Research Chair No 01DG15010.

##  1\. Introduction

In this work, we consider the following system of stochastic integral
equations on the plane with additive noise:

(1.1) $\displaystyle
X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}=\int_{0}^{t}\int_{0}^{s}b(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta+W_{s,t}\text{
for }(s,t)\in\mathbb{R}_{+}^{2},$

where $b:\mathbb{R}_{+}^{2}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ is Borel
measurable satisfying some conditions that will be specified later and
$W=(W_{s,t},(s,t)\in\mathbb{R}_{+}^{2})$ is a $d$-dimensional Brownian sheet
given on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in\mathbb{R}_{+}^{2}),\mathbb{P})$
with $\partial W=0$, where $\partial W$ stands for the restriction of $W$ to
the boundary $\partial
D=\\{0\\}\times\mathbb{R}_{+}\cup\mathbb{R}_{+}\times\\{0\\}$ of
$D=\mathbb{R}_{+}^{2}$. We endow $D$ with the partial order “$\preceq$”
(respectively “$\prec$”) defined by

$(s,t)\preceq(s^{\prime},t^{\prime})\text{ when }s\leq s^{\prime}\text{ and
}t\leq t^{\prime},$

respectively

$(s,t)\prec(s^{\prime},t^{\prime})\text{ when }s<s^{\prime}\text{ and
}t<t^{\prime}.$

Observe that (1.1) is a particular case of the more general non-Markovian type
equation

$\displaystyle X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}$ (1.2)
$\displaystyle=\int_{0}^{t}\int_{0}^{s}b(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\,\mathrm{d}\zeta+\int_{0}^{t}\int_{0}^{s}a(\xi,\zeta,X_{\xi,\zeta})\,\mathrm{d}W_{\xi,\zeta}\text{
for }(s,t)\in\mathbb{R}_{+}^{2},$

where
$a:\,\mathbb{R}_{+}^{2}\times\mathbb{R}^{d}\to\mathbb{R}^{d}\times\mathbb{R}^{d}$
is a Borel measurable matrix function. Note that (1) appears as an integral
equation when one rewrites the following quasilinear stochastic hyperbolic
differential equation

(1.3) $\displaystyle\frac{\partial^{2}X_{s,t}}{\partial s\partial
t}=b(s,t,X_{s,t})+a(s,t,X_{s,t})\frac{\partial^{2}W_{s,t}}{\partial s\partial
t},$

where the notation “$\frac{\partial^{2}W_{s,t}}{\partial s\partial t}$”
designates a white noise on $D$. As pointed out by Farré and Nualart [18] (see
also [31]), a formal $\frac{\pi}{4}$ rotation transforms (1.3) into a
nonlinear stochastic wave equation. This idea, thanks to Walsh [37], has been
used by Carmona and Nualart [10] to provide existence and uniqueness results
for the 1-dimension stochastic wave equation

(1.4) $\displaystyle\frac{\partial^{2}Y}{\partial
t^{2}}(t,x)-\frac{\partial^{2}Y}{\partial
x^{2}}(t,x)=a(Y(t,x))\dot{W}(t,x)+b(Y(t,x))$

with some initial conditions $Y(0,\cdot)$ and $\dfrac{\partial Y}{\partial
t}(0,\cdot)$, where $t$ varies in $\mathbb{R}_{+}$, $x$ varies in $\mathbb{R}$
and $\dot{W}$ denotes a white noise in time as well as in space. Reformulation
of (1.4) using a $\frac{\pi}{4}$ rotation allows use of the rectangular
increments of both $t$ and $x$ (see e.g. [31, Section 1]).

The problem (1.1) can also be interpreted as a noisy analog of the so-called
Darboux problem given by

(1.5) $\displaystyle\frac{\partial^{2}y}{\partial s\partial
t}=b\Big{(}s,t,y,\frac{\partial y}{\partial s},\frac{\partial y}{\partial
t}\Big{)}\quad\text{for }(s,t)\in[0,T]\times[0,T],$

with the initial conditions

(1.6) $\displaystyle y(0,t)=\sigma(t)\,\text{ on }[0,T]\,\text{ and
}\,y(s,0)=\tau(s)\,\text{ on }[0,T],$

where $\sigma$ and $\tau$ are absolutely continuous on $[0,T]$. Using
Caratheodory’s theory of differential equations, Deimling [15] proved an
existence theorem for the system (1.5)-(1.6) when $b$ is Borel measurable in
the first two variables and bounded and continuous in the last three
variables.

Existence and uniqueness of solutions to stochastic differential equations
(SDEs) driven by a Brownian sheet has been widely studied. In the time
homogeneous case, Cairoli [8] proved that (1) has a unique strong solution
when the coefficients are Lipschitz continuous. This result was generalised to
the time dependent coefficients by Yeh [39] under an additional growth
condition. Weak existence of solutions to (1) was derived in [40] assuming
that the coefficients are continuous, satisfy a growth condition and the
initial value has moment of order six. In all of the above mentioned works,
the coefficients are at least continuous. Nualart and Tindel [26], show that
(1) has a unique strong solution when the drift is nondecreasing and bounded.
Their results were extended to SDEs driven by two parameter martingales in
[27] (see also [10; 18] for further extensions). In [17], the authors
generalised the above results to SDEs driven by a fractional Brownian sheet.

In this work we are concerned with a different uniqueness question. In
particular, we look at the notion of path-by-path uniqueness introduced by
Davie [13] (see also Flandoli [19]). Let $\mathcal{V}$, resp.
$\partial\mathcal{V}$ be the space of continuous $\mathbb{R}^{d}$-valued
functions on $D$, resp. $\partial D$. The following definition can be seen as
a counterpart of [19, Definition 1.5] in the case of two parameter processes.

######  Definition 1.1.

We say that the path-by-path uniqueness of solutions to (1.1) holds when there
exists a full $\mathbb{P}$-measure set $\Omega_{0}\subset\Omega$ such that for
all $\omega\in\Omega_{0}$ the following statement is true: there exists at
most one function $y\in\mathcal{V}$ which satisfies

$\int_{0}^{T}\int_{0}^{T}|b(\xi,\zeta,y_{\xi,\zeta})|\mathrm{d}\xi\mathrm{d}\zeta<\infty,\text{
}\partial y=x,\text{ for some }x\in\partial\mathcal{V}\text{ and }T>0$

and

(1.7) $\displaystyle
y_{s,t}=x+\int_{0}^{s}\int_{0}^{t}b(\xi,\zeta,y_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta+W_{s,t}(\omega),\text{
}\forall\,(s,t)\in[0,T]^{2}.$

One of the motivations for studying path-by-path uniqueness comes from the
regularisation by noise of random ODEs. For instance, let $v$ be a continuous
function and let us consider the following one parameter equation in
$\mathbb{R}^{d}$

$X_{t}=X_{0}+\int_{0}^{t}b(s,X_{s})\mathrm{d}s+v_{t}.$

We know that there exists a unique solution to the above equation when $b$ is
Lipschitz in $x$, uniformly in $t$, with uniform linear growth. Observe that
uniqueness also holds when $b$ is only locally Lipschitz. Under some weak
conditions on $b$ the corresponding equation without $v$ might be ill-posed or
uniqueness could not be valid. For example when $b$ is merely bounded and
measurable one may ask if there is a notion of uniqueness if $v$ has some
specific features. In other words, can we find a path $v$ that regularises the
equation? The result obtained in [13] shows that when $b\in L^{\infty}$, the
Brownian path regularises the drift $b$ in the sense of Definition 1.1. In
addition, as shown in [4, Section 1.8.5], path-by-path uniqueness is much
stronger than pathwise uniqueness. Indeed Shaposhnikov and Wresch [34, Section
4] exhibit SDEs such that strong solutions exist, pathwise uniqueness holds
and path-by-path uniqueness fails to hold. This is another motivation for
studying path-by-path uniqueness even when pathwise uniqueness holds.

In the case of Brownian motion, when the drift is bounded and measurable, and
the diffusion is reduced to the identity, the path-by-path uniqueness of
equation (1) was proved by Davie in [13]. This result was extended in several
directions. For non-constant diffusion, Davie in [14] proved path-by-path
uniqueness of solution to (1), interpreting the equation in the rough path
sense. In [11], the authors showed that path-by-path uniqueness holds if the
Brownian motion is replaced by a $d$-dimensional fractional Brownian motion of
Hurst parameter $H\in(0,1)$. It is also assumed that the drift $b$ can be
merely a distribution as long as $H$ is sufficiently small. In [29], Priola
considered equations driven by a Lévy process assuming that the drift is
Hölder continuous (see [30] for the non-constant diffusion coefficient case).

Path-by-path uniqueness is closely related to the regularisation by noise
problem for ordinary (or partial) differential equations (ODEs or PDEs) which
has recently drawn a lot of attention. Beck, Flandoli, Gubinelli and Maurelli
[4] proved a Sobolev regularity of solutions to the linear stochastic
transport and continuity equations with drift in critical $L^{p}$ spaces. Such
a result does not hold for the corresponding deterministic equations.
Butkovsky and Mytnik [6] analysed the regularisation by noise phenomenon for a
non-Lipschitz stochastic heat equation and proved path-by-path uniqueness for
any initial condition in a certain class of a set of probability one. Amine,
Mansouri and Proske [2] investigated path-by-path uniqueness for transport
equations driven by the fractional Brownian motion of Hurst index $H<1/2$ with
bounded vector-fields. In [11; 20] the authors solved the regularisation by
noise problem from the point of view of additive perturbations. In particular,
Galeati and Gubinelli [11] considered generic perturbations without any
specific probabilistic setting. Amine, Baños and Proske [1] constructed a new
Gaussian noise of fractional nature and proved that it has a strong
regularising effect on a large class of ODEs. More recently, Harang and
Perkowski [21] studied the regularisation by noise problem for ODEs with
vector fields given by Schwartz distributions and proved that if one perturbs
such an equation by adding an infinitely regularising path, then it has a
unique solution. Kremp and Perkowski [23] looked at multidimensional SDEs with
distributional drift driven by symmetric $\alpha$-stable Lévy processes for
$\alpha\in(1,2]$. In all of the above mentioned works, the driving noise
considered are one parameter processes.

In what follows, we make use of the Girsanov theorem to show that the path-by-
path uniqueness in our setting is equivalent to the uniqueness of a random ODE
on the plane. For any $x\in\partial\mathcal{V}$ and any $\omega$ such that the
path $(s,t)\longmapsto W_{s,t}$ is continuous, we denote by $S(x,\omega)$ the
set of functions in $\mathcal{V}$ that solve (1.7). Under linear growth and
monotonicity conditions on $b$, we prove that $S(x,\omega)$ has at most one
element. By a vector translation argument, it suffices to show that
$S(0,\omega)$ has no more than one element.

As in [13, Section 1], we show the path-by-path uniqueness on
$D^{1}=[0,1]^{2}$. Precisely, we consider the integral equation

(1.8) $\displaystyle
X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}=\int_{0}^{t}\int_{0}^{s}b(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta+W_{s,t}\text{
for }(s,t)\in D^{1},$

where the drift is of spatial linear growth. There is no loss of generality in
reducing the problem to $D^{1}$ since we can repeat the argument on any square
$[m,m+1]\times[\ell,\ell+1]$, $(m,\ell)\in\mathbb{N}^{2}$, $m>0$. We first
suppose that $b$ is bounded and monotone. Let $\mathcal{V}^{1}$ be the space
of continuous $\mathbb{R}^{d}$-valued functions on $D^{1}$ and let
$\mathcal{V}^{1}_{0}$ be the space of functions $y\in\mathcal{V}^{1}$ with
$\partial y=0$, where $\partial y$ is the restriction of $y$ to $\partial
D^{1}$ ($\partial D^{1}=\\{0\\}\times[0,1]\cup[0,1]\times\\{0\\}$). Let
$\mathbb{P}$ be the law of an $\mathbb{R}^{d}$-valued Brownian sheet on
$D^{1}$ which vanishes on $\partial D^{1}$.

The function $L$ given by

$\displaystyle
L(y)=\exp\Big{(}\int_{0}^{1}\int_{0}^{1}b(\xi,\zeta,y_{\xi,\zeta})\mathrm{d}y_{\xi,\zeta}-\frac{1}{2}\int_{0}^{1}\int_{0}^{1}|b(\xi,\zeta,y_{\xi,\zeta})|^{2}\mathrm{d}\xi\mathrm{d}\zeta\Big{)}$

is well-defined for $\mathbb{P}$-a.e. $y\in\mathcal{V}^{1}_{0}$. Moreover, if
$y\in\mathcal{V}^{1}_{0}$ is chosen random, with law
$\mathrm{d}\widetilde{\mathbb{P}}=L\mathrm{d}\mathbb{P}$, then, by Girsanov
theorem (see for example [8; 12; 22]), the path $W$ defined by

(1.9) $\displaystyle
W_{s,t}=y_{s,t}-\int_{0}^{s}\int_{0}^{t}b(\xi,\zeta,y_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta$

has law $\mathbb{P}$. This means that $y$ is a solution to (1.8) with $W$
defined by (1.9). Path-by-path uniqueness of solutions to (1.8) holds if and
only if for $\widetilde{\mathbb{P}}$-a.e. $y\in\mathcal{V}^{1}_{0}$, $z=y$ is
the only solution to

$\displaystyle
W_{s,t}=z_{s,t}-\int_{0}^{s}\int_{0}^{t}b(\xi,\zeta,z_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta$

with $W$ given by (1.9), which is equivalent to saying that for
$\widetilde{\mathbb{P}}$-a.e. $y\in\mathcal{V}^{1}_{0}$, the only solution to

(1.10) $\displaystyle
u(s,t)=\int_{0}^{s}\int_{0}^{t}\\{b(\xi,\zeta,y_{\xi,\zeta}+u(\xi,\zeta))-b(\xi,\zeta,y_{\xi,\zeta})\\}\mathrm{d}\xi\mathrm{d}\zeta$

is $u=0$ (see e.g. [13, Section 1]). Since $\widetilde{\mathbb{P}}$ is
absolutely continuous with respect to $\mathbb{P}$, it is enough to show that,
if $W$ is an $\mathbb{R}^{d}$-valued Brownian sheet, then, with probability
one, there is no nontrivial solution $u\in\mathcal{V}^{1}_{0}$ of

$\displaystyle
u(s,t)=\int_{0}^{s}\int_{0}^{t}\\{b(\xi,\zeta,W_{\xi,\zeta}+u(\xi,\zeta))-b(\xi,\zeta,W_{\xi,\zeta})\\}\mathrm{d}\xi\mathrm{d}\zeta.$

This is the statement of Theorem 3.4 which is extended to unbounded monotone
drifts in Theorem 3.2. Our proof of Theorem 3.4 relies on some estimates for
an averaging operator along the sheet (see Lemma 3.6). This result plays a key
role in the proof of a Gronwall type lemma (see Lemma 3.9) which enables us to
prove path-by-path uniqueness of solutions to (1.1).

The Yamada-Watanabe principle for one dimensional SDEs driven by Brownian
sheets was derived in [28] (see also [41]). More precisely, the authors show
that combining weak existence and pathwise uniqueness yields existence of a
unique strong solution in the two parameter setting. This result can be
extended to the multidimensional case (see e.g. [36, Remark 2]). When $b$ is
of linear growth, we can show (see Lemma A.1) that the SDE (1) has a weak
solution. The latter together with path by path uniqueness (and thus pathwise
uniqueness) implies the existence of a unique strong solution to the SDE (1)
and therefore generalises some results in [17; 26] to the multidimensional
case. To the best of our knowledge, such a result has not been derived in the
multidimensional case.

