# Uniform large deviation principle for the solutions of two-dimensional
stochastic Navier-Stokes equations in vorticity form 111 1,2Department of
Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar
Highway, Roorkee, Uttarakhand 247667, INDIA.e-mail: Manil T. Mohan:
<EMAIL_ADDRESS><EMAIL_ADDRESS>e-mail: Ankit Kumar:
<EMAIL_ADDRESS>*Corresponding author. Keywords: Stochastic Navier-
Stokes equation, vorticity, uniform large deviation principle, Wiener process.
Mathematics Subject Classification (2020): Primary 60H15, 60F10; Secondary
35Q30, 37L55, 60H20.

Ankit Kumar1 and Manil T. Mohan2*

###### Abstract.

The main objective of this paper is to demonstrate the uniform large deviation
principle (UDLP) for the solutions of two-dimensional stochastic Navier-Stokes
equations (SNSE) in the vorticity form when perturbed by two distinct types of
noises. We first consider an infinite-dimensional additive noise that is white
in time and colored in space and then consider a finite-dimensional Wiener
process with linear growth coefficient. In order to obtain the ULDP for 2D
SNSE in the vorticity form, where the noise is white in time and colored in
space, we utilize the existence and uniqueness result from _B. Ferrario et.
al., Stochastic Process. Appl., 129 (2019), 1568–1604,_ and the uniform
contraction principle. For the finite-dimensional multiplicative Wiener noise,
we first prove the existence of a unique local mild solution to the vorticity
equation using a truncation and fixed point arguments. We then establish the
global existence of the truncated system by deriving a uniform energy estimate
for the local mild solution. By applying stopping time arguments and a version
of Skorokhod’s representation theorem, we conclude the global existence and
uniqueness of a solution to our model. We employ the weak convergence approach
to establish the ULDP for the law of the solutions in two distinct topologies.
We prove ULDP in the $\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$
topology, for $p>2$, taking into account the uniformity of the initial
conditions contained in bounded subsets of $\mathrm{L}^{p}(\mathbb{T}^{2})$.
Finally, in $\mathrm{C}([0,T]\times\mathbb{T}^{2})$ topology, the uniformity
of initial conditions lying in bounded subsets of $\mathrm{C}(\mathbb{T}^{2})$
is considered.

##  1\. Introduction

We consider the following two-dimensional stochastic Navier-Stokes equations
(SNSE):

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\partial\boldsymbol{u}^{\epsilon}}{\partial
t}(t,x)+(\boldsymbol{u}^{\epsilon}(t,x)\cdot\nabla)\boldsymbol{u}^{\epsilon}(t,x)-&\nu\Delta\boldsymbol{u}^{\epsilon}(t,x)+\nabla
p^{\epsilon}(t,x)\\\ &=\sqrt{\epsilon}\boldsymbol{\eta},\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\
\nabla\cdot\boldsymbol{u}^{\epsilon}(t,x)&=0,\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\
\boldsymbol{u}^{\epsilon}(0,x)&=\boldsymbol{u}_{0}(x),\
x\in{\mathbb{T}^{2}},\end{aligned}}\right.$ (1.1)

which describe the motion of a viscous incompressible fluid in a domain
${\mathbb{T}^{2}}\subset\mathbb{R}^{2}$. Here,
$\boldsymbol{u}^{\epsilon}(t,x)=(u_{1}^{\epsilon},u_{2}^{\epsilon})(t,x)\in\mathbb{R}^{2}$
represents the velocity field at time $t$ and position $x$,
$p^{\epsilon}(t,x)\in\mathbb{R}$ denotes the pressure field, $\nu>0$ is the
viscosity, $\boldsymbol{u}_{0}$ us the initial velocity and
$\boldsymbol{\eta}$ is the stochastic forcing term. We consider our domain to
be a torus, with ${\mathbb{T}^{2}}=[0,2\pi]^{2}$ and periodic boundary
conditions in place.

###  1.1. Vorticity formulation

In this article, we discuss the vorticity formulation of the model in (1.1)
with two distinct types of noises.