The remainder of the paper is structured as follows. In Section 2, we recall
some basic definitions and concepts. The main results are stated and proved in
Section 3. In section 4, we prove some preliminary results whereas Section 5
is devoted to the proof of a number of auxiliary results.

##  2\. Basic definitions and concepts

In this section we recall some basic definitions and concepts for SDEs on the
plane. We start with the definitions of filtered probability space and
$d$-dimensional Brownian sheet that can be found in [28; 39].

######  Definition 2.1.

We call a filtered probability space any probability space
$(\Omega,\mathcal{F},\mathbb{P})$ with a family $(\mathcal{F}_{s,t},(s,t)\in
D)$ of sub-$\sigma$-algebras of $\mathcal{F}$ such that

  1. (1)

$\mathcal{F}_{0,0}$ contains all null sets in
$(\Omega,\mathcal{F},\mathbb{P})$,

  2. (2)

$\\{\mathcal{F}_{s,t},(s,t)\in D\\}$ is nondecreasing in the sense that
$\mathcal{F}_{s,t}\subset\mathcal{F}_{s^{\prime},t^{\prime}}$ when
$(s,t)\prec(s^{\prime},t^{\prime})$,

  3. (3)

$(\mathcal{F}_{s,t},(s,t)\in D)$ is a right-continuous system in the sense
that

$\mathcal{F}_{s,t}=\bigcap\limits_{(s^{\prime},t^{\prime})\prec(s,t)}\mathcal{F}_{s^{\prime},t^{\prime}}.$

######  Definition 2.2.

We call a one-dimensional $(\mathcal{F}_{s,t})$-Brownian sheet on a filtered
probability space $(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in
D),\mathbb{P})$ any real valued two-parameter stochastic process
$W^{(0)}=(W^{(0)}_{s,t},(s,t)\in D)$ satisfying the following conditions:

  1. (1)

$W^{(0)}$ is $(\mathcal{F}_{s,t},(s,t)\in D)$-adapted, i.e. $W^{(0)}_{s,t}$ is
$\mathcal{F}_{s,t}$-measurable for every $(s,t)\in D$.

  2. (2)

Every sample function $(s,t)\longmapsto W^{(0)}_{s,t}(\omega)$ of $W^{(0)}$ is
continuous on $D$.

  3. (3)

For every finite rectangle of the type
$\Pi=]s,s^{\prime}]\times]t,t^{\prime}]\subset D$, the random variable

$W^{(0)}(\Pi):=W^{(0)}_{s^{\prime},t^{\prime}}-W^{(0)}_{s,t^{\prime}}-W^{(0)}_{s^{\prime},t}+W^{(0)}_{s,t}$

is centered, Gaussian with variance $(s^{\prime}-s)(t^{\prime}-t)$ and
independent of $\mathcal{F}_{s,1}\vee\mathcal{F}_{1,t}$.

We call a $d$-dimensional Brownian sheet any $\mathbb{R}^{d}$-valued two-
parameter process $W=(W^{(1)},\ldots,W^{(d)})$ such that $W^{(i)}$,
$i=1,\ldots,d$ are independent one-dimensional Brownian sheets.

In the following, we discuss the notions of weak and strong solutions to the
SDE (1) (see for example [39, Section 2]). We start with the definition of a
weak solution.

######  Definition 2.3.

A weak solution to the SDE (1) is a system
$(\Omega,\mathcal{F},(\mathcal{F}_{s,t}),W,X=(X_{s,t}),\mathbb{P})$ such that

  1. (1)

$(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in D),\mathbb{P})$ is a filtered
probability space,

  2. (2)

$W=(W_{s,t},(s,t)\in D)$ is a $d$-dimensional $(\mathcal{F}_{s,t})$-Brownian
sheet with $\partial W=0$,

  3. (3)

$X$ is $(\mathcal{F}_{s,t})$-adapted, has continuous sample paths and,
$\mathbb{P}$-a.s.,

$\displaystyle X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}b(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta+\int_{0}^{t}\int_{0}^{s}a(\xi,\zeta,X_{\xi,\zeta})\,\mathrm{d}W_{\xi,\zeta},\quad\forall\,(s,t)\in
D.$

######  Remark 2.4.

When the drift $b$ satisfies a linear growth condition, weak existence holds
for (1.8) (see Theorem A.1).

We now turn to the notion of strong solution. Let $\mathcal{B}(\mathcal{V})$
(respectively $\mathcal{B}(\partial\mathcal{V})$) be the $\sigma$-algebra of
Borel sets in the space $\mathcal{V}$ (respectively $\partial\mathcal{V}$) of
all continuous $\mathbb{R}^{d}$-valued functions on $D$ (respectively
$\partial D$) with respect to the metric topology of uniform convergence on
compact subsets of $D$. The subsequent definitions are borrowed from [28].

######  Definition 2.5.

Let $\overline{\mathcal{B}(\mathcal{V})}$ be the completion of
$\mathcal{B}(\mathcal{V})$ with respect to the Wiener measure
$m_{\mathcal{V}}$ on $(\mathcal{V},\mathcal{B}(\mathcal{V}))$ concentrated on
$\mathcal{V}_{0}$. For every $(s,t)\in D$, we denote by
$\mathcal{B}_{s,t}(\mathcal{V})$ the $\sigma$-algebra generated by the
cylinder sets of the type $\\{w\in\mathcal{V};\,w(\xi,\zeta)\in E\\}$ for some
$(\xi,\zeta)\preceq(s,t)$ and $E\in\mathcal{B}(R^{d})$ and by
$\overline{\mathcal{B}_{s,t}(\mathcal{V})}$ the $\sigma$-algebra generated by
$\mathcal{B}_{s,t}(\mathcal{V})$ and all the null sets in
$(\mathcal{V},\overline{\mathcal{B}(\mathcal{V})},m_{\mathcal{V}})$. Let
$\overline{\mathcal{B}(\partial\mathcal{V}\times\mathcal{V})}^{\lambda\times
m_{\mathcal{V}}}$ be the completion of
$\mathcal{B}(\partial\mathcal{V}\times\mathcal{V})$ with respect to the
product measure $\lambda\times m_{\mathcal{V}}$ for any probability measure
$\lambda$ on $\partial\mathcal{V}$.

######  Definition 2.6.

Let $\mathbf{T}(\partial\mathcal{V}\times\mathcal{V})$ be the class of
transformations $F$ of $\partial\mathcal{V}\times\mathcal{V}$ into
$\mathcal{V}$ which satisfies the condition that for every probability measure
$\lambda$ on $(\partial\mathcal{V},\mathcal{B}(\partial\mathcal{V}))$, there
exists a transformation $F_{\lambda}$ of
$\partial\mathcal{V}\times\mathcal{V}$ into $\mathcal{V}$ such that

  1. (1)

$F_{\lambda}$ is
$\overline{\mathcal{B}(\partial\mathcal{V}\times\mathcal{V})}^{\lambda\times
m_{\mathcal{V}}}/\mathcal{B}(\mathcal{V})$ measurable,

  2. (2)

For every $x\in\partial\mathcal{V}$, $F_{\lambda}[x,\cdot]$ is
$\overline{\mathcal{B}_{s,t}(\mathcal{V})}/\mathcal{B}_{s,t}(\mathcal{V})$
measurable, for every $(s,t)\in D$,

  3. (3)

There exists a null set $N_{\lambda}$ in
$(\partial\mathcal{V},\mathcal{B}(\partial\mathcal{V}),\lambda)$ such that
$F[x,w]=F_{\lambda}[x,w]$ for almost all $w$ in
$(\mathcal{V},\overline{\mathcal{B}(\mathcal{V})},m_{\mathcal{V}})$ and all
$x\in\partial\mathcal{V}\setminus N_{\lambda}$.

######  Definition 2.7.

Let $(X,W)$ be a weak solution to the SDE (1) on a filtered probability space
$(\Omega,\mathcal{F},\\{\mathcal{F}_{s,t},(s,t)\in D\\},\mathbb{P})$ and let
$\lambda$ be the probability distribution of $\partial X$. We call $(X,W)$ a
strong solution to (1) if there exists a transformation $F_{\lambda}$ of
$\partial\mathcal{V}\times\mathcal{V}$ into $\mathcal{V}$ satisfying
Conditions 1 and 2 of Definition 2.6 such that

$\displaystyle X=F_{\lambda}[\partial X,W]\,\text{ }\mathbb{P}\text{-a.s. on
}\Omega.$

Here is a well known concept of uniqueness associated to strong solutions of
(1) provided such solutions exist.

######  Definition 2.8.

We say that the SDE (1) has a unique strong solution if there exists
$F\in\mathbf{T}(\partial\mathcal{V}\times\mathcal{V})$ such that,

  1. (1)

if $(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in D),\mathbb{P})$ is a
filtered probability space on which an $\mathbb{R}^{d}$-valued
$(\mathcal{F}_{s,t},(s,t)\in D)$-Brownian sheet $W$ with $\partial W=0$
exists, then for every continuous $(\mathcal{F}_{s,t},(s,t)\in D)$-adapted
boundary process $Z$ on $(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in
D),\mathbb{P})$ whose probability distribution is denoted by $\lambda$,
$(X,W)$ with $X=F(Z,W)$ is a weak solution of (1) with $\partial X=Z$
$\mathbb{P}$-a.s. on $\Omega$.

  2. (2)

if $(X,W)$ is a weak solution of (1) on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in D),\mathbb{P})$ and the
probability distribution of $\partial X$ is denoted by $\lambda$, then
$X=F_{\lambda}[\partial X,W]$ $\mathbb{P}$-a.s. on $\Omega$.

There are two classical notions of uniqueness associated to weak solutions
(see e.g. [28, Definitions 1.2 and 1.7]).

######  Definition 2.9.

We say that the solution to the SDE (1) is unique in the sense of probability
distribution if whenever $(X,W)$ and $(X^{\prime},W^{\prime})$ are two
solutions of (1) on two possibly different filtered probability spaces and
$\partial X=x=\partial X^{\prime}$ for some $x\in\partial\mathcal{V}$, then
$X$ and $X^{\prime}$ have the same probability distribution on
$(\mathcal{V},\mathcal{B}(\mathcal{V}))$.

######  Definition 2.10.

We say that the pathwise uniqueness of solutions to the SDE (1) holds if
whenever $(X,W)$ and $(X^{\prime},W)$ with the same $W$ are two solutions to
(1) on the same probability space and $\partial X=\partial X^{\prime}$, then
$X=X^{\prime}$ for $\mathbb{P}$-a.e. $\omega\in\Omega$.

##  3\. Main results

In this section, we present the main results of this paper. We assume the
following conditions on the drift:

######  Hypothesis 3.1.

  1. (1)

$b:[0,1]^{2}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is a Borel
measurable function satisfying the spatial linear growth condition, that is,
there exists a constant $M$ such that

$|b(t,s,x)|\leq M(1+|x|)\text{ for all }x\in\mathbb{R}^{d}.$

  2. (2)

$b$ is componentwise nondecreasing in space, that is, each component
$(b_{i})_{1\leq i\leq d}$ is componentwise nondecreasing in space. More
precisely for $x,y\in\mathbb{R}^{d}$, we have:

$x\preceq y\Rightarrow b_{i}(x)\leq b_{i}(y),1\leq i\leq d,$

where $x\preceq y$ means $x_{i}\leq y_{i}$ for all $i\in\\{1,\ldots,d\\}$.

######  Theorem 3.2.

Suppose $b$ satisfies Hypothesis 3.1. Then for almost every Brownian sheet
path $W$, there exists a unique continuous function
$X:[0,1]^{2}\rightarrow\mathbb{R}^{d}$ satisfying (1.8).

######  Corollary 3.3.

Suppose $b$ is as in Theorem 3.2. Then the SDE (1.8) admits a unique strong
solution.

###### Proof.

It follows from the fact that under the conditions of the Corollary, (1.8) has
a weak solution. In addition, since path-by-path uniqueness implies pathwise
uniqueness (see [4, Page 9, Section 1.8.4]), the result follows from the
Yamada-Watanabe type principle for SDEs driven by Brownian sheets (see e.g.
Nualart and Yeh [28]). ∎

The proof of Theorem 3.2 relies on the following theorem

######  Theorem 3.4.

Suppose $b$ is as in Theorem 3.2. Suppose in addition that $b$ is uniformly
bounded. Then for almost every Brownian sheet path $W$, there exists a unique
continuous function $X:[0,1]^{2}\rightarrow\mathbb{R}^{d}$ satisfying (1.8).

Recall that using the Girsanov theorem, path-by-path uniqueness holds if there
exists $\Omega_{1}\subset\Omega$ with $\mathbb{P}(\Omega_{1})=1$ such that for
any $\omega\in\Omega_{1}$, there is no nontrivial solution $u\in
C([0,1]^{2},\mathbb{R}^{d})$ to the following system of integral equations

(3.1) $\displaystyle
u(s,t)=\int_{0}^{t}\int_{0}^{s}\\{b(\xi,\zeta,W_{\xi,\zeta}(\omega)+u(\xi,\zeta))-b(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta,\text{
}\forall\,(s,t)\in[0,1]^{2}.$

Let us also consider the set $\mathbf{Q}=[-1,1]^{d}$ and its dyadic
decomposition. Recall that $x\in\mathbf{Q}$ is called a dyadic number if it is
a rational with denominator a power of $2$. The next theorem is equivalent to
Theorem 3.4.

######  Theorem 3.5.

Let $W:=\left(W_{s,t},(s,t)\in[0,1]^{2}\right)$ be a $d$-dimensional Brownian
sheet defined on a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where
$\mathbb{F}=(\mathcal{F}_{s,t};s,t\in[0,1])$. Let $b$ be as in Theorem 3.4.
Then there exists $\Omega_{1}\subset\Omega$ with $\mathbb{P}(\Omega_{1})=1$
such that for any $\omega\in\Omega_{1}$, $u=0$ is the unique solution in
$\mathcal{V}_{0}$ to the system of integral equations (3.1).

The proof of Theorem 3.5 is carried out in two main steps. In the first step,
we use a two-parameter Wiener process to regularise (3.1) on dyadic intervals.
In the second step we show a Gronwall type lemma (see Lemma 3.9). The
regularisation is as follows: For any positive integer $n$, we divide $[0,1]$
into $2^{n}$ intervals $I_{n,k}=]k2^{-n},(k+1)2^{-n}]$ and define
$\varrho_{nkk^{\prime}}$ by

$\displaystyle\varrho_{nkk^{\prime}}(x,y):=\int_{I_{n,k^{\prime}}}\int_{I_{n,k}}\\{b(s,t,W_{s,t}+x)-b(s,t,W_{s,t}+y)\\}\,\mathrm{d}t\mathrm{d}s.$

The next three lemmas whose proofs are given in Section 5 provide an estimate
for $\varrho_{nkk^{\prime}}(x,y)$ and $\varrho_{nkk^{\prime}}(0,x)$ for every
dyadic numbers $x,y\in\mathbf{Q}$. Lemmas 3.6 and 3.7 are counterparts of
Lemmas 3.1 and 3.2 in [13] for the Brownian sheet. The proof of Lemma 3.6 uses
the local time-space integration formula for the Brownian sheet as given in
[5]. Lemma 3.8 follows from Lemma 3.7 using the fact that the set of dyadic
numbers is dense in $\mathbb{R}$.

######  Lemma 3.6.

Suppose $b:\,[0,1]^{2}\times\mathbb{R}^{d}\to\mathbb{R}$ is a Borel measurable
function such that $|b(s,t,x)|\leq 1$ everywhere on
$[0,1]^{2}\times\mathbb{R}^{d}$. Then there exists a subset $\Omega_{1}$ of
$\Omega$ with $\mathbb{P}(\Omega_{1})=1$ such that for all
$\omega\in\Omega_{1}$,

$\displaystyle|\varrho_{nkk^{\prime}}(x,y)(\omega)|\leq
C_{1}(\omega)2^{-n}\Big{[}\sqrt{n}+\Big{(}\log^{+}\frac{1}{|x-y|}\Big{)}^{1/2}\Big{]}|x-y|\,\text{
on }\Omega_{1}$

for all dyadic numbers $x,\,y\in\mathbf{Q}$ and all choices of integers
$n,\,k,\,k^{\prime}$ with $n\geq 1$, $0\leq k,k^{\prime}\leq 2^{n}-1$, where
$C_{1}(\omega)$ is a positive random constant that does not depend on $x$,
$y$, $n$, $k$ and $k^{\prime}$.

######  Lemma 3.7.

Suppose $b$ is as in Lemma 3.6. Then there exists a subset $\Omega_{2}$ of
$\Omega$ with $\mathbb{P}(\Omega_{2})=1$ such that for all
$\omega\in\Omega_{2}$, for any choice of $n,\,k,\,k^{\prime}$, and any choice
of a dyadic number $x\in\mathbf{Q}$

(3.2) $\displaystyle\left|\varrho_{nkk^{\prime}}(0,x)(\omega)\right|\leq
C_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)},$

where $C_{2}(\omega)$ is a positive random constant that does not depend on
$x$, $n$, $k$ and $k^{\prime}$.

Observe that the proofs of the above two results do not require the monotonic
argument on the drift $b$.

######  Lemma 3.8.

Suppose $b$ is as in Theorem 3.4. Let $\Omega_{2}$ be a subset of $\Omega$
such that, for any $\omega\in\Omega_{2}$, (3.2) holds for every
$n,\,k,\,k^{\prime}$, and every dyadic number $x\in\mathbf{Q}$. Then

$\displaystyle\left|\varrho_{nkk^{\prime}}(0,x)(\omega)\right|\leq\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)}$

for any $\omega\in\Omega_{2}$, any $n$, $k$, $k^{\prime}$, and any
$x\in\mathbf{Q}$, where $\widetilde{C}_{2}(\omega)$ is a positive random
constant that does not depend on $x$, $n$, $k$ and $k^{\prime}$.

The subsequent result is a Gronwall type lemma and constitutes the main result
in the second step of the proof of Theorem 3.5. Its proof is found in Section
5.

######  Lemma 3.9.

Let $W:=\Big{(}W^{(1)}_{s,t},\ldots,W^{(d)}_{s,t};(s,t)\in[0,1]^{2}\Big{)}$ be
a $d$-dimensional Brownian sheet defined on a filtered probability space
$(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where
$\mathbb{F}=(\mathcal{F}_{s,t};(s,t)\in[0,1]^{2})$ and let the drift $b$ be as
in Theorem 3.4. There exists $\Omega_{1}\subset\Omega$ with
$\mathbb{P}(\Omega_{1})=1$ and a positive random constant $C_{1}$ such that
for any $\omega\in\Omega_{1}$, any sufficiently large positive integer $n$,
any $(k,k^{\prime})\in\\{1,2,\ldots,2^{n}\\}^{2}$, any
$\beta(n)\in\Big{[}2^{-4^{3n/4}},2^{-4^{2n/3}}\Big{]}$, and any solution $u$
to the system of integral equations

$\displaystyle u_{i}(s,t)-u_{i}(s,0)-u_{i}(0,t)+u_{i}(0,0)$ (3.3)
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\\{b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega)+u(\xi,\zeta))-b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta,\text{
}\forall\,(s,t)\in[0,1]^{2},\,\forall\,i,$

satisfying

(3.4)
$\max\\{|u(s,0)|,|u(0,t)|\\}\leq\beta(n),\quad\forall\,(s,t)\in[0,1]^{2},$

we have

(3.5)
$\displaystyle\max\\{|\overline{u}^{(n)}(k,k^{\prime})|,|\underline{u}^{(n)}(k,k^{\prime})|\\}\leq(3\sqrt{d})^{k+k^{\prime}-1}\Big{(}1+C_{1}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+k^{\prime}}\beta(n)\,\text{
on }\Omega_{1},$

where
$\overline{u}^{(n)}(k,k^{\prime})=(\overline{u}^{(n)}_{1}(k,k^{\prime}),\ldots,\overline{u}^{(n)}_{d}(k,k^{\prime}))$,
$\underline{u}^{(n)}(k,k^{\prime})=(\underline{u}^{(n)}_{1}(k,k^{\prime}),\ldots,\underline{u}^{(n)}_{\,d}(k,k^{\prime}))$,

$\displaystyle\overline{u}^{(n)}_{i}(k,k^{\prime}):=\sup\limits_{(s,t)\in
I_{n,k-1}\times
I_{n,k^{\prime}-1}}\max\\{0,u_{i}(s,t)\\}=\sup\limits_{(s,t)\in
I_{n,k-1}\times I_{n,k^{\prime}-1}}u_{i}^{+}(s,t),\quad\forall\,i$

and

$\displaystyle\underline{u}^{(n)}_{i}(k,k^{\prime}):=\sup\limits_{(s,t)\in
I_{n,k-1}\times
I_{n,k^{\prime}-1}}\max\\{0,-u_{i}(s,t)\\}=\sup\limits_{(s,t)\in
I_{n,k-1}\times I_{n,k^{\prime}-1}}u_{i}^{-}(s,t),\quad\forall\,i.$

We are now ready to prove Theorem 3.5.

###### Proof of Theorem 3.5.

We choose $\Omega_{1}$, $C_{1}$, $\omega$, $n$ and $\beta(n)$ as in Lemma 3.9.
Let $u$ be a solution to (3.1). We have
$\max\\{|u|(s,0),|u|(0,t)\\}=0\leq\beta(n)\text{ for all }(s,t)\in[0,1]^{2}.$
Moreover, we deduce from (3.5) that

(3.6)
$\displaystyle\sup\limits_{k,k^{\prime}\in\\{0,1,2,\cdots,2^{n}\\}}\max\left\\{|\overline{u}^{(n)}(k,k^{\prime})|,|\underline{u}^{(n)}(k,k^{\prime})|\right\\}\leq(4\sqrt{d})^{2^{n+1}}\beta(n)$

for all $n$ satisfying $C_{1}(\omega)\sqrt{dn}2^{-n}\leq 1/3$. Since the right
side of (3.6) converges to $0$ as $n$ goes to $\infty$, it holds $u(s,t)=0$ on
$\Omega_{1}$ for all $(s,t)$. ∎

###### Proof of Theorem 3.2.

For every positive integer $n$, consider the bounded and nondecreasing
function $g_{n}:\,\mathbb{R}\to\mathbb{R}$ defined by

$g_{n}(a)=\begin{cases}a\text{ for }|a|<n,\\\ n\text{ for }a\geq n,\\\
-n\text{ for }a\leq-n.\end{cases}$

Then for every $n$ and $i$, $g_{n}\circ b_{i}$ is a bounded and nondecreasing
function. Let $b^{(n)}:\,[0,1]^{2}\times\mathbb{R}^{d}\to\mathbb{R}^{d}$ be
the bounded and componentwise nondecreasing function given by
$b^{(n)}_{i}(s,t,x)=g_{n}(b_{i}(s,t,x))$ for all $i$, all $(s,t)\in[0,1]^{2}$
and all $x\in\mathbb{R}^{d}$. We have

$|b^{(n)}(s,t,x)|\leq
M(1+|x|),\quad\forall\,n,\,(s,t)\in[0,1]^{2},\,x\in\mathbb{R}^{d}.$

It follows from Theorem 3.4 that for any $n$, there exists an event
$\Omega_{n}$ of full measure, such that for any $\omega\in\Omega_{n}$, the
system of integral equations

(3.7) $\displaystyle
X_{s,t}(\omega)=\int_{0}^{t}\int_{0}^{s}b^{(n)}(\xi,\zeta,X_{\xi,\zeta}(\omega))\mathrm{d}\xi\mathrm{d}\zeta+W_{s,t}(\omega),\quad(s,t)\in[0,1],$

has a unique solution $(X_{s,t}^{(n)}(\omega),0\leq s,t\leq 1)$. Moreover, we
have

(3.8) $\displaystyle\Big{|}X_{s,t}^{(n)}(\omega)\Big{|}\leq
M\int_{0}^{t}\int_{0}^{s}\Big{|}X_{\xi,\zeta}^{(n)}(\omega)\Big{|}\mathrm{d}\xi\mathrm{d}\zeta+M+|W_{s,t}(\omega)|.$

By the Gronwall inequality for integrals in the plane provided in [35, Section
2] (see also [32, Section 1]), it holds

(3.9) $\displaystyle\Big{|}X_{s,t}^{(n)}(\omega)\Big{|}\leq
M+|W_{s,t}(\omega)|+M\int_{0}^{t}\int_{0}^{s}\Big{(}M+|W_{\xi,\zeta}(\omega)|\Big{)}h(\xi,\zeta,s,t)\mathrm{d}\xi\mathrm{d}\zeta$

for all $(s,t)\in[0,1]^{2}$, where $h$ is the unique solution to

(3.10) $\displaystyle
h(\xi,\zeta,s,t)=1+M\int_{\xi}^{t}\int_{\zeta}^{s}h(\eta,\gamma,s,t)\mathrm{d}\eta\mathrm{d}\gamma,\quad(\xi,\gamma)\in[0,s]\times[0,t].$

It is known (see for example [24, Page 145]) that for every $(\xi,\zeta,s,t)$,
$h(\xi,\zeta,s,t)=I_{0}\Big{(}2\sqrt{M(s-\xi)(t-\zeta)}\Big{)}$, where $I_{0}$
is the modified Bessel function of order zero. Since $h$ is nonnegative, we
have

$\displaystyle\sup\limits_{(s,t)\in[0,1]^{2}}\Big{|}X^{(n)}_{s,t}(\omega)\Big{|}\leq
C_{1}^{\ast}\Big{(}M+\sup\limits_{(s,t)\in[0,1]^{2}}|W_{s,t}(\omega)|\Big{)},$

with

$\displaystyle C_{1}^{\ast}=1+MI_{0}(2\sqrt{M}).$

Let $\Omega_{\infty}=\bigcap\limits_{n\geq 1}\Omega_{n}$, then
$\mathbb{P}(\Omega_{\infty})=1$. Fix $\omega\in\Omega_{\infty}$ and $n\geq 1$
such that

(3.11)
$\displaystyle(C_{1}^{\ast})^{2}M\Big{(}1+\sup\limits_{(s,t)\in[0,1]^{2}}|W_{s,t}(\omega)|\Big{)}\leq
n.$

Since $C_{1}^{\ast}\geq 1+M$, we obtain

$\displaystyle\sup\limits_{(s,t)\in[0,1]^{2}}\Big{|}b(s,t,X^{(n)}_{s,t}(\omega))\Big{|}$
$\displaystyle\leq
M\Big{(}1+\sup\limits_{(s,t)\in[0,1]^{2}}\Big{|}X^{(n)}_{s,t}(\omega)\Big{|}\Big{)}$
$\displaystyle\leq
M\Big{(}1+C_{1}^{\ast}M+C_{1}^{\ast}\sup\limits_{(s,t)\in[0,1]^{2}}|W_{s,t}(\omega)|\Big{)}\leq
n.$

As a consequence,
$b^{(n)}(s,t,X^{(n)}_{s,t}(\omega))=b(s,t,X^{(n)}_{s,t}(\omega))$ for every
$(s,t)\in[0,1]^{2}$. Hence $(X^{(n)}_{s,t}(\omega),0\leq s,t\leq 1)$ is a
solution to the system

(3.12) $\displaystyle
X_{s,t}(\omega)=\int_{0}^{t}\int_{0}^{s}b(\xi,\zeta,X_{\xi,\zeta}(\omega))\mathrm{d}\xi\mathrm{d}\zeta+W_{s,t}(\omega),\quad(s,t)\in[0,1].$

Let $(y_{s,t},0\leq s,t\leq 1)$ be another solution to (3.12) for the same
$\omega\in\Omega_{\infty}$. Then, using once more Gronwall inequality for
integrals on the plane, we obtain

$\displaystyle\sup\limits_{(s,t)\in[0,1]^{2}}|y_{s,t}|\leq
C_{1}^{\ast}\Big{(}M+\sup\limits_{(s,t)\in[0,1]^{2}}|W_{s,t}(\omega)|\Big{)}.$

This implies that for $n$ in (3.11), $(y_{s,t},0\leq s,t\leq 1)$ is also a
solution to (3.7). Since $\omega\in\Omega_{n}$, the system (3.7) has a unique
solution for this $\omega$. Thus, $y_{s,t}=X^{(n)}_{s,t}(\omega)$ for every
$(s,t)\in[0,1]^{2}$ and uniqueness is proved. ∎

##  4\. Preliminary results

In order to prove the auxiliary lemmas provided in the previous section, we
need some preliminary results that have been obtained by applying a local
time-space integration formula for Brownian sheets (see [5] for related
results). Let us first recall the notion of local time in the plane of the
Brownian sheet. Let $(W^{(0)}_{s,t},(s,t)\in D)$ be a one dimensional Brownian
sheet given on a filtered probability space. For $s$ fixed,
$(W^{(0)}_{s,t},t\in[0,1])$ is a one dimensional Brownian motion and its local
time process $(L_{1}^{x}(s,t);x\in\mathbb{R},t\geq 0)$ is given by the
Tanaka’s formula (see for example [37, Section 1]):

(4.1) $\displaystyle\int_{0}^{t}\mathbf{1}_{\\{W^{(0)}_{s,u}\leq
x\\}}\mathrm{d}_{u}W^{(0)}_{s,u}=\frac{s}{2}L^{x}_{1}(s,t)-(W^{(0)}_{s,t}-x)^{-}+x^{+}.$

For $s\in[0,1]$ fixed, let $\widehat{W}^{(0)}_{s,\cdot}$ be the time reversal
process on $[0,1]$ of the Brownian motion $\widehat{W}^{(0)}_{s,\cdot}$
(i.e.,$\widehat{W}^{(0)}_{s,t}=W^{(0)}_{s,1-t}$) and let
$(\widehat{L}^{x}_{1}(s,t);x\in\mathbb{R},0\leq t\leq 1)$ be the local time
process of $(\widehat{W}^{(0)}_{s,t},0\leq t\leq 1)$. Then the following holds

$\displaystyle\widehat{L}^{x}_{1}(s,t)=L^{x}_{1}(s,1)-L^{x}_{1}(s,1-t).$

Next, we consider the local time process in the plane
$L:=(L_{s,t}^{x};x\in\mathbb{R},s\geq 0,t\geq 0)$ as defined in [37, Section
2] (see also [38, Section 6, Page 157]) by

$\displaystyle
L^{x}_{s,t}:=\int_{0}^{s}L_{1}^{x}(\xi,t)\mathrm{d}\xi,\quad\forall\,x\in\mathbb{R},\,\forall\,(s,t)\in\mathbb{R}_{+}^{2}.$