####  1.1.1. Additive infinite-dimensional noise

Taking curl on both sides of the first equation of the system (1.1), one
obtains the vorticity formulation, where the unknown is the vorticity

$\xi^{\epsilon}(t,x)=\nabla^{\perp}\cdot\boldsymbol{u}^{\epsilon}(t,x)\equiv\partial_{x_{1}}u_{2}^{\epsilon}(t,x)-\partial_{x_{2}}u_{1}^{\epsilon}(t,x),\
(t,x)\in[0,T]\times{\mathbb{T}^{2}}.$

We know that the curl of a planar vector field has only one significant
component $\xi^{\epsilon}$, which is orthogonal to the plane. Therefore, with
enough regularity of solutions, the system (1.1) is equivalent to the
following system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\partial\xi^{\epsilon}}{\partial
t}(t,x)+\boldsymbol{u}^{\epsilon}(t,x)\cdot\nabla\xi^{\epsilon}(t,x)-&\nu\Delta\xi^{\epsilon}(t,x)\\\
&=\sqrt{\epsilon}\mathrm{W}(\mathrm{d}t,\mathrm{d}x),\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\
\nabla\cdot\boldsymbol{u}^{\epsilon}(t,x)&=0,\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\ \xi^{\epsilon}(0,x)&=\xi_{0}(x),\
x\in{\mathbb{T}^{2}},\end{aligned}}\right.$ (1.2)

with the periodic boundary conditions. For the sake of convenience, we are
assuming $\nu=1$. Here, $\mathrm{W}(\cdot,\cdot)$ is the formal notation for
some Gaussian perturbation defined on a filtered probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$. The Biot-
Savart law provides a relationship between the velocity field
$\boldsymbol{u}^{\epsilon}$ and the vorticity $\xi^{\epsilon}$, expressed as
$\boldsymbol{u}^{\epsilon}=\boldsymbol{k}*\xi^{\epsilon}$, wherein
$\boldsymbol{k}$ stands for the Biot-Savart kernel (see Subsection 2.3).

We interpret the system (1.2) in the sense of Walsh (see [51]). Let
$G(\cdot,\cdot,\cdot)$ be the fundamental solution to the heat equation on the
flat torus ${\mathbb{T}^{2}}$. Then, the random field
$\xi^{\epsilon}=\\{\xi(t,x):(t,x)\in[0,T]\times{\mathbb{T}^{2}}\\}$ is a
solution to the system (1.2) if it satisfies the following integral equation

$\displaystyle\xi^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{u}^{\epsilon}(s,y)\xi^{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),\
\ \mathbb{P}\text{-a.s.},$ (1.3)

for all $(t,x)\in[0,T]\times{\mathbb{T}^{2}}$.

####  1.1.2. Multiplicative finite-dimensional noise

Let us now assume that the stochastic perturbation be a finite-dimensional
Wiener process with linear growth coefficient (see Hypothesis 4.1 below). With
the above change in the noise term, one can rewrite the system (1.2) as
follows:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\partial\xi^{\epsilon}}{\partial
t}(t,x)&+\boldsymbol{u}^{\epsilon}(t,x)\cdot\nabla\xi^{\epsilon}(t,x)-\Delta\xi^{\epsilon}(t,x)\\\
&=\sqrt{\epsilon}\sum_{j=1}^{n}\sigma_{j}(t,x,\xi^{\epsilon}(t,x))\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{W}^{j}(t),\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\
\nabla\cdot\boldsymbol{u}^{\epsilon}(t,x)&=0,\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\ \xi^{\epsilon}(0,x)&=\xi_{0}(x),\
x\in{\mathbb{T}^{2}},\end{aligned}}\right.$ (1.4)

where $\mathrm{W}(\cdot)$ is an $n$-dimensional Wiener process. We have a
similar formulation to (1.1.1) for the solution to the system (1.4) in sense
of Walsh (see [51]) as

$\displaystyle\xi^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{u}^{\epsilon}(s,y)\xi^{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s),\
\ \mathbb{P}\text{-a.s.},$ (1.5)

for all $(t,x)\in[0,T]\times{\mathbb{T}^{2}}$.

###  1.2. Literature review

The analysis of stochastic partial differential equations (SPDEs) deals with
the existence and uniqueness results, and various properties of solution
processes such as the large deviation principle (LDP), uniform large deviation
principle (ULDP), invariant measures, random attractors, and much more. In
this work, we are able to formulate our system as a two-dimensional stochastic
parabolic nonlinear equation with a distinct nonlinear term compared to most
papers related to higher-dimensional spatial domains concerning stochastic
parabolic nonlinear equations (cf. [23, 38], etc. and the references therein).
We assume our problems in the vorticity form (1.2) and (1.4) by taking the
curl of the system (1.1), similarly to the approach used by [23], but with
different types of noises. The main focus of this work is to analyze the
asymptotic behavior of the solutions to the system (1.1) in the vorticity form
with different types of stochastic forcing terms (as seen in (1.2) and (1.4)).

The intriguing research topic of large deviation theory in probability theory
is examined to analyze the limiting behavior of rare event probabilities. The
outstanding works by Donsker and Vardhan [20, 49, 50], as well as Freidlin and
Wentzell [24], brought the large deviation theory to the forefront of
probability theory research. Numerous authors, including [2, 5, 7, 8, 11, 12,
14, 19, 30, 31, 33, 34, 42, 53, 54], have contributed to the field by
establishing LDP for several types of SPDEs. Our goal in this article is to
demonstrate the Freidlin-Wentzell Uniform Large Deviation Principle (FWULDP),
also known as the ULDP from Freidlin and Wentzell. The Contraction Principle,
as stated in Theorem 4.2.1 of [18], provides a way to address the LDP in the
setting of additive Gaussian noise. This principle, which states that the LDP
is preserved under continuous mappings, has been employed by the authors of
[18, 39, 40], and other papers, to obtain LDP for various types of SPDEs.
Recently, the idea of Contraction Principle has been extended by the author in
[52] to Uniform Contraction Principle in order to prove ULDP, considering
uniformity across initial conditions that lie within a bounded subsets of an
appropriate space.

We cannot apply the approach developed by [10] directly to establish the ULDP
for SPDEs that have non-compact sets of initial conditions, as this approach
requires the convergence of initial conditions. For several applications, such
as the study of exit time and place of a stochastic process $\eta$ from its
domain, it is not necessarily required for the large deviations of $\eta$ to
be uniform over bounded, non-compact subsets of the space (for more details,
we refer to [43]). Several authors have worked towards establishing ULDP for
various types of SPDEs, with notable contributions from [13, 31, 43, 45, 52]
and their cited references. The authors of [43] obtained a ULDP for a general
class of SPDEs, but it was limited to the application on SPDEs defined in
bounded domains with compact Sobolev embeddings. An LDP for a general class of
Banach space valued SPDEs that is uniform with respect to initial conditions
in bounded subsets of the Banach space is established in [44]. Recently, the
author in [52] established the uniform contraction principle to obtain the
ULDP for solutions of fractional stochastic reaction-diffusion equations,
which is uniformly valid for initial conditions in bounded subsets of the
$\mathrm{L}^{2}(\mathbb{R}^{d})$-norm.

Under Hypothesis 4.1 (see below), the second main result of this article is
the existence and uniqueness of a solution to the system (1.4) driven by a
finite dimensional Wiener process with a linear growth coefficient. We adopted
the idea from [25, 27, 28, 29] for different types of SPDEs. The heat kernel
estimates given by Lemma 2.1 and Lemma 2.2 play a crucial role in our proof.
We choose the method of localization to handle the non-Lipschitz nonlinear
term in (1.2), as suggested in [25, 27, 29, 28, 31, 32]. First we show that
the truncated integral equation (4.1) has a unique global solution, which
implies the local existence and uniqueness of the solution upto some stopping
time $\tau$ to the integral equation (1.1.2). We establish the global
solvability results by showing that $\tau=T$, $\mathbb{P}$-a.s. Furthermore,
we assume our initial data $\xi_{0}\in\mathrm{C}({\mathbb{T}^{2}})$ and prove
that the solution $\xi$ admits a modification which is a space-time continuous
process,

The final goal of this article is to establish ULDP for the solution of the
integral equation (1.1.2). Different definitions of ULDP can be shown to be
equivalent with appropriate assumptions, as demonstrated in [43]. The authors
in [9] used a variational representation to establish the FWULDP for a diverse
family of infinite-dimensional stochastic dynamical systems. They have
established the weak convergence approach for proving the ULDP with the help
of uniform Laplace principle (ULP). They established the ULDP for the law of
solution to certain SPDEs by proving a sufficient condition, and to verify it
one needs to explore more basic properties such as well-posedness and
tightness of skeleton integral equations ((5.1) and (5.1) below), which are
similar to the original integral equation (1.1.2).

In a recent work [43], the author established the equivalency between four
different kind of definitions of ULDP and ULP. He formulated the
Equicontinuous Uniform Laplace Principle (EULP) and demonstrated it is
equivalent to the FWULDP without any additional or compactness assumptions. He
provided a sufficient condition for the proof of EULP under which the law of
the solution follows ULDP with uniformity considered over initial conditions
belonging to a bounded, but not necessarily compact, set. In the work [45],
the authors established FWULDP for the law of the solution to the stochastic
Burgers type equation, assuming a polynomial nonlinearity of any order driven
by a finite-dimensional Wiener process. The authors in [46] derived the ULDP
for the law of solutions to a class of semilinear SPDEs with bounded noise
coefficient, which are driven by space-time white noise. Recently, the authors
in [31] used the approach of [43] to demonstrate the UDLP for a stochastic
generalized Burgers-Huxley equation that is driven by either a
$\mathrm{Q}$-Wiener process with a linear growth of noise coefficient or a
space-time white noise with a bounded noise coefficient. In the works [31, 45,
46], it is assumed that the uniformity with respect to the initial conditions
that are in a bounded subsets does not need to be compact.