Then it holds

(4.2) $\displaystyle
L^{x}_{s,t}=\int_{0}^{s}\int_{1-t}^{1}\mathbf{1}_{\\{W^{(0)}_{\xi,u}\leq
x\\}}\frac{\mathrm{d}_{u}W^{(0)}_{\xi,u}}{\xi}\mathrm{d}\xi+\int_{0}^{s}\int_{0}^{t}\mathbf{1}_{\\{\widehat{W}^{(0)}_{\xi,u}\leq
x\\}}\frac{\mathrm{d}_{u}\widehat{W}^{(0)}_{s,u}}{\xi}\mathrm{d}\xi,\quad\forall\,(s,t)\in[0,1]^{2}.$

Let us now consider the norm $\|\cdot\|$ defined by

$\displaystyle\|f\|:=$ $\displaystyle
2\Big{(}\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}f^{2}(s,t,x)\exp\Big{(}-\frac{x^{2}}{2st}\Big{)}\frac{\mathrm{d}x\mathrm{d}s\mathrm{d}t}{\sqrt{2\pi
st}}\Big{)}^{1/2}$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}|xf(s,t,x)|\exp\Big{(}-\frac{x^{2}}{2st}\Big{)}\frac{\mathrm{d}x\mathrm{d}s\mathrm{d}t}{st\sqrt{2\pi
st}}$ $\displaystyle=$ $\displaystyle
2\Big{(}\int_{0}^{1}\int_{0}^{1}\mathbb{E}\Big{[}f^{2}(s,t,W^{(0)}_{s,t})\Big{]}\mathrm{d}s\mathrm{d}t\Big{)}^{1/2}+\int_{0}^{1}\int_{0}^{1}\mathbb{E}\Big{[}\Big{|}f(s,t,W^{(0)}_{s,t})\frac{W^{(0)}_{s,t}}{st}\Big{|}\Big{]}\mathrm{d}s\mathrm{d}t.$

Consider the set $\mathcal{H}$ of measurable functions $f$ on
$[0,1]^{2}\times\mathbb{R}$ such that $\|f\|<\infty$. Endowed with
$\|\cdot\|$, the space $\mathcal{H}$ is a Banach space. In the following, we
define a stochastic integral over the space with respect to the local time for
the elements of $\mathcal{H}$. This extends the definition in [16]. We say
that $f_{\Delta}:\,[0,1]^{2}\times\mathbb{R}\to\mathbb{R}$ is an elementary
function if there exist two sequences of real numbers $(x_{i})_{0\leq i\leq
n}$, $(f_{ijk};0\leq i\leq n,0\leq j\leq m,0\leq k\leq\ell)$ and two
subdivisions of $[0,1]$ $(s_{j})_{0\leq j\leq m}$, $(t_{k})_{0\leq k\leq\ell}$
such that

(4.3) $\displaystyle
f_{\Delta}(s,t,x)=\sum\limits_{(x_{i},s_{j},t_{k})\in\Delta}f_{ijk}\mathbf{1}_{(x_{i},x_{i+1}]}(x)\mathbf{1}_{(s_{j},s_{j+1}]}(s)\mathbf{1}_{(t_{k},t_{k+1}]}(t),$

where $\Delta=\\{(x_{i},s_{j},t_{k});0\leq i\leq n,0\leq j\leq m,0\leq
k\leq\ell\\}$.

######  Definition 4.1.

For a simple function $f_{\Delta}$ given in (4.3), we define its integral with
respect to $L$ as

$\displaystyle\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}f_{\Delta}(s,t,x)\mathrm{d}L^{x}_{s,t}:=$
$\displaystyle\sum\limits_{(x_{i},s_{j},t_{k})\in\Delta}f_{ijk}\Big{(}L^{x_{i+1}}_{s_{j+1},t_{k+1}}-L^{x_{i+1}}_{s_{j},t_{k+1}}-L^{x_{i}}_{s_{j+1},t_{k+1}}+L^{x_{i}}_{s_{j},t_{k+1}}$
$\displaystyle\text{
}-L^{x_{i+1}}_{s_{j+1},t_{k}}+L^{x_{i+1}}_{s_{j},t_{k}}+L^{x_{i}}_{s_{j+1},t_{k}}-L^{x_{i}}_{s_{j},t_{k}}\Big{)}.$

######  Remark 4.2.

Let $f$ be an element of $\mathcal{H}$ and let $(f_{n})_{n\in\mathbb{N}}$ be a
sequence of elementary functions converging to $f$ in $\mathcal{H}$. It is
proved in [5, Proposition 2.1] that the sequence
$\left(\int_{0}^{1}\int_{0}^{1}\int_{\mathbb{R}}f_{n}(s,t,x)\mathrm{d}L^{x}_{s,t}\right)_{n\in\mathbb{N}}$
converges in $L^{1}(\Omega,\mathbb{P})$ and that the limit does not depend on
the choice of the sequence $(f_{n})_{n\in\mathbb{N}}$. This limit is called
integral of $f$ with respect to $L$. Similar results were obtained in [16].

Let $f:\,[0,1]^{2}\times\mathbb{R}^{d}\to\mathbb{R}$ be a continuous function
such that for any $(s,t)\in[0,1]^{2}$, $f(s,t,\cdot)$ is differentiable and
for any $i\in\\{1,\cdots,d\\}$, the partial derivative $\partial_{x_{i}}f$ is
continuous. We also know from [5, Proposition 3.1] that for a $d$-dimensional
Brownian sheet $\Big{(}W_{s,t}:=(W_{s,t}^{(1)},\cdots,W_{s,t}^{(d)});s\geq
0,t\geq 0\Big{)}$ defined on a filtered probability space and for any
$(s,t)\in[0,1]^{2}$ and any $i\in\\{1,\cdots,d\\}$, we have

$\displaystyle\int_{0}^{s}\int_{0}^{t}\partial_{x_{i}}f(\xi,u,W_{\xi,u})\mathrm{d}u\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle-\int_{0}^{s}\int_{0}^{t}f(\xi,u,W_{\xi,u})\frac{d_{u}W^{(i)}_{\xi,u}}{\xi}\mathrm{d}\xi-\int_{0}^{s}\int_{1-t}^{1}f(\xi,1-u,\widehat{W}_{\xi,u})\frac{d_{u}B^{(i)}_{\xi,u}}{\xi}\mathrm{d}\xi$
(4.4)
$\displaystyle+\int_{0}^{s}\int_{1-t}^{1}f(\xi,1-u,\widehat{W}_{\xi,u})\frac{\widehat{W}^{(i)}_{\xi,u}}{\xi(1-u)}\mathrm{d}u\mathrm{d}\xi,$

where $\widehat{W}^{(i)}:=(\widehat{W}^{(i)}_{\xi,u};0\leq\xi,u\leq 1)$ and
$B^{(i)}:=(B^{(i)}_{\xi,u};0\leq\xi,u\leq 1)$ is a standard Brownian sheet
with respect to the filtration of $\widehat{W}^{(i)}$, independent of
$(W^{(i)}_{s,1},s\geq 0)$.

The following result will be extensively used in this work and corresponds to
[33, Proposition 2.1] for the standard Wiener process.

######  Proposition 4.3.

Let $W:=\left(W^{(1)}_{s,t},\ldots,W^{(d)}_{s,t};(s,t)\in[0,1]^{2}\right)$ be
a $\mathbb{R}^{d}$-valued Brownian sheet defined on a filtered probability
space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$, where
$\mathbb{F}=(\mathcal{F}_{s,t};s,t\in[0,1])$. Let
$b\in\mathcal{C}\left([0,1]^{2},\mathcal{C}^{1}(\mathbb{R}^{d})\right)$,
$\|b\|_{\infty}\leq 1$. Let
$(a,a^{\prime},\varepsilon,\varepsilon^{\prime})\in[0,1]^{4}$. Then there
exist positive constants $\alpha$ and $C$ (independent of $\nabla_{y}b$, $a$,
$a^{\prime}$, $\varepsilon$ and $\varepsilon^{\prime}$) such that

(4.5)
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon^{\prime}\varepsilon\Big{|}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b\left(s,t,\widetilde{W}^{\varepsilon,\varepsilon^{\prime}}_{s,t}\right)\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}\leq
C.$

Here $\nabla_{y}b$ denotes the gradient of $b$ with respect to the third
variable, $|\cdot|$ is the usual norm on $\mathbb{R}^{d}$ and the
$\mathbb{R}^{d}$-valued two-parameter Gaussian process
$\widetilde{W}^{\varepsilon,\varepsilon^{\prime}}:=\Big{(}\widetilde{W}^{(\varepsilon,\varepsilon^{\prime},1)}_{s,t},\ldots,\widetilde{W}^{(\varepsilon,\varepsilon^{\prime},d)}_{s,t};(s,t)\in[0,1]^{2}\Big{)}$
is given by

$\widetilde{W}^{(\varepsilon,\varepsilon^{\prime},i)}_{s,t}=W^{(i)}_{a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon
t}-W^{(i)}_{a^{\prime},a+\varepsilon
t}-W^{(i)}_{a^{\prime}+\varepsilon^{\prime}s,a}+W^{(i)}_{a^{\prime},a}\quad\text{for
all }i\in\\{1,\ldots,d\\}.$

###### Proof.

The proof of (4.5) is based on the local time-space integration formula (4)
and the Barlow-Yor inequality. Fix
$(a,a^{\prime},\varepsilon,\varepsilon^{\prime})\in[0,1]^{4}$. Since
$x\longmapsto e^{\alpha\varepsilon\varepsilon^{\prime}dx^{2}}$ is a convex
function, we deduce from the Jensen inequality that

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon^{\prime}\varepsilon\Big{|}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b\Big{(}s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}\Big{)}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon^{\prime}\varepsilon\sum\limits_{i=1}^{d}\Big{|}\int_{0}^{1}\int_{0}^{1}\partial_{y_{i}}b\Big{(}s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}\Big{)}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{d}\sum\limits_{i=1}^{d}\mathbb{E}\Big{[}\exp\Big{(}\alpha
d\varepsilon^{\prime}\varepsilon\Big{|}\int_{0}^{1}\int_{0}^{1}\partial_{y_{i}}b\Big{(}s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}\Big{)}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}.$

In order to obtain (4.5) it suffices to prove that for every
$i\in\\{1,2,\ldots,d\\}$, there exist positive constants $\alpha=\alpha_{i}$
and $C=C_{i}$ such that

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon\varepsilon^{\prime}\left|\int_{0}^{1}\int_{0}^{1}\partial_{y_{i}}b\Big{(}s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}\Big{)}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\right]\leq
C.$

For every $i\in\\{1,\ldots,d\\}$, we apply (4) to the standard $d$-dimensional
Brownian sheet
$\Big{(}Y_{s,t}:=(\varepsilon\varepsilon^{\prime})^{-1/2}\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t},(s,t)\in[0,1]^{2}\Big{)}$
and the function $f:\,[0,1]^{2}\times\mathbb{R}^{d}\to\mathbb{R}$ given by
$f(s,t,y)=b\Big{(}s,t,\sqrt{\varepsilon\varepsilon^{\prime}}\,y\Big{)}$ to
obtain

$\displaystyle\int_{0}^{1}\int_{0}^{1}\partial_{y_{i}}f(s,t,Y_{s,t})\mathrm{d}t\mathrm{d}s$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}\int_{0}^{1}f(s,t,Y_{s,t})\frac{\mathrm{d}_{t}Y^{(i)}_{s,t}}{s}\mathrm{d}s-\int_{0}^{1}\int_{0}^{1}f(s,1-t,Y_{s,1-t})\frac{\mathrm{d}_{t}B^{(i)}_{s,t}}{s}\mathrm{d}s$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}f(s,1-t,Y_{s,1-t})\frac{Y^{(i)}_{s,1-t}}{s(1-t)}\mathrm{d}t\mathrm{d}s,$

where $(B^{(i)}_{s,t},0\leq s,t\leq 1)$ denotes a standard Brownian sheet
independent of the process $(Y^{(i)}_{s,1},0\leq s\leq 1)$. Hence,

$\displaystyle\sqrt{\varepsilon\varepsilon^{\prime}}\int_{0}^{1}\int_{0}^{1}\partial_{y_{i}}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t})\mathrm{d}t\mathrm{d}s$
$\displaystyle=$
$\displaystyle-\int_{0}^{1}\int_{0}^{1}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t})\frac{\mathrm{d}_{t}Y^{(i)}_{s,t}}{s}\mathrm{d}s-\int_{0}^{1}\int_{0}^{1}b(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})\frac{\mathrm{d}_{t}B^{(i)}_{s,t}}{s}\mathrm{d}s$
$\displaystyle+\int_{0}^{1}\int_{0}^{1}b(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})\frac{Y^{(i)}_{s,1-t}}{s(1-t)}\mathrm{d}t\mathrm{d}s$
$\displaystyle=$ $\displaystyle I_{1}+I_{2}+I_{3}.$

Using once more the convexity of the function $x\longmapsto e^{3\alpha x^{2}}$
for any $\alpha>0$, we obtain

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon\varepsilon^{\prime}\Big{|}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b\Big{(}s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}\Big{)}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha(I_{1}+I_{2}+I_{3})^{2}\Big{)}\Big{]}$
$\displaystyle\leq$
$\displaystyle\frac{1}{3}\Big{(}\mathbb{E}\Big{[}\exp(3\alpha
I_{1}^{2})\Big{]}+\mathbb{E}\Big{[}\exp(3\alpha
I_{2}^{2})\Big{]}+\mathbb{E}\Big{[}\exp(3\alpha I_{3}^{2})\Big{]}\Big{)}.$

Hence to get the desired estimate, we need to prove that for every
$k\in\\{1,2,3\\}$, there exist positive constants $\alpha_{k}$ and $C_{k}$
such that $\mathbb{E}\left[\exp(\alpha_{k}I_{k}^{2})\right]\leq C_{k}.$
For every $s\in]0,1]$, $\left(s^{-1/2}Y_{s,v},0\leq v\leq 1\right)$ is a
standard Brownian motion with respect to the filtration
$\mathcal{F}_{1,\cdot}:=(\mathcal{F}_{1,t},t\in[0,1])$. Therefore the process

$\Big{(}M_{s,t}:=\int_{0}^{t}b(s,v,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,v})\mathrm{d}_{v}\left[\frac{Y_{s,v}}{\sqrt{s}}\right],0\leq
t\leq 1\Big{)}$

is an Itô integral with respect to $\mathcal{F}_{1,\cdot}$ and thus a square-
integrable $\mathcal{F}_{1,\cdot}$-martingale. In addition, for any constant
$\alpha\in\mathbb{R}_{+}$, the following expansion formula holds

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha
I_{1}^{2}\Big{)}\Big{]}=\mathbb{E}\Big{[}\exp\Big{(}\alpha\Big{|}\int_{0}^{1}M_{s,1}\frac{\mathrm{d}s}{\sqrt{s}}\Big{|}^{2}\Big{)}\Big{]}=\sum\limits_{m=0}^{\infty}\frac{\alpha^{m}\mathbb{E}\Big{[}\Big{|}\displaystyle\int_{0}^{1}M_{s,1}\frac{\mathrm{d}s}{\sqrt{s}}\Big{|}^{2m}\Big{]}}{m!}.$

Moreover, by the Jensen inequality and the Barlow-Yor inequality applied to
the martingale $(M_{s,t},t\in[0,1])$ (see for example [3, Proposition 4.2] and
[9, Appendix]), there exists a universal constant $c_{1}$ (not depending on
$m$) such that,

$\displaystyle\mathbb{E}\Big{[}\Big{|}\int_{0}^{1}M_{s,1}\frac{\mathrm{d}s}{\sqrt{s}}\Big{|}^{2m}\Big{]}$
$\displaystyle\leq
4^{m}\int_{0}^{1}\mathbb{E}\Big{[}|M_{s,1}|^{2m}\Big{]}\frac{\mathrm{d}s}{2\sqrt{s}}\leq
4^{m}\int_{0}^{1}\mathbb{E}\Big{[}\Big{(}\sup\limits_{0\leq t\leq
1}|M_{s,t}|\Big{)}^{2m}\Big{]}\frac{\mathrm{d}s}{2\sqrt{s}}$
$\displaystyle\leq c_{1}^{2m}(8m)^{m}\int_{0}^{1}\mathbb{E}\Big{[}\langle
M_{s,\cdot}\rangle_{1}^{m}\Big{]}\frac{\mathrm{d}s}{2\sqrt{s}}$
$\displaystyle\leq
c_{1}^{2m}(8m)^{m}\int_{0}^{1}\mathbb{E}\Big{[}\Big{(}\int_{0}^{1}b^{2}(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t})\mathrm{d}t\Big{)}^{m}\Big{]}\frac{\mathrm{d}s}{2\sqrt{s}}\leq
c_{1}^{2m}(8m)^{m},$

since $\|b\|_{\infty}\leq 1$. Thus,

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha
I_{1}^{2}\Big{)}\Big{]}=\mathbb{E}\Big{[}\exp\Big{(}\alpha\Big{|}\int_{0}^{1}M_{s,1}\frac{\mathrm{d}s}{\sqrt{s}}\Big{|}^{2}\Big{)}\Big{]}=\sum\limits_{m=0}^{\infty}\frac{\Big{(}8\alpha
c_{1}^{2}\Big{)}^{m}m^{m}}{m!}.$

The above expression if finite for $\alpha$ such that $8\alpha c_{1}^{2}e<1$,
i.e. $\alpha<1/8c_{1}^{2}e$ (by ratio test). Hence, there exists positive
constants $\alpha_{1}$ and $C_{1}$ such that

$\mathbb{E}\Big{[}\exp\Big{(}\alpha_{1}I_{1}^{2}\Big{)}\Big{]}\leq C_{1}.$

Similarly for

$\displaystyle
I_{2}=-\int_{0}^{1}\int_{0}^{1}b(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})\mathrm{d}_{t}B^{(i)}_{s,t}\frac{\mathrm{d}s}{s},$

there exists positive constants $\alpha_{2}$ and $C_{2}$ such that

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\alpha_{2}I_{2}^{2}\Big{)}\Big{]}\leq\mathbb{E}\Big{[}\exp\Big{(}\alpha_{2}\Big{|}\int_{0}^{1}\int_{0}^{1}b(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})\mathrm{d}_{t}B^{(i)}_{s,t}\frac{\mathrm{d}s}{s}\Big{|}^{2}\Big{)}\Big{]}\leq
C_{2}.$

It remains to estimate the term $I_{3}$. By the Jensen inequality, we have

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{I_{3}^{2}}{64}\Big{)}\Big{]}=$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{\\{}\frac{1}{4}\Big{(}\int_{0}^{1}\int_{0}^{1}\frac{b(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})Y^{(i)}_{s,1-t}}{\sqrt{s(1-t)}}\frac{\mathrm{d}t\mathrm{d}s}{4\sqrt{s(1-t)}}\Big{)}^{2}\Big{\\}}\Big{]}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{1}\int_{0}^{1}\mathbb{E}\Big{[}\exp\Big{\\{}\frac{1}{4}b^{2}(s,1-t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,1-t})\Big{|}\frac{Y^{(i)}_{s,1-t}}{\sqrt{s(1-t)}}\Big{|}^{2}\Big{\\}}\Big{]}\frac{\mathrm{d}t\mathrm{d}s}{4\sqrt{s(1-t)}}$
(4.6) $\displaystyle\leq$
$\displaystyle\int_{0}^{1}\int_{0}^{1}\mathbb{E}\Big{[}\exp\Big{(}\frac{1}{4}\Big{|}\frac{Y^{(i)}_{s,1-t}}{\sqrt{s(1-t)}}\Big{|}^{2}\Big{)}\Big{]}\frac{\mathrm{d}t\mathrm{d}s}{4\sqrt{s(1-t)}}.$

Note that for every $(s,t)\in]0,1]\times[0,1[$,
$\dfrac{Y^{(i)}_{s,1-t}}{\sqrt{s(1-t)}}$ is a standard normal random variable.
Therefore (4.6) yields

$\mathbb{E}\Big{[}\exp\Big{(}\frac{I_{3}^{2}}{64}\Big{)}\Big{]}\leq C_{3}.$

The proof of (4.5) is completed by taking
$\alpha=\min(\frac{1}{64},\alpha_{2},\alpha_{3})$. ∎

For every $0\leq a<h\leq 1$, $0\leq a^{\prime}<h^{\prime}\leq 1$ and for
$(x,y)\in\mathbb{R}^{d}$ define the function $\varrho$ by:

$\varrho(x,y):=\int_{a^{\prime}}^{h^{\prime}}\int_{a}^{h}\Big{\\{}b(\xi,\zeta,W_{\xi,\zeta}+x)-b(\xi,\zeta,W_{\xi,\zeta}+y)\Big{\\}}\mathrm{d}\zeta\mathrm{d}\xi.$

As a consequence of Proposition 4.3, we have:

######  Corollary 4.4.

Let $b:[0,1]^{2}\times\mathbb{R}^{d}\to\mathbb{R}$ be a bounded and Borel
measurable function such that $\|b\|_{\infty}\leq 1$. Let $\alpha$, $C$ and
$\widetilde{W}^{\varepsilon,\varepsilon^{\prime}}$ be defined as in
Proposition 4.3. Then the following two bounds are valid:

  1. (1)

For every $(x,y)\in\mathbb{R}^{2d}$, $x\neq y$ and every
$(\varepsilon,\varepsilon^{\prime})\in[0,1]^{2}$, we have

(4.7)
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon^{\prime}\varepsilon}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\left\\{b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+x)-b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y)\right\\}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}\leq
C.$

  2. (2)

For any $(x,y)\in\mathbb{R}^{2}$ and any $\eta>0$, we have

(4.8)
$\displaystyle\mathbb{P}\left(|\varrho(x,y)|\geq\eta\sqrt{(h-a)(h^{\prime}-a^{\prime})}|x-y|\right)\leq
Ce^{-\alpha\eta^{2}}.$

###### Proof.

We start by showing (4.7). Note that it is enough to show this when $b$ is
compactly supported and differentiable. Indeed, if $b$ is not differentiable,
then, since the set of compactly supported and differentiable functions is
dense in $L^{\infty}([0,1]^{2}\times\mathbb{R}^{d})$, there exists a sequence
$(b_{n},n\in\mathbb{N})$ of compactly supported and differentiable functions
which converges a.e. to $b$ on $[0,1]^{2}\times\mathbb{R}^{d}$ and the desired
result will follow from the Vitali’s convergence theorem.
Using the mean value theorem and the Cauchy-Schwartz inequality, we have

$\displaystyle\Big{|}\int_{0}^{1}\int_{0}^{1}\Big{\\{}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+x)-b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s\Big{|}^{2}$
$\displaystyle=$
$\displaystyle\Big{|}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y+\xi(x-y))\cdot(x-y)\mathrm{d}\xi\mathrm{d}t\mathrm{d}s\Big{|}^{2}$
$\displaystyle\leq$
$\displaystyle|x-y|^{2}\Big{|}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y+\xi(x-y))\mathrm{d}\xi\mathrm{d}t\mathrm{d}s\Big{|}^{2}.$

Using the Minkowski inequality, the Jensen inequality and Proposition 4.3
applied to the function $(s,t,z)\longmapsto b(s,t,z+y+\xi(x-y))$, we obtain

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\Big{\\{}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+x)-b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y+\xi(x-y))\cdot(x-y)\mathrm{d}\xi\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle\leq$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\int_{0}^{1}\Big{|}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y+\xi(x-y))\cdot(x-y)\mathrm{d}t\mathrm{d}s\Big{|}^{2}\mathrm{d}\xi\Big{)}\Big{]}$
(4.9) $\displaystyle\leq$
$\displaystyle\int_{0}^{1}\mathbb{E}\Big{[}\exp\Big{(}\alpha\varepsilon\varepsilon^{\prime}\Big{|}\int_{0}^{1}\int_{0}^{1}\nabla_{y}b(s,t,\widetilde{W}^{\varepsilon^{\prime},\varepsilon}_{s,t}+y+\xi(x-y))\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}\mathrm{d}\xi\leq
C.$

This ends the proof of (4.7).

As for the proof of (4.8), let $(x,y)\in\mathbb{R}^{2d}$ such that $x\neq y$
and set $\varepsilon=h-a$ and$\varepsilon^{\prime}=h^{\prime}-a^{\prime}$.
Define $\widehat{b}$ by
$\widehat{b}(s,t,x):=b(a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon t,x)$
and additionally define the processes
$\widetilde{W}^{\varepsilon^{\prime},\varepsilon}:=\Big{(}\widetilde{W}_{s,t}^{\varepsilon,\varepsilon^{\prime}},(s,t)\in[0,1]^{2}\Big{)}$
and
$Z^{\varepsilon^{\prime},\varepsilon}:=\Big{(}Z_{s,t}^{\varepsilon^{\prime},\varepsilon},(s,t)\in[0,1]^{2}\Big{)}$,
respectively by

$\displaystyle\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}=W_{a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon
t}-W_{a^{\prime},a+\varepsilon
t}-W_{a^{\prime}+\varepsilon^{\prime}s,a}+W_{a^{\prime},a},$

and

$\displaystyle
Z_{s,t}^{\varepsilon^{\prime},\varepsilon}=W_{a^{\prime},a+\varepsilon
t}+W_{a^{\prime}+\varepsilon^{\prime}s,a}-W_{a^{\prime},a}.$

Then $\widetilde{W}^{\varepsilon^{\prime},\varepsilon}$ and
$Z^{\varepsilon^{\prime},\varepsilon}$ are independent processes. To see this,
observe that $Z^{\varepsilon^{\prime},\varepsilon}_{s,t}$ is
$\mathcal{F}_{1,a}\vee\mathcal{F}_{a^{\prime},1}$-measurable for every
$(s,t)\in[0,1]^{2}$ and $\widetilde{W}^{\varepsilon^{\prime},\varepsilon}$ is
independent of $\mathcal{F}_{1,a}\vee\mathcal{F}_{a^{\prime},1}$. Using the
change of variable
$(\xi,\zeta):=(a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon t)$, we obtain

$\displaystyle\varrho(x,y)$
$\displaystyle=\int_{a^{\prime}}^{h^{\prime}}\int_{a}^{h}\Big{\\{}b(\xi,\zeta,W_{\xi,\zeta}+x)-b(\xi,\zeta,W_{\xi,\zeta}+y)\Big{\\}}\mathrm{d}\zeta\mathrm{d}\xi$
$\displaystyle=$
$\displaystyle\varepsilon\varepsilon^{\prime}\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{b}(s,t,W_{a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon
t}+x)-\widehat{b}(s,t,W_{a^{\prime}+\varepsilon^{\prime}s,a+\varepsilon
t}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s$ $\displaystyle=$
$\displaystyle\varepsilon\varepsilon^{\prime}\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{b}(s,t,\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+Z_{s,t}^{\varepsilon^{\prime},\varepsilon}+x)-\widehat{b}(s,t,\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+Z_{s,t}^{\varepsilon,\varepsilon^{\prime}}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s.$

Taking the expectation on both sides after some operations and using the fact
that $\widetilde{W}^{\varepsilon^{\prime},\varepsilon}$ and
$Z^{\varepsilon^{\prime},\varepsilon}$ are independent, we have

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha|\varrho(x,y)|^{2}}{\varepsilon\varepsilon^{\prime}|x-y|^{2}}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\left\\{\widehat{b}(s,t,\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+Z_{s,t}^{\varepsilon,\varepsilon^{\prime}}+x)-\widehat{b}(s,t,\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+Z_{s,t}^{\varepsilon,\varepsilon^{\prime}}+y)\right\\}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\int\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{b}^{z}_{s,t}(\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+x)-\widehat{b}^{z}_{s,t}(\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}\mathbb{P}_{Z^{\varepsilon,\varepsilon^{\prime}}}(\mathrm{d}z),$

where $\widehat{b}^{z}_{s,t}(w)=\widehat{b}(s,t,w+z_{s,t})$. Since
$|\widehat{b}(s,t,w)|\leq 1$ almost everywhere on
$[0,1]^{2}\times\mathbb{R}^{d}$, we deduce from (4.7) that

$\displaystyle\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha|\varrho(x,y)|^{2}}{\varepsilon\varepsilon^{\prime}|x-y|^{2}}\Big{)}\Big{]}$
$\displaystyle=$
$\displaystyle\int\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha\varepsilon\varepsilon^{\prime}}{|x-y|^{2}}\Big{|}\int_{0}^{1}\int_{0}^{1}\Big{\\{}\widehat{b}^{z}_{s,t}(\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+x)-\widehat{b}^{z}_{s,t}(\widetilde{W}_{s,t}^{\varepsilon^{\prime},\varepsilon}+y)\Big{\\}}\mathrm{d}t\mathrm{d}s\Big{|}^{2}\Big{)}\Big{]}\mathbb{P}_{Z^{\varepsilon,\varepsilon^{\prime}}}(\mathrm{d}z)\leq
C.$

Therefore, by Chebychev inequality, we obtain

$\displaystyle\mathbb{P}\Big{(}|\varrho(x,y)|\geq\eta\sqrt{\varepsilon\varepsilon^{\prime}}|x-y|\Big{)}\leq
e^{-\alpha\eta^{2}}\mathbb{E}\Big{[}\exp\Big{(}\frac{\alpha|\varrho(x,y)|^{2}}{\varepsilon\varepsilon^{\prime}|x-y|^{2}}\Big{)}\Big{]}\leq
Ce^{-\alpha\eta^{2}}$

for any $\eta>0$. This concludes the proof. ∎

##  5\. Proof of the auxiliary results

In this section we prove the auxiliary results stated in Section 3. Consider
$\mathbf{Q}=[-1,1]^{d}$ and its dyadic decomposition.