###  1.3. Objectives and novelties of the paper

This work aims to analyze the asymptotic behavior of the two-dimensional
Navier-Stokes equations in the vorticity form with a stochastic forcing term
(to be specified later). To the best of our understanding, this analysis
presented in this article has not been previously investigated in available
literature. This paper is structured into two parts, which focus on the noise
structure:

In the first part, we assume that the equation is driven by an infinite-
dimensional additive noise, which is white in time and colored in space.

  * •

The noise structure of this part resembles that of the work [23]. Minor
modifications to the well-posed result presented in [23] can be used to obtain
the global solvability of the system (1.2). Our objective is to demonstrate
that ULDP holds in two distinct topologies:
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$ with $p>2$, where the
uniformity is measured over initial conditions belonging to bounded subsets of
$\mathrm{L}^{p}({\mathbb{T}^{2}})$, and
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$ with uniformity over initial
conditions belonging to bounded subsets of $\mathrm{C}({\mathbb{T}^{2}})$. The
proof of the ULDP in both cases are based on the uniform contraction principle
established in [52].

We assume in the second part that the equation is driven by a multiplicative
finite-dimensional Wiener process with a linear growth of the noise
coefficient.

  * •

We use the approach from [25, 28, 29, 32], etc. to ensure the well-posedness
of the system (1.4). We prove the existence of a local solution to the
integral equation (1.1.2) by utilizing a truncation and fixed point argument.
With the help of stopping time arguments, tightness properties and Skorokhod’s
representation theorem, we establish the uniform global energy estimates of
the local solution and consequently, the global existence of the solution.

  * •

Our final goal is to verify the ULDP for the system (1.4). Given the initial
conditions in a bounded subsets of an appropriate space, it is not sufficient
to establish ULP to obtain UDLP without any additional assumption as discussed
in [43]. The motivation for the ULDP comes from the work of [43], which
provides a sufficient condition for the verification of EULP, leading to ULDP
in two distinct topologies:
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$ with $p>2$, where the
uniformity is measured over initial conditions belonging to bounded subsets of
$\mathrm{L}^{p}({\mathbb{T}^{2}})$, and
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$ with uniformity over initial
conditions belonging to bounded subsets of $\mathrm{C}({\mathbb{T}^{2}})$. We
use the embedding
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})\hookrightarrow\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$,
in the proof of ULDP in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$ topology.

###  1.4. Organization of the paper

The following structure is used to organize this article: We start Section 2
by introducing basic function spaces, heat kernel estimates, Biot-Savart law,
and noise structure that assist us in formulating our problem. Then, we
analyze the global solvability of the system (1.2). Section 3 begins by
introducing the basics of LDP and ULDP, followed by recalling the uniform
contraction principle (Proposition 3.4) and proceeding to the proof of local
Lipschitz continuity of solutions of (3.10) (Lemma 3.7). Subsequently, we
present and demonstrate our main results of this section (Theorems 3.8 and
3.10), which are the ULDP for the law of the solutions of the system (1.2). We
analyze the perturbation caused by a finite-dimensional Wiener process in
Sections 4 and 5, as shown in the equation (1.4). We obtain global solvability
of the integral equation (1.1.2) under Hypothesis 4.1 via the use of the
truncation method and uniform energy estimates (Lemmas 4.3 and 4.5,
Proposition 4.4, and Theorem 4.2). The ULDP for the law of the solutions to
the integral equation (1.1.2) are presented in the final Section 5. We first
recall some useful notations and results from [43] including a sufficient
condition 5.1 that, according to [43, Theorem 2.10], ensures EULP and hence
ULDP. We establish intermediate results (Theorems 5.4 and 5.5, Proposition
5.9, and Corollary 5.10) later that are useful to obtain our main results
(Theorems 5.7 and 5.8) in this section. We use uniform convergence in
probability (Corollary 5.11) to prove Theorem 5.7. We conclude the article
with the proof of Theorem 5.8, relying on Lemma 5.12 for tightness and Theorem
5.13 for convergence in probability in supremum norm.

##  2\. Mathematical formulation

We will begin this section by considering the vorticity form of the system
(1.1), and then discuss the required function spaces needed, along with the
heat kernel and its associated estimates, Biot-Savrat law, and a brief
overview of the noise structure which will lead us to our model and its
solvability results.