###  5.1. Regularization by noise

###### Proof of Lemma 3.6.

Let $\mathcal{Q}$ denote the set of couples $(x,y)$ of dyadic numbers in
$\mathbf{Q}$. For every integer $m\geq 0$, we define

$\mathcal{Q}_{m}=\left\\{(x,y)\in\mathcal{Q}:\,2^{m}x\in\mathbb{Z}^{d},\,2^{m}y\in\mathbb{Z}^{d}\right\\}.$

Then $\mathcal{Q}_{m}$ has no more than $2^{2d(m+2)}$ elements and we have
$\mathcal{Q}=\bigcup\limits_{m\in\mathbb{N}}\mathcal{Q}_{m}$. Consider the set
$\mathbf{E}_{\delta,n}$ defined by

$\displaystyle\mathbf{E}_{\delta,n}:=$
$\displaystyle\left\\{\omega\in\Omega:\,\text{there exist
}k,k^{\prime}\in\\{0,1,\ldots,2^{n}-1\\},\text{ }m\in\mathbb{N}^{\ast},\text{
}\text{ and }(x,y)\in\mathcal{Q}_{m}\right.$ $\displaystyle\qquad\text{ such
that
}|\varrho_{nkk^{\prime}}(x,y)|(\omega)\geq\delta(1+\sqrt{n}+\sqrt{m})2^{-n}|x-y|\\}$

for every $n\in\mathbb{N}$, $\delta\in\mathbb{Q}_{+}$. Observe that

$\displaystyle\mathbf{E}_{\delta,n}$ $\displaystyle=$
$\displaystyle\bigcup_{k=0}^{2^{n}-1}\bigcup_{k^{\prime}=0}^{2^{n}-1}\bigcup_{m=0}^{\infty}\Big{(}\bigcup_{(x,y)\in\mathcal{Q}_{m}}\left\\{\omega\in\Omega:\,|\varrho_{nkk^{\prime}}(x,y)|(\omega)\geq\delta(1+\sqrt{n}+\sqrt{m})2^{-n}|x-y|\right\\}\Big{)}.$

Define also $\mathbf{E}_{\delta}$ by
$\mathbf{E}_{\delta}:=\bigcup\limits_{n=0}^{\infty}\mathbf{E}_{\delta,n}$.
Then, we deduce from (4.8) that

$\displaystyle\mathbb{P}(\mathbf{E}_{\delta})$
$\displaystyle\leq\sum\limits_{n=0}^{\infty}\sum\limits_{k=0}^{2^{n}-1}\sum\limits_{k^{\prime}=0}^{2^{n}-1}\sum\limits_{m=0}^{\infty}\sum\limits_{(x,y)\in\mathcal{Q}_{m}}\mathbb{P}\left(|\varrho_{nkk^{\prime}}(x,y)|\geq\delta(1+\sqrt{n}+\sqrt{m})2^{-n}|x-y|\right)$
$\displaystyle\leq
2^{4d}C\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty}2^{2n}2^{2dm}e^{-\alpha\delta^{2}(1+n+m)},$

where $C$ and $\alpha$ are the deterministic constants in Proposition 4.3.
Moreover, $(\mathbf{E}_{\delta},\delta\in\mathbb{Q}_{+})$ is a nonincreasing
family and for $\delta\geq\delta_{0}:=\sqrt{2(d+1)\alpha^{-1}}$, we have

(5.1) $\displaystyle\mathbb{P}(\mathbf{E}_{\delta})\leq
2^{4d}Ce^{-\alpha\delta^{2}}\,\sum\limits_{n=0}^{\infty}\sum\limits_{m=0}^{\infty}2^{-dn-2m}\leq
2^{4d}C\,e^{-\alpha\delta^{2}}.$

Therefore

$\displaystyle\mathbb{P}\Big{(}\bigcap\limits_{\delta\in[\delta_{0},\infty[\cap\mathbb{Q}}\mathbf{E}_{\delta}\Big{)}=\lim\limits_{\delta\to\infty}\mathbb{P}(\mathbf{E}_{\delta})=0.$

Let us now define $\Omega_{1}$ by

$\Omega_{1}:=\bigcup\limits_{\delta\in[\delta_{0},\infty[\cap\mathbb{Q}}\left(\Omega\setminus\mathbf{E}_{\delta}\right).$

Then $\mathbb{P}(\Omega_{1})=1$ and for every $\omega\in\Omega_{1}$, there
exists $\delta_{\omega}>0$ such that

(5.2)
$\displaystyle|\varrho_{nkk^{\prime}}(x,y)(\omega)|<\delta_{\omega}2^{-n}(1+\sqrt{n}+\sqrt{m})|x-y|$

for all choices of $n$, $k$, $k^{\prime}$, $m$, and all choices of couples
$(x,y)$ in $\mathcal{Q}_{m}$.
Now, fix $\omega\in\Omega_{1}$, choose two dyadic numbers $x$, $y$ in
$\mathbf{Q}$ and define $m$ by

(5.3) $\displaystyle m:=\inf\Big{\\{}n\in\mathbb{N}\text{ s.t.
}2^{-n}\leq\max_{1\leq i\leq d}|x_{i}-y_{i}|\Big{\\}}.$

For $r\geq m$, define $x_{r}=(2^{-r}\left\lfloor
2^{r}x_{1}\right\rfloor,\ldots,2^{-r}\left\lfloor 2^{r}x_{d}\right\rfloor)$
and $y_{r}=(2^{-r}\left\lfloor
2^{r}y_{1}\right\rfloor,\ldots,2^{-r}\left\lfloor 2^{r}y_{d}\right\rfloor)$,
where $\left\lfloor\cdot\right\rfloor$ is the integer part function. Observe
that $(x_{m},y_{m})\in\mathcal{Q}_{m}$, $(x_{r},x_{r+1})\in\mathcal{Q}_{r+1}$
and $(y_{r},y_{r+1})\in\mathcal{Q}_{r+1}$. Since for any
$i\in\\{1,\ldots,d\\}$, $|\left\lfloor 2^{m}x_{i}\right\rfloor-\left\lfloor
2^{m}y_{i}\right\rfloor|\leq 1+2^{m}|x_{i}-y_{i}|$ and $(\left\lfloor
2x_{i}\right\rfloor-2\left\lfloor x_{i}\right\rfloor,\left\lfloor
2y_{i}\right\rfloor-2\left\lfloor y_{i}\right\rfloor)\in\\{0,1\\}^{2}$, we
have $|x_{m}-y_{m}|\leq 3\sqrt{d}\,2^{-m}$,
$|x_{r+1}-x_{r}|\leq\sqrt{d}\,2^{-r-1}$ and
$|y_{r+1}-y_{r}|\leq\sqrt{d}\,2^{-r-1}$.

It follows from the definition of $\rho_{nk}$ that

(5.4)
$\displaystyle\varrho_{nkk^{\prime}}(x,y)=\varrho_{nkk^{\prime}}(x,x_{m})+\varrho_{nkk^{\prime}}(x_{m},y_{m})+\varrho_{nkk^{\prime}}(y_{m},y).$

Moreover since $\varrho_{nkk^{\prime}}(x_{m},x_{m})=0$ and
$\varrho_{nkk^{\prime}}(y_{m},y_{m})=0$, we obtain

$\displaystyle\varrho_{nkk^{\prime}}(x_{q+1},x_{m})=\sum\limits_{r=m}^{q}\rho_{nkk^{\prime}}(x_{r+1},x_{r})\,\text{
and
}\varrho_{nkk^{\prime}}(y_{m},y_{q+1})=\sum\limits_{r=m}^{q}\varrho_{nkk^{\prime}}(y_{r},y_{r+1})$

for every integer $q\geq m+1$. In addition for some integer $q\geq m+1$, we
have $x_{r}=x$ and $y_{r}=y$ for all $r\geq q$, therefore

$\displaystyle\varrho_{nkk^{\prime}}(x,x_{m})=\sum\limits_{r=m}^{\infty}\varrho_{nkk^{\prime}}(x_{r},x_{r+1})\,\text{
and
}\varrho_{nkk^{\prime}}(y_{m},y)=\sum\limits_{r=m}^{\infty}\varrho_{nkk^{\prime}}(y_{r+1},y_{r}).$

It follows from (5.4) and (5.2) that

$\displaystyle 2^{n}|\varrho_{nkk^{\prime}}(x,y)(\omega)|$ $\displaystyle\leq
3\sqrt{d}\delta_{\omega}(1+\sqrt{n}+\sqrt{m})2^{-m}+2\sqrt{d}\delta_{\omega}\sum\limits_{r=m}^{\infty}(1+\sqrt{n}+\sqrt{r+1})2^{-r-1}$
$\displaystyle\leq
3\sqrt{d}\delta_{\omega}(1+\sqrt{n}+\sqrt{m})2^{-m}+2\sqrt{d}\delta_{\omega}\sqrt{n}\sum\limits_{r=m}^{\infty}2^{-r-1}+\frac{4\sqrt{d}\delta_{\omega}}{\sqrt{m+1}}\sum\limits_{r=m}^{\infty}(r+1)2^{-r-1}.$

Using the following facts: $\sum\limits_{r=m}^{\infty}2^{-r-1}=2^{-m}$,
$\sum\limits_{r=m}^{\infty}r2^{-r+1}=(m+1)2^{-m+2}$,
$2^{-m}\leq\max\limits_{1\leq i\leq d}|x_{i}-y_{i}|\leq 2^{-m+1}$,
$|x-y|\leq\sqrt{d}\max\limits_{1\leq i\leq d}|x_{i}-y_{i}|\leq\sqrt{d}\,|x-y|$
and $n\geq 1$, we obtain

$\displaystyle 2^{n}|\varrho_{nkk^{\prime}}(x,y)(\omega)|$ $\displaystyle\leq$
$\displaystyle 24\sqrt{d}\delta_{\omega}(1+\sqrt{n}+\sqrt{m})2^{-m}\leq
48d\delta_{\omega}\Big{[}\sqrt{n}+\Big{(}\log^{+}\frac{1}{|x-y|}\Big{)}^{1/2}\Big{]}|x-y|.$

Choosing $C_{1}(\omega)=48d\delta_{\omega}$ yields the desired result. ∎

###### Proof of Lemma 3.7.

It suffices to show that there exists a subset $\widetilde{\Omega}_{1}$ of
$\Omega$ with $\mathbb{P}(\widetilde{\Omega}_{1})=1$ and a positive random
constant $\widetilde{C}_{1}$ such that for any
$\omega\in\widetilde{\Omega}_{1}$, any $n\in\mathbb{N}^{\ast}$, any
$k,\,k^{\prime}\in\\{0,1,\cdots,2^{n}-1\\}$ and any dyadic number
$x\in\mathbf{Q}$ with $\max\limits_{1\leq i\leq d}|x_{i}|\geq 2^{-2^{2n+1}}$,

(5.5)
$\displaystyle|\varrho_{nkk^{\prime}}(x,0)(\omega)|\leq\widetilde{C}_{1}(\omega)\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)}.$

In fact, using Lemma 3.6, there exist a subset $\Omega_{1}$ of $\Omega$ with
$\mathbb{P}(\Omega_{1})=1$ and a positive random constant $C_{1}$ such that
for any $\omega\in\Omega_{1}$, any $n$, $k$, $k^{\prime}$ and any dyadic
number $x\in\mathbf{Q}$ with $\max\limits_{1\leq i\leq
d}|x_{i}|<2^{-2^{2n+1}}$, we have

$\displaystyle|\varrho_{nkk^{\prime}}(x,0)(\omega)|\leq$ $\displaystyle
C_{1}(\omega)2^{-n}\Big{(}\sqrt{n}+\Big{(}\log^{+}\frac{1}{|x|}\Big{)}^{1/2}\Big{)}|x|$
$\displaystyle\leq$ $\displaystyle
C_{1}(\omega)2^{-n}\sqrt{n}\,|x|+\sqrt[4]{d}C_{1}(\omega)2^{-n}\,2^{-4^{n}}\Big{(}|x|\log^{+}\frac{1}{|x|}\Big{)}^{1/2}$
$\displaystyle\leq$ $\displaystyle
C_{1}(\omega)\Big{(}1+\sqrt[4]{dC_{0}^{2}}\Big{)}\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)}\leq
C_{1,1}(\omega)\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)},$

where $C_{0}=\sup_{\xi\in]0,1]}\xi\log^{+}(1/\xi)$ and
$C_{1,1}(\omega)=2C_{1}(\omega)\left(1+\sqrt[4]{dC_{0}^{2}}\right)$. The
required result follows by choosing
$\Omega_{2}=\Omega_{1}\cap\widetilde{\Omega}_{1}$ and
$C_{2}=\max\\{C_{1},\widetilde{C}_{1}\\}$.
Let us now prove that (5.5) holds. Define the following three sets:

$\displaystyle\mathcal{O}_{q}:=$
$\displaystyle\\{y\in\mathbf{Q}:\,\max\limits_{1\leq i\leq d}|y_{i}|\leq
2^{-q}\\},$ $\displaystyle\mathcal{D}_{q}:=$
$\displaystyle\left\\{(y,z)\in\mathcal{O}_{q}^{2}\text{ such that there exists
}r\in\mathbb{N},\,r\geq q\text{ s.t.
}2^{r}y\in\mathbb{Z}^{d},2^{r}z\in\mathbb{Z}^{d}\right\\},$

and for $r\geq q$,

$\mathcal{D}_{q,r}=\left\\{(y,z)\in\mathcal{O}_{q}^{2}:\,2^{r}y\in\mathbb{Z}^{d},\,2^{r}z\in\mathbb{Z}^{d}\right\\}.$

Observe that $\mathcal{D}_{q,r}$ has no more than $2^{4d}\times 2^{2d(r-q)}$
elements and

$\mathcal{D}_{q}=\bigcup\limits_{r=q}^{\infty}\mathcal{D}_{q,r}.$

In addition, define

$\displaystyle\mathbf{F}_{\delta,n}:=$
$\displaystyle\left\\{\omega\in\Omega:\,\text{for some
}k\in\\{0,1,\cdots,2^{n}-1\\},\text{ }q\in\\{0,1,\cdots,2^{2n+1}\\},\text{
}r\geq q,\text{ and }\right.$
$\displaystyle\quad\left.(y,z)\in\mathcal{D}_{q,r}\text{ one has
}\left|\varrho_{nkk^{\prime}}(y,z)\right|(\omega)\geq\delta
2^{-n}(\sqrt{n}+\sqrt{r-q})|y-z|\right\\}$

for every $n\in\mathbb{N}^{\ast}$ and $\delta>0$. Then

$\displaystyle\mathbf{F}_{\delta,n}=$
$\displaystyle\bigcup_{k=0}^{2^{n}-1}\bigcup_{k^{\prime}=0}^{2^{n}-1}\bigcup\limits_{q=0}^{2^{2n+1}}\bigcup_{r=q}^{\infty}\Big{(}\bigcup_{(y,z)\in\mathcal{D}_{q,r}}\left\\{\omega\in\Omega:\,\left|\varrho_{nkk^{\prime}}(y,z)\right|(\omega)\geq\delta
2^{-n}(\sqrt{n}+\sqrt{r-q})|y-z|\right\\}\Big{)}.$

Let $\mathbf{F}_{\delta}$ be defined by
$\mathbf{F}_{\delta}:=\bigcup\limits_{n=1}^{\infty}\mathbf{F}_{\delta,n}$.
Next we show that $\mathbb{P}(\mathbf{F}_{\delta})$ tends to $0$ as $\delta$
goes to $\infty$. By the definition of $\mathbf{F}_{\delta}$ and using (4.8),
we have

$\displaystyle\mathbb{P}(\mathbf{F}_{\delta})\leq$
$\displaystyle\sum\limits_{n=1}^{\infty}\sum\limits_{k=0}^{2^{n}-1}\sum\limits_{k^{\prime}=0}^{2^{n}-1}\sum\limits_{q=0}^{2^{2n+1}}\sum\limits_{r=q}^{\infty}\sum\limits_{(y,z)\in\mathcal{D}_{q,r}}\mathbb{P}\left(\left|\varrho_{nkk^{\prime}}(y,z)\right|\geq\delta
2^{-n}(\sqrt{n}+\sqrt{r-q})|y-z|\right)$ $\displaystyle\leq$ $\displaystyle
2^{4d}C\sum\limits_{n=1}^{\infty}\sum\limits_{q=0}^{2^{2n+1}}\sum\limits_{r=q}^{\infty}2^{2n}2^{2d(r-q)}e^{-\alpha\delta^{2}(n+r-q)}$
$\displaystyle\leq$ $\displaystyle
2^{4d+2}C\sum\limits_{n=1}^{\infty}\sum\limits_{r=0}^{\infty}2^{4n+2dr}e^{-\alpha\delta^{2}(n+r)}$
$\displaystyle=$ $\displaystyle
4^{2d+1}C\sum\limits_{n=0}^{\infty}\sum\limits_{r=0}^{\infty}2^{4n+2dr}e^{-\alpha\delta^{2}(1+n+r)},$

where $C$ and $\alpha$ are the deterministic constants in Proposition 4.3. For
$\delta\geq\delta^{\ast}:=\sqrt{2(d+2)\alpha^{-1}}$,

$\displaystyle\mathbb{P}(\mathbf{F}_{\delta})$ $\displaystyle\leq
4^{2d+1}Ce^{-\alpha\delta^{2}}\sum\limits_{n=0}^{\infty}\sum\limits_{r=0}^{\infty}2^{4n+r}2^{-\alpha\delta^{2}(n+r)}$
$\displaystyle\leq
4^{2d+1}Ce^{-\alpha\delta^{2}}\sum\limits_{n=0}^{\infty}\sum\limits_{r=0}^{\infty}2^{-2dn-4r}\leq
4^{2d+1}Ce^{-\alpha\delta^{2}}.$

It follows that $\mathbb{P}(\mathbf{F}_{\delta})$ tends to $0$ as $\delta$
goes to $\infty$. Define $\widetilde{\Omega}_{1}4=$ by

$\widetilde{\Omega}_{1}:=\bigcup\limits_{\delta\in[\delta^{\ast},\infty[\cap\mathbb{Q}}(\Omega\setminus\mathbf{F}_{\delta}).$

Observe that
$(\mathbf{F}_{\delta},\delta\in[\delta^{\ast},\infty[\cap\mathbb{Q})$ is a
nonincreasing family. Then

$\displaystyle\mathbb{P}\left(\widetilde{\Omega}_{1}\right)=1-\lim\limits_{\delta\to\infty}\mathbb{P}(F_{\delta})=1$

and for every $\omega\in\widetilde{\Omega}_{1}$, there exists
$\overline{\delta}_{\omega}>0$ such that

(5.6)
$\displaystyle\left|\varrho_{nkk^{\prime}}(y,z)(\omega)\right|<\overline{\delta}_{\omega}(\sqrt{n}+\sqrt{r-q})2^{-n}|y-z|$

for all choices of $n$, $k$, $k^{\prime}$, $q\in\\{0,1,\ldots,2^{n+1}\\}$,
$r\geq q$ and couples $(y,z)$ in $\mathcal{D}_{q,r}$.