###  2.1. Function spaces

We denote by $x=(x_{1},x_{2})$ a point in $\mathbb{R}^{2}$ and the scalar
product and the norm in $\mathbb{R}^{2}$ by $x\cdot
y=x_{1}y_{1}+x_{2}y_{2},\text{ and }\ |x|=\sqrt{x\cdot x},\
x,y\in\mathbb{R}^{2},$ respectively. Given
$z=\mathrm{Re}(z)+i\mathrm{Im}(z)\in\mathbb{C}$, we denote the absolute value
and complex conjugate of $z$ by $|z|$ and $\overline{z}$, respectively, and
$|z|=\sqrt{(\mathrm{Re}(z))^{2}+(\mathrm{Im}(z))^{2}},\text{ and
}\overline{z}=\mathrm{Re}(z)-i\mathrm{Im}(z).$ We define
$\mathbb{Z}_{0}^{2}=\mathbb{Z}^{2}\backslash\\{(0,0)\\}$ and
$\mathbb{Z}_{+}^{2}=\\{\eta=(\eta_{1},\eta_{2})\in\mathbb{Z}^{2}:\eta_{1}>0\\}\cup\\{\eta=(0,\eta_{2})\in\mathbb{Z}^{2}:\eta_{2}>0\\}$.
Let us denote the space of all complex-valued $2\pi$-periodic functions in
$x_{1}$ and $x_{2}$ which are measurable and square integrable on
${\mathbb{T}^{2}}$ by $\mathrm{L}_{\\#}^{2}({\mathbb{T}^{2}})$. The space
$\mathrm{L}_{\\#}^{2}({\mathbb{T}^{2}})$ is equipped with the scalar product
and norm given by

$\displaystyle(f_{1},f_{2})_{\mathrm{L}^{2}({\mathbb{T}^{2}})}=\int_{\mathbb{T}^{2}}f_{1}(x)\overline{f_{2}(x)}\mathrm{d}x\
\text{ and }\
\|\cdot\|_{\mathrm{L}^{2}({\mathbb{T}^{2}})}^{2}=(\cdot,\cdot)_{\mathrm{L}^{2}({\mathbb{T}^{2}})},$

respectively. We define the space
$\mathbb{L}_{\\#}^{2}({\mathbb{T}^{2}})=[\mathrm{L}_{\\#}^{2}({\mathbb{T}^{2}})]^{2}$
consisting of all pairs $\boldsymbol{u}=(u_{1},u_{2})$ of complex-valued
periodic functions equipped with the inner product

$\displaystyle(\boldsymbol{u},\boldsymbol{v})_{\mathbb{L}_{\\#}^{2}({\mathbb{T}^{2}})}:=\int_{\mathbb{T}^{2}}\boldsymbol{u}(x)\cdot\overline{\boldsymbol{v}(x)}\mathrm{d}x=\int_{\mathbb{T}^{2}}\big{[}u_{1}(x)\overline{v_{1}(x)}+u_{2}(x)\overline{v_{2}(x)}\big{]}\mathrm{d}x,\
\boldsymbol{u},\boldsymbol{v}\in\mathbb{L}_{\\#}^{2}({\mathbb{T}^{2}}).$

An orthonormal basis for the space $\mathrm{L}_{\\#}^{2}({\mathbb{T}^{2}})$ is
given by $\\{e_{\eta}\\}_{\eta\in\mathbb{Z}^{2}}$, where

$\displaystyle e_{\eta}(x)=\frac{1}{2\pi}e^{i\eta\cdot x},\ \
x\in{\mathbb{T}^{2}},\text{ and }\ \eta\in\mathbb{Z}^{2}.$ (2.1)

One can also consider non-zero mean value vectors in the same manner as
discussed in [48], even though mean value zero vectors are more commonly used
and mathematically simpler to consider in periodic cases. We denote the space
with the zero-mean condition on the space
$\mathrm{L}_{\\#}^{2}({\mathbb{T}^{2}})$ by
$\mathring{\mathrm{L}}_{\\#}^{2}({\mathbb{T}^{2}})$. An orthonormal system of
eigenfunctions $\\{e_{\eta}\\}$ defined in (2.1), with associated eigenvalues
$\lambda_{\eta}=|\eta|^{2}$, is provided for the space
$\mathring{\mathrm{L}}_{\\#}^{2}({\mathbb{T}^{2}})$ by the operator $-\Delta$.

The real-valued functions in
$\mathring{\mathrm{L}}_{\\#}^{2}({\mathbb{T}^{2}})$ can be characterized by
their Fourier series expansion as follows:

$\displaystyle\mathring{\mathrm{L}}_{\\#}^{2}({\mathbb{T}^{2}})=\bigg{\\{}g(x)=\sum_{\eta\in\mathbb{Z}_{0}^{2}}g_{\eta}e_{\eta}(x):\overline{g_{\eta}}=g_{-\eta},\
\text{ for any }\
\eta,\sum_{\eta\in\mathbb{Z}_{0}^{2}}|g_{\eta}|^{2}<\infty\bigg{\\}}.$

For every $p>2$, we denote the subspace of $\mathrm{L}^{p}({\mathbb{T}^{2}})$
consisting of zero mean and periodic scalar functions by
$\mathring{\mathrm{L}}_{\\#}^{p}({\mathbb{T}^{2}})$. Moreover, the spaces
$\mathring{\mathrm{L}}_{\\#}^{p}({\mathbb{T}^{2}})$ are Banach spaces with the
norms inherited from the spaces $\mathrm{L}^{p}({\mathbb{T}^{2}})$.

The Laplacian operator $-\Delta$ with periodic boundary conditions is
represented by $\mathrm{A}$. For every $a\in\mathbb{R}$, we define the powers
of the operator $\mathrm{A}$ for
$g=\sum\limits_{\eta\in\mathbb{Z}_{0}^{2}}g_{\eta}e_{\eta}$ as

$\displaystyle\mathrm{A}^{a}g=\sum_{\eta\in\mathbb{Z}_{0}^{2}}|\eta|^{2a}g_{\eta}e_{\eta},$

and the domain of $\mathrm{A}^{a}$ is denoted by
$\mathrm{D}(\mathrm{A}^{a}):=\bigg{\\{}g=\sum\limits_{\eta\in\mathbb{Z}_{0}^{2}}g_{\eta}e_{\eta}:\sum\limits_{\eta\in\mathbb{Z}_{0}^{2}}|\eta|^{4a}|g_{\eta}|^{2}<\infty\bigg{\\}}$.

For any $a\in\mathbb{R}^{+}$ and $p\geq 1$, we define
$\mathrm{H}^{a,p}=\\{g\in\mathring{\mathrm{L}}_{\\#}^{p}({\mathbb{T}^{2}}):\mathrm{A}^{\frac{a}{2}}g\in\mathring{\mathrm{L}}_{\\#}^{p}({\mathbb{T}^{2}})\\}$,
as Banach spaces with the usual norm. For $p=2$, $\mathrm{H}^{a,2}$ become
Hilbert spaces and we denote them by $\mathrm{H}^{a}$. For $a<0$, we define
$\mathrm{H}^{a}$ as the dual space of $\mathrm{H}^{-a}$ with respect to the
$\mathrm{L}^{2}$-scalar product.

We define the space regularity of periodic vector fields, zero mean value, and
divergence-free through Laplace operator’s corresponding action on each vector
component. Therefore, we define the space
$\mathbb{H}=\\{\boldsymbol{u}\in\mathring{\mathbb{L}}_{\\#}^{2}({\mathbb{T}^{2}}):\nabla\cdot\boldsymbol{u}=0\\}$,
here the divergence free condition has to be understood in the distributional
sense. The scalar product from $[\mathrm{L}^{2}({\mathbb{T}^{2}})]^{2}$ allows
$\mathbb{H}$ to be considered as a Hilbert space. The norm on the space
$\mathbb{H}$ is given by $\|\cdot\|_{\mathbb{H}}$,
$\|\boldsymbol{u}\|_{\mathbb{H}}=\sqrt{(\boldsymbol{u},\boldsymbol{u})_{\mathbb{H}}}$.
Let
$\mathopen{}\mathclose{{}\left\\{\frac{\eta^{\perp}}{|\eta|}e_{\eta}}\right\\}_{\eta\in\mathbb{Z}_{0}^{2}}$
be the basis of the space $\mathbb{H}$, where
$\eta^{\perp}=(-\eta_{2},\eta_{1})$ and $e_{\eta}$ is defined in (2.1). For
$p>2$, set
$\mathbb{L}_{\sigma}^{p}(\mathbb{T}^{2}):=\mathbb{H}\cap\mathbb{L}_{\\#}^{p}(\mathbb{T}^{2})$,
which forms Banach spaces with the norms inherited from the space
$\mathbb{L}_{\\#}^{2}(\mathbb{T}^{2})$. Similarly, we set the vector space,
$\mathbb{H}^{a,p}=\\{\boldsymbol{u}\in\mathbb{L}_{\sigma}^{p}({\mathbb{T}^{2}}):\mathrm{A}^{\frac{a}{2}}\boldsymbol{u}\in\mathbb{L}_{\sigma}^{p}({\mathbb{T}^{2}})\\}$.
These are Banach spaces with the usual norm, and for $p=2$ they become Hilbert
spaces and we denote them by $\mathbb{H}^{a}$. For $a<0$, we define
$\mathbb{H}^{a}$ as the dual space of $\mathbb{H}^{-a}$ with respect to the
$\mathbb{H}$-scalar product. Moreover, the Poincaré inequality holds. The zero
mean value assumption provides that the norms
$\|\boldsymbol{u}\|_{\mathbb{H}^{a,p}}=\|\mathrm{A}^{\frac{a}{2}}\boldsymbol{u}\|_{\mathbb{L}^{p}}$
and
$\big{(}\|\boldsymbol{u}\|_{\mathbb{L}_{\sigma}^{p}}^{p}+\|\boldsymbol{u}\|_{\mathbb{H}^{a,p}}^{p}\big{)}^{\frac{1}{p}}$
are equivalent.