Now, let $m$ be the largest nonnegative integer $m$ such that
$x\in\mathcal{O}_{m}$, i.e.$2^{-m-1}<\max\limits_{1\leq i\leq d}|x_{i}|\leq
2^{-m}$. For $r\geq m$ define $x_{r}=(x_{1,r},\ldots,x_{d,r})$, where
$x_{i,r}=2^{-r}\left\lfloor 2^{r}x_{i}\right\rfloor$. Then
$(0,x_{m})\in\mathcal{D}_{m,m}$ and for every $r\geq m$,
$(x_{r},x_{r+1})\in\mathcal{D}_{m,r+1}$. Moreover,
$|x_{m}-0|\leq\sqrt{d}2^{-m}$ and for every $r\geq m$,
$|x_{r}-x_{r+1}|\leq\sqrt{d}\,2^{-r-1}$. We deduce from (5.4), (5.6) and the
inequality $\sum\limits_{r=0}^{\infty}\sqrt{r}2^{-r}\leq 4$ that

$\displaystyle 2^{n}\Big{|}\varrho_{nkk^{\prime}}(x,0)(\omega)\Big{|}$
$\displaystyle\leq
2^{n}\Big{(}\Big{|}\varrho_{nkk^{\prime}}(x_{m},0)(\omega)\Big{|}+\sum\limits_{r=m}^{\infty}\Big{|}\varrho_{nkk^{\prime}}(x_{r+1},x_{r})(\omega)\Big{|}\Big{)}$
$\displaystyle\leq$
$\displaystyle\overline{\delta}_{\omega}\sqrt{dn}2^{-m}+\sqrt{d}\overline{\delta}_{\omega}\sum\limits_{r=m}^{\infty}\left(\sqrt{n}+\sqrt{r-m}\right)2^{-r-1}$
$\displaystyle\leq$ $\displaystyle
6\overline{\delta}_{\omega}\sqrt{dn}2^{-m}\leq
12\overline{\delta}_{\omega}\sqrt{dn}|x|\leq
12\overline{\delta}_{\omega}\sqrt{dn}\Big{(}|x|+2^{-4^{n}}\Big{)}.$

The proof of (5.5) is completed by choosing
$\widetilde{C}_{1}=12\sqrt{d}\,\overline{\delta}_{\omega}$. ∎

###### Proof of Lemma 3.8.

Define $\varrho_{nkk^{\prime}}^{+}:=\max\\{0,\varrho_{nkk^{\prime}}\\}$,
$\varrho_{nkk^{\prime}}^{-}:=\max\\{0,-\varrho_{nkk^{\prime}}\\}$,
$x^{+}:=(x^{+}_{1},\ldots,x^{+}_{d})$ and
$x^{-}:=(x^{-}_{1},\ldots,x^{-}_{d})$, where $x_{i}^{+}=\max\\{0,x_{i}\\}$ and
$x_{i}^{-}=\max\\{0,-x_{i}\\}$ for every $i\in\\{1,\cdots,d\\}$. Consider the
sequence
$(x^{+}_{r})_{r\in\mathbb{N}}=(x^{+}_{r,1},\ldots,x^{+}_{r,d})_{r\in\mathbb{N}}$
in $\mathbf{Q}$ defined by
$x^{+}_{r,i}:=1-\left\lfloor(1-x^{+}_{i})2^{r}\right\rfloor
2^{-r},\,\,i=1,\ldots,d$. Observe that $(x^{+}_{r},r\in\mathbb{N})$ is a
coordinatewise nonincreasing sequence of dyadic numbers in $\mathbf{Q}$
converging to $x^{+}$. It follows from the hypothesis on $b$ and (3.2) that

$\displaystyle\varrho^{+}_{nkk^{\prime}}(x,0)(\omega)\leq\varrho_{nkk^{\prime}}(x^{+}_{r},0)(\omega)\leq
C_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x^{+}_{r}|+2^{-4^{n}}\Big{)}\text{ for all
}n,\,k,\,k^{\prime},\,r.$

Similarly for
$(x^{-}_{r})_{r\in\mathbb{N}}=(x^{-}_{r,1},\ldots,x^{-}_{r,d})_{r\in\mathbb{N}}$
defined by $x^{-}_{r,i}:=1-\left\lfloor(1-x^{-}_{i})2^{r}\right\rfloor
2^{-r},\,\,i=1,\ldots,d$, we also have

$\displaystyle\varrho^{-}_{nkk^{\prime}}(x,0)(\omega)\leq-\varrho_{nkk^{\prime}}(-x^{-},0)(\omega)\leq-\varrho_{nkk^{\prime}}(-x^{-}_{r},0)(\omega)\leq
C_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x^{-}_{r}|+2^{-4^{n}}\Big{)}\text{ for all
}n,\,k,\,k^{\prime},\,r.$

Hence,

$\displaystyle|\varrho_{nkk^{\prime}}(x,0)(\omega)|=\varrho^{+}_{nkk^{\prime}}(x,0)(\omega)+\varrho^{-}_{nkk^{\prime}}(x,0)(\omega)\leq
C_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x^{+}_{r}|+|x^{-}_{r}|+2\times
2^{-4^{n}}\Big{)}\text{ for all }n,\,k,\,k^{\prime},\,r.$

Thus, by taking the limit as $r$ goes to $\infty$, we get

$\displaystyle|\varrho_{nkk^{\prime}}(x,0)(\omega)|\leq
2C_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|x|+2^{-4^{n}}\Big{)}\text{ for all
}n,\,k,\,k^{\prime}.$

Taking $\widetilde{C}_{2}(\omega)=2C_{2}(\omega)$ gives the desired result.

∎

###  5.2. A Gronwall type inequality

The second and last step to prove the uniqueness result is the version of the
Gronwall inequality given in Lemma 3.9. Recall that

$u_{i}^{+}=\max\\{0,u_{i}\\},\,\,u_{i}^{-}=\max\\{0,-u_{i}\\}$

and set $u^{+}=\left(u_{1}^{+},\ldots,u_{d}^{+}\right)$ and
$u^{-}=\left(u_{1}^{-},\ldots,u_{d}^{-}\right)$.

###### Proof of Lemma 3.9.

Using Lemmas 3.7 and 3.8, there exists a subset $\Omega_{2}$ of $\Omega$ with
$\mathbb{P}(\Omega_{2})=1$ such that for all $\omega\in\Omega_{2}$, we have

$\displaystyle\left|\varrho_{nkk^{\prime}}^{(i)}(0,x)(\omega)\right|:=$
$\displaystyle\Big{|}\int_{I_{n,k}}\int_{I_{n,k^{\prime}}}\\{b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega)+x)-b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta\Big{|}$
(5.7)
$\displaystyle\leq\widetilde{C}_{2,i}(\omega)\sqrt{n}2^{-n}\left(|x|+\beta(n)\right)\leq\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}\left(|x|+\beta(n)\right)\,\text{
on }\Omega_{2}$

for all $i\in\\{1,\ldots,d\\}$, all integers $n,\,k,\,k^{\prime}$ with $n\geq
1$, $0\leq k,k^{\prime}\leq 2^{n}-1$ and all $x\in\mathbf{Q}$, where
$\widetilde{C}_{2}(\omega)=\max\\{\widetilde{C}_{2,i}(\omega),\,i=1,\ldots,d\\}$.
Let $\omega\in\Omega_{2}$ and let $u$ be a solution to (3.9) satisfying (3.4).
Choose $n\in\mathbb{N}^{\ast}$ such that
$\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\leq 1/2$ and split the set
$[0,1]\times[0,1]$ onto $4^{n}$ squares $I_{n,k}\times I_{n,k^{\prime}}$.
Since $u\preceq u^{+}$ and for $i\in\\{1,\cdots,d\\},\,b_{i}$, is
componentwise nondecreasing, we deduce from (3.9) that for every $(s,t)\in
I_{n,k}\times I_{n,k^{\prime}}$ and every $i\in\\{1,\cdots,d\\}$:

$\displaystyle
u_{i}(s,t)-u_{i}(s,k^{\prime}2^{-n})-u_{i}(k2^{-n},t)+u_{i}(2^{-n}(k,k^{\prime}))$
$\displaystyle=\int_{k2^{-n}}^{s}\int_{k^{\prime}2^{-n}}^{t}\\{b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega)+u(\xi,\zeta))-b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta$
$\displaystyle\leq\int_{k2^{-n}}^{s}\int_{k^{\prime}2^{-n}}^{t}\\{b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega)+u^{+}(\xi,\zeta))-b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta.$

Then, using the fact that $\max\\{0,x+y\\}\leq\max\\{0,x\\}+\max\\{0,y\\}$ and
using once more 2. in Hypothesis 3.1, we obtain

$\displaystyle u_{i}^{+}(s,t)$
$\displaystyle\leq\max\\{0,u_{i}(s,k^{\prime}2^{-n})+u_{i}(k2^{-n},t)-u_{i}(2^{-n}(k,k^{\prime}))\\}$
$\displaystyle\qquad+\int_{k2^{-n}}^{s}\int_{k^{\prime}2^{-n}}^{t}\\{b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega)+u^{+}(\xi,\zeta))-b_{i}(\xi,\zeta,W_{\xi,\zeta}(\omega))\\}\mathrm{d}\xi\mathrm{d}\zeta.$

As a consequence,

(5.8) $\displaystyle u_{i}^{+}(s,t)\leq
u_{i}^{+}(s,k^{\prime}2^{-n})+u_{i}^{+}(k2^{-n},t)+u_{i}^{-}(2^{-n}(k,k^{\prime}))+\varrho^{(i)}_{nkk^{\prime}}\Big{(}0,\overline{u}^{(n)}(k+1,k^{\prime}+1)\Big{)}$

for all $(s,t)\in I_{n,k}\times I_{n,k^{\prime}}$, where
$\overline{u}^{(n)}=\Big{(}\overline{u}^{(n)}_{1},\ldots,\overline{u}^{(n)}_{d}\Big{)}$.

Similarly, it can be shown that

(5.9) $\displaystyle u_{i}^{-}(s,t)\leq
u_{i}^{-}(s,k^{\prime}2^{-n})+u_{i}^{-}(k2^{-n},t)+u_{i}^{+}(2^{-n}(k,k^{\prime}))-\varrho^{(i)}_{nkk^{\prime}}\Big{(}0,-\underline{u}^{(n)}(k+1,k^{\prime}+1)\Big{)}$

for all $(s,t)\in I_{n,k}\times I_{n,k^{\prime}}$, where
$\underline{u}^{(n)}=\Big{(}\underline{u}^{(n)}_{1},\ldots,\underline{u}^{(n)}_{d}\Big{)}$.
We also have the following claim:

Claim: For all $k\in\\{1,2,\cdots,2^{n}\\}$, we have

(5.10)
$\displaystyle\max\Big{\\{}|\overline{u}^{(n)}(k,1)|,|\overline{u}^{(n)}(1,k)|\Big{\\}}\leq
3^{k}d^{k/2}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+1}\beta(n)$

and

(5.11)
$\displaystyle\max\Big{\\{}|\underline{u}^{(n)}(k,1)|,|\underline{u}^{(n)}(1,k)|\Big{\\}}\leq
3^{k}d^{k/2}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+1}\beta(n),$

where $|\cdot|$ denotes the usual norm in $\mathbb{R}^{d}$

###### Proof of the Claim.

We will only prove (5.10) by induction and the proof of (5.11) will follow
analogously. Let $(s,t)\in I_{n,0}\times I_{n,0}$. Since
$\max\\{|u|(0,0),|u|(s,0),|u|(0,t)\\}\leq\beta(n)$, using (5.8) and (5.2), we
have

$\displaystyle u_{i}^{+}(s,t)\leq
3\beta(n)+\varrho^{(i)}_{n11}\Big{(}0,\overline{u}^{(n)}(1,1)\Big{)}\leq
3\beta(n)+\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|\overline{u}^{(n)}(1,1)|+\beta(n)\Big{)}.$

By the definition of $\overline{u}^{(n)}$ and the Euclidean norm

$\displaystyle|\overline{u}^{(n)}(1,1)|\leq
3\sqrt{d}\beta(n)+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\left(|\overline{u}^{(n)}(1,1)|+\beta(n)\right).$

Then, since $(1-\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})^{-1}\leq
1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}$, we have

$\displaystyle|\overline{u}^{(n)}(1,1)|\leq$
$\displaystyle\Big{(}3\sqrt{d}+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}\beta(n)$
(5.12) $\displaystyle\leq$ $\displaystyle
3\sqrt{d}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{2}\beta(n).$

Now, let $k\in\\{1,\ldots,2^{n}-1\\}$ and by induction suppose

(5.13)
$\displaystyle\max\Big{\\{}|\overline{u}^{(n)}(k,1)|,|\overline{u}^{(n)}(1,k)|\Big{\\}}\leq
3^{k}d^{k/2}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+1}\beta(n).$

Let $(s,t)\in I_{n,k}\times I_{n,0}$. Since
$\max\\{|u|(s,0),|u|(q2^{-n},0)\\}\leq\beta(n)$, and using once more (5.8), we
have

$\displaystyle u_{i}^{+}(s,t)$ $\displaystyle\leq
2\beta(n)+u_{i}^{+}(k2^{-n},t)+\varrho^{(i)}_{nk1}(0,\overline{u}^{(n)}(k+1,1))$
$\displaystyle\leq
2\beta(n)+\overline{u}^{(n)}_{i}(k,1)+\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}(|\overline{u}^{(n)}(k+1,1)|+\beta(n))\text{
for every }i.$

Hence

$\displaystyle\overline{u}^{(n)}_{i}(k+1,1)\leq
2\beta(n)+\overline{u}^{(n)}_{i}(k,1)+\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}(|\overline{u}^{(n)}(k+1,1)|+\beta(n))\text{
for every }i.$

As a consequence,

$\displaystyle|\overline{u}^{(n)}(k+1,1)|\leq
2\sqrt{d}\beta(n)+|\overline{u}^{(n)}(k,1)|+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}(|\overline{u}^{(n)}(k+1,1)|+\beta(n)),$

yielding to

$\displaystyle|\overline{u}^{(n)}(k+1,1)|\leq\left(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\right)\left(2\sqrt{d}\beta(n)+|\overline{u}^{(n)}(k,1)|+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\beta(n)\right).$

Using $\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}\leq 1$, we deduce from (5.13)
that

$\displaystyle|\overline{u}^{(n)}(k+1,1)|$
$\displaystyle\leq\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}\Big{[}3^{k}d^{k/2}(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})^{k+1}+3\sqrt{d}\Big{]}\beta(n)$
$\displaystyle\leq\Big{(}3^{k}d^{k/2}+3\sqrt{d}\Big{)}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+2}\beta(n)$
(5.14) $\displaystyle\leq
3^{k+1}d^{(k+1)/2}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+2}\beta(n).$

It can also be shown analogously that

(5.15) $\displaystyle|\overline{u}^{(n)}(1,k+1)|\leq
3^{k+1}d^{(k+1)/2}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+2}\beta(n).$

This ends the proof of (5.10). ∎

Now we prove by induction that $k,k^{\prime}\in\\{1,\ldots,2^{n}\\}$,

$\displaystyle\max\left\\{|\overline{u}^{(n)}(k,k^{\prime})|,|\underline{u}^{(n)}(k,k^{\prime})|\right\\}\leq\left(3\sqrt{d}\right)^{k+k^{\prime}-1}\left(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\right)^{k+k^{\prime}}\beta(n).$

We deduce from (5.10) and (5.11) that (3.5) holds for all couples $(1,k)$,
$(k,1)$, $k\in\\{1,2,\ldots,2^{n}\\}$. Fix
$(k,k^{\prime})\in\\{1,2,\ldots,2^{n}\\}$ and suppose (3.5) holds for
$(k,k^{\prime})$, $(k+1,k^{\prime})$ and $(k,k^{\prime}+1)$. It follows from
(5.8) that for every $(s,t)\in I_{n,k}\times I_{n,k^{\prime}}$ and every
$i\in\\{1,\ldots,d\\}$,

$\displaystyle u_{i}^{+}(s,t)\leq
u_{i}^{+}(2^{-n}k,t)+u_{i}^{+}(s,2^{-n}k^{\prime})+u_{i}^{-}(2^{-n}k,2^{-n}k^{\prime})+\varrho^{(i)}_{nkk^{\prime}}\left(0,\overline{u}^{(n)}(k+1,k^{\prime}+1)\right)$

and by (5.2),

$\displaystyle\overline{u}^{(n)}_{i}(k+1,k^{\prime}+1)$
$\displaystyle\leq\overline{u}^{(n)}_{i}(k,k^{\prime}+1)+\overline{u}^{(n)}_{i}(k+1,k^{\prime})+\underline{u}^{(n)}_{i}(k,k^{\prime})+\widetilde{C}_{2}(\omega)\sqrt{n}2^{-n}\Big{(}|\overline{u}^{(n)}(k+1,k^{\prime}+1)|+\beta(n)\Big{)}.$

Hence

$\displaystyle|\overline{u}^{(n)}(k+1,k^{\prime}+1)|$
$\displaystyle\leq|\overline{u}^{(n)}(k,k^{\prime}+1)|+|\overline{u}^{(n)}(k+1,k^{\prime})|+|\underline{u}^{(n)}(k,k^{\prime})|+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{(}|\overline{u}^{(n)}(k+1,k^{\prime}+1)|+\beta(n)\Big{)},$

that is

$\displaystyle(1-\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})|\overline{u}^{(n)}(k+1,k^{\prime}+1)|$
$\displaystyle\leq|\overline{u}^{(n)}(k,k^{\prime}+1)|+|\overline{u}^{(n)}(k+1,k^{\prime})|+|\underline{u}^{(n)}(k,k^{\prime})|+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\beta(n).$

Since (3.5) holds for $(k,k^{\prime})$, $(k+1,k^{\prime})$ and
$(k,k^{\prime}+1)$, we obtain

$\displaystyle(1-\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})|\overline{u}^{(n)}(k+1,k^{\prime}+1)|$
$\displaystyle\leq$
$\displaystyle\Big{\\{}2(3\sqrt{d})^{k+k^{\prime}}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+k^{\prime}+1}$
$\displaystyle+(3\sqrt{d})^{k+k^{\prime}-1}\Big{(}1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{)}^{k+k^{\prime}}+\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\Big{\\}}\beta(n)$
$\displaystyle\leq$
$\displaystyle\left(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\right)^{k+k^{\prime}+1}\Big{(}2(3\sqrt{d})^{k+k^{\prime}}+(3\sqrt{d})^{k+k^{\prime}-1}+1\Big{)}\beta(n).$

Using the inequalities

$(1-\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})^{-1}\leq(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n})$

and

$2(3\sqrt{d})^{k+k^{\prime}}+(3\sqrt{d})^{k+k^{\prime}-1}+1\leq(3\sqrt{d})^{k+k^{\prime}+1},$

we have

$\displaystyle|\overline{u}^{(n)}(k+1,k^{\prime}+1)|\leq\left(3\sqrt{d}\right)^{k+k^{\prime}+1}\left(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\right)^{k+k^{\prime}+2}\beta(n).$

Similarly, we show that

$\displaystyle|\underline{u}^{(n)}(k+1,k^{\prime}+1)|\leq\left(3\sqrt{d}\right)^{k+k^{\prime}+1}\left(1+2\widetilde{C}_{2}(\omega)\sqrt{dn}2^{-n}\right)^{k+k^{\prime}+2}\beta(n).$

The proof is completed by choosing $C_{1}=2\widetilde{C}_{2}$. ∎

##  Appendix A Appendix

In this section we provide a weak existense result for SDEs driven by Brownian
sheet under the linear growth condition

######  Theorem A.1.

Suppose there exists $M>0$ such that

$\displaystyle|b(s,t,x)|\leq
M(1+|x|),\quad\forall\,(s,t,x)\in[0,1]^{2}\times\mathbb{R}^{d}.$

Then (1.8) has a weak solution.

The above result is a direct consequence of the Cameron-Martin-Girsanov
theorem for two-parameter processes (see e.g. [12, Theorem 3.5], [25,
Proposition 1.6]). Indeed if $X=(X_{s,t},(s,t)\in[0,1]^{2})$ is a
$d$-dimensional Brownian sheet given on a filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{s,t},(s,t)\in[0,1]^{2}),\mathbb{P})$, then,
using Doob-Cairoli-Imkeller’s maximal inequalities for two-parameter
martingales (see e.g. [7, Theorem 1], [22, Chapter II, Section 8, Theorem 1]),
one can show that the process $(\mathcal{Z}_{t},t\in[0,1])$ defined by

$\displaystyle\mathcal{Z}_{t}=\exp\Big{(}\int_{0}^{1}\int_{0}^{t}b(s,\zeta,X_{s,\zeta})\mathrm{d}X_{s,\zeta}-\dfrac{1}{2}\int_{0}^{1}\int_{0}^{t}|b(s,\zeta,X_{s,\zeta})|^{2}\mathrm{d}s\mathrm{d}\zeta\Big{)}$

is a martingale with respect to the filtration
$(\mathcal{F}_{1,t},t\in[0,1])$. Then, by [12, Theorem 3.5], the process
$(W_{s,t},(s,t)\in[0,1]^{2})$ given by

$\displaystyle
W_{s,t}=X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}-\int_{0}^{s}\int_{0}^{t}b(\xi,\zeta,X_{\xi,\zeta})\mathrm{d}\xi\mathrm{d}\zeta,\quad\forall\,(s,t)\in[0,1]^{2},$

is a $\mathbb{R}^{d}$-valued $(\mathcal{F}_{s,t})$-Brownian sheet with
$\partial W=0$ under the probability $\widetilde{\mathbb{P}}$ defined by

$\displaystyle\frac{\mathrm{d}\widetilde{\mathbb{P}}}{\mathrm{d}\mathbb{P}}=\mathcal{Z}_{1}.$

## Ackowledgment

The authors wish to thank an anonymous referee and the editor for their
valuable comments and suggestions.

## References

  * [1] O. Amine, David Baños, and Frank Proske.  $C^{\infty}$ regularization by noise of singular ODE’s.  arXiv preprint arXiv:1710.05760, 2017.
  * [2] O. Amine, A.-R. Mansouri, and F. Proske.  Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional brownian paths.  arXiv preprint arXiv:2003.06200, 2020.
  * [3] M. T. Barlow and M. Yor.  Semi-martingale inequalities via the garsia-rodemich-rumsey lemma, and applications to local times.  Journal of Functional Analysis, 49(2):198–229, 1982.
  * [4] L. Beck, F. Flandoli, M. Gubinelli, and M. Maurelli.  Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness.  Electronic Journal of Probability, 24, 2019.
  * [5] A.-M. Bogso, M. Dieye, and O. Menoukeu Pamen.  Stochastic integration with respect to local time of the brownian sheet and regularising properties of brownian sheet paths.  arXiv preprint arXiv:2112.00401, 2021.
  * [6] O. Butkovsky and L. Mytnik.  Regularization by noise and flows of solutions for a stochastic heat equation.  Annals of Probability, 47:169–212, 2019.
  * [7] R. Cairoli.  Une inégalité pour martingales à indices multiples et ses applications.  Séminaire de probabilités de Strasbourg, 4:1–27, 1970.
  * [8] R. Cairoli.  Sur une équation différentielle stochastique.  C.R. Acad. Sci. Paris Série A, 274:1739–1742, 1972.
  * [9] E. Carlen and P. Kree.  Lp estimates on iterated stochastic integrals.  The Annals of Probability, pages 354–368, 1991.
  * [10] R. Carmona and D. Nualart.  Random non-linear wave equations: smoothness of the solutions.  Probability Theory and Related Fields, 79(4):469–508, 1988.
  * [11] R. Catellier and M. Gubinelli.  Averaging along irregular curves and regularisation of ODEs.  Stochastic Process. Appl., 126:2323–2366, 2016.
  * [12] R. C. Dalang and C. Mueller.  Multiple points of the brownian sheet in critical dimensions.  The Annals of Probability, 43(4):1577–1593, 2015.
  * [13] A. M. Davie.  Uniqueness of solutions of stochastic differential equations.  International Mathematics Research Notices, Vol. 2007, 2007.
  * [14] A. M. Davie.  Individual path uniqueness of solutions of stochastic differential equations.  In In Stochastic Analysis, pages 213–225. Springer-Heidelberg, 2011\.
  * [15] K. Deimling.  A carathéodory theory for systems of integral equations.  Annali di Matematica Pura ed Applicata, 86(1):217–260, 1970.
  * [16] N. Eisenbaum.  Integration with respect to local time.  Potential analysis, 13(4):303–328, 2000.
  * [17] M. Erraoui, Y. Ouknine, and D. Nualart.  Hyperbolic stochastic partial differential equations with additive fractional brownian sheet.  Stochastics and Dynamics, 3(02):121–139, 2003.
  * [18] M. Farré and D. Nualart.  Nonlinear stochastic integral equations in the plane.  Stochastic processes and their applications, 46(2):219–239, 1993\.
  * [19] F. Flandoli.  Random Perturbation of PDE’s and Fluid Dynamic Models: École d’été de Probabilités de Saint-Flour XL 2010, volume 2015.  Springer Science & Business Media, 2010.
  * [20] L. Galeati and M. Gubinelli.  Noiseless regularisation by noise.  Revista Matemática Iberoamericana, 38(2):433–502, 2021.
  * [21] F. A. Harang and N. Perkowski.  C∞ regularization of odes perturbed by noise.  Stochastics and Dynamics, 21(08):2140010, 2021.
  * [22] P. Imkeller.  Two-parameter martingales and their quadratic variation, volume 1308\.  Springer, 1988.
  * [23] H. Kremp and N. Perkowski.  Multidimensional SDE with distributional drift and lévy noise.  Bernoulli, 28(3):1757–1783, 2022.
  * [24] D. Nualart.  The Malliavin calculus and related topics, volume 1995.  Springer, 2006.
  * [25] D. Nualart and E. Pardoux.  Markov field properties of solutions of white noise driven quasi-linear parabolic pdes.  Stochastics: An International Journal of Probability and Stochastic Processes, 48(1-2):17–44, 1994.
  * [26] D. Nualart and S. Tindel.  Quasilinear stochastic hyperbolic differential equations with nondecreasing coefficient.  Potential Analysis, 7(3):661–680, 1997.
  * [27] D. Nualart and S. Tindel.  On two-parameter non-degenerate brownian martingales.  Bulletin des sciences mathematiques, 122(4):317–335, 1998.
  * [28] D. Nualart and J. Yeh.  Existence and uniqueness of a strong solution to stochastic differential equations in the plane with stochastic boundary process.  Journal of Multivariate Analysis, 28(1):149–171, 1989.
  * [29] E. Priola.  Davie’s type uniqueness for a class of SDEs with jumps.  Ann. Inst Henri Poincaré: Prob. Stat, 54:694–725, 2018.
  * [30] E. Priola.  On davie’s uniqueness for some degenerate SDEs.  arXiv:1912.02776, 2019.
  * [31] L. Quer-Sardanyons and S. Tindel.  The 1-d stochastic wave equation driven by a fractional brownian sheet.  Stochastic processes and their applications, 117(10):1448–1472, 2007.
  * [32] D. L. Rasmussen.  Gronwall’s inequality for functions of two independent variables.  Journal of Mathematical Analysis and Applications, 55(2):407–417, 1976.
  * [33] A. V. Shaposhnikov.  Some remarks on davie’s uniqueness theorem.  Proceedings of the Edinburgh Mathematical Society, 59(4):1019–1035, 2016.
  * [34] Alexander Shaposhnikov and Lukas Wresch.  Pathwise vs. path-by-path uniqueness.  arXiv preprint arXiv:2001.02869, 2020.
  * [35] D. R. Snow.  Gronwall’s inequality for systems of partial differential equations in two independent variables.  Proceedings of the American Mathematical Society, 33(1):46–54, 1972\.
  * [36] C. Tudor.  On the two parameter Itô equations.  Publications mathématiques et informatique de Rennes, (1):1–17, 1983.
  * [37] J. B. Walsh.  The local time of the brownian sheet.  Astérisque, 52(53):47–61, 1978.
  * [38] R Walsh and Cairoli J. B.  Stochastic integrals in the plane.  Acta Math, 134:111–183, 1975.
  * [39] J. Yeh.  Existence of strong solutions for stochastic differential equations in the plane.  Pacific Journal of Mathematics, 97(1):217–247, 1981.
  * [40] J. Yeh.  Existence of weak solutions to stochastic differential equations in the plane with continuous coefficients.  Transactions of the American Mathematical Society, 290(1):345–361, 1985.
  * [41] J. Yeh.  Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process.  Pacific journal of mathematics, 128(2):391–400, 1987.