We use the following Rellich-Kondrachov compactness Theorem in the sequel (see
[4, Theorem 9.16]):

  1. (1)

For every $p\in(2,\infty)$, the space $\mathbb{H}^{1,p}$ is compactly embedded
in $\mathbb{L}_{\sigma}^{\infty}({\mathbb{T}^{2}})$, that is, there exists a
constant $C_{p}$ such that

$\displaystyle\|\boldsymbol{u}\|_{\mathbb{L}_{\sigma}^{\infty}}\leq
C_{p}\|\boldsymbol{u}\|_{\mathbb{H}^{1,p}}.$ (2.2)

  2. (2)

The space $\mathrm{H}^{a}$ is compactly embedded in
$\mathrm{L}^{\infty}({\mathbb{T}^{2}})$ for $a>1$.

Let $(\mathbb{U},\|\cdot\|_{\mathbb{U}})$ and
$(\mathbb{V},\|\cdot\|_{\mathbb{V}})$ be any two normed vector spaces. The
space of all linear bounded operators from $\mathbb{U}$ into $\mathbb{V}$ is
denoted by $\mathcal{L}(\mathbb{U},\mathbb{V})$. We use the notation
$(\cdot,\cdot)$ and
$\langle\cdot,\cdot\rangle_{\mathbb{U}^{\prime}\times\mathbb{U}}$ for scalar
product and duality pairing between $\mathbb{U}^{\prime}$ and $\mathbb{U}$,
respectively.

###  2.2. The heat kernel

We interpreted system (1.2) in the sense of Walsh (see [51]) as expressed in
(1.1.1), with the heat kernel $G$. We need to make accurate estimations of $G$
to complete our calculations, as it is a critical factor.

The operator $\mathrm{A}$ generates a semigroup
$\mathrm{S}(t)=e^{-t\mathrm{A}}$, for
$\xi\in\mathring{\mathrm{L}}_{\\#}^{2}(\mathbb{T}^{2})$ and $t\in[0,T]$, we
have

$\displaystyle[\mathrm{S}(t)\xi](x)=\sum_{\eta\in\mathbb{Z}^{2}}e^{-t|\eta|^{2}}(\xi,e_{\eta})_{\mathrm{L}^{2}}e_{\eta}(x)=\frac{1}{2\pi}\sum_{\eta\in\mathbb{Z}^{2}}(\xi,e_{\eta})_{\mathrm{L}^{2}}e^{-t|\eta|^{2}+i\eta\cdot
x}.$ (2.3)

Moreover, the action of the semigroup $\mathrm{S}(\cdot)$ on the vorticity
$\xi$ can be represented by the convolution

$\displaystyle[\mathrm{S}(t)\xi](x)=\int_{\mathbb{T}^{2}}G(t,x,y)\xi(y)\mathrm{d}y,$
(2.4)

where $G$ is the heat kernel of the following problem

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial v}{\partial
t}(t,x)-\Delta v(t,x)&=0,\ \ (t,x)\in(0,T]\times{\mathbb{T}^{2}},\\\
v(t,\cdot)&\text{ is periodic },\ \ t\in[0,T],\\\ v(0,x)&=\delta_{0}(x-y),\ \
x,y\in{\mathbb{T}^{2}},\end{aligned}}\right.$ (2.5)

where $\delta_{0}(\cdot)$ is the Dirac delta distribution centered at $(0,0)$.
Using Fourier series expansion, we obtain the following explicity form of heat
kernel $G$

$\displaystyle
G(t,x,y)=\frac{1}{(2\pi)^{2}}\sum_{\eta\in\mathbb{Z}^{2}}e^{-t|\eta|^{2}+i\eta\cdot(x-y)}.$
(2.6)

For our calculations, we need another form of heat kernel which can be derived
using the method of images (see [22, Chapter 2.7, Section 5 or Chapter 2.11,
Section 3] or [47, Chapter 7, Section 2])

$\displaystyle G(t,x,y)=\frac{1}{4\pi
t}\sum_{\eta\in\mathbb{Z}^{2}}e^{-\frac{|x-y+2\eta\pi|^{2}}{4t}}.$ (2.7)

From (2.6) or (2.7), one can observe the following properties of the heat
kernel $G$ for all
$(t,x,y)\in[0,T]\times{\mathbb{T}^{2}}\times{\mathbb{T}^{2}}$:

  1. (1)

Symmetry: $G(t,x,y)=G(t,y,x)$,

  2. (2)

$G(t,x,y)=G(t,0,x-y)$.

Let us recall a result from [23] which gives useful estimates on the heat
kernel and its gradient in the two-dimensional case.

######  Lemma 2.1 ([23, Theorem 4]).

For fixed $s\in(0,t)$ and $x\in\mathbb{T}^{2},$ the following estimates hold:

  1. (1)

For $\beta\in(0,4/3)$, there exists a positive constant $C_{\beta}$ such that

$\displaystyle\int_{\mathbb{T}^{2}}|\nabla_{y}G(s,x,y)|^{\beta}\mathrm{d}y$
$\displaystyle\leq C_{\beta}s^{1-\frac{3\beta}{2}},$ (2.8)
$\displaystyle\int_{0}^{t}\int_{\mathbb{T}^{2}}|\nabla_{y}G(s,x,y)|^{\beta}\mathrm{d}y\mathrm{d}s$
$\displaystyle\leq C_{\beta}t^{2-\frac{3\beta}{2}}.$ (2.9)

  2. (2)

For every $\beta\in(0,2)$, there exists a positive constant $C_{\beta}$ such
that

$\displaystyle\int_{\mathbb{T}^{2}}|G(s,x,y)|^{\beta}\mathrm{d}y$
$\displaystyle\leq C_{\beta}s^{1-\beta},$ (2.10)
$\displaystyle\int_{0}^{t}\int_{\mathbb{T}^{2}}|G(s,x,y)|^{\beta}\mathrm{d}y\mathrm{d}s$
$\displaystyle\leq C_{\beta}t^{2-\beta}.$ (2.11)

The following result shows the regularizing effect of convolution with the
gradient of the heat kernel $G$. Let $J$ be the linear operator defined as

$\displaystyle(J\boldsymbol{\phi})(t,x)=\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{\phi}(s,y)\mathrm{d}y\mathrm{d}s,\
\ \text{ for }\ \ (t,x)\in[0,T]\times{\mathbb{T}^{2}},$ (2.12)

and $\boldsymbol{\phi}$ smooth enough. We have the following result from [23].

######  Lemma 2.2 ([23, Lemma 6]).

The following hold:

  1. (1)

Let $p\geq 1,\ \alpha\geq 1,\
\beta\in\mathopen{}\mathclose{{}\left[1,\frac{4}{3}}\right),\
\gamma>\frac{2\beta}{2-\beta}$ such that
$\frac{1}{\beta}=1+\frac{1}{p}-\frac{1}{\alpha}$. Then, $J$ is a bounded
linear operator from
$\mathrm{L}^{\gamma}(0,T;\mathbb{L}_{\sigma}^{\alpha}(\mathbb{T}^{2}))$ into
$\mathrm{L}^{\infty}(0,T;\mathrm{L}^{p}(\mathbb{T}^{2}))$. Moreover, there
exists a positive constant $C_{\beta}$ such that

$\displaystyle\|(J\boldsymbol{\phi})(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C_{\beta}\int_{0}^{t}(t-s)^{\frac{1}{\beta}-\frac{3}{2}}\|\boldsymbol{\phi}(s)\|_{\mathbb{L}_{\sigma}^{\alpha}}\mathrm{d}s$
$\displaystyle\leq
C_{\beta}t^{\frac{1}{\beta}-\frac{3}{2}+\frac{\gamma-1}{\gamma}}\bigg{(}\int_{0}^{t}\|\boldsymbol{\phi}(s)\|_{\mathbb{L}_{\sigma}^{\alpha}}^{\gamma}\mathrm{d}s\bigg{)}^{\frac{1}{\gamma}},$
(2.13)

for all $t\in[0,T]$.

  2. (2)

Let $p>4$ and $\gamma>\frac{2p}{p-2}$. Then, $J$ is a bounded linear operator
from
$\mathrm{L}^{\gamma}(0,T;\mathbb{L}_{\sigma}^{p}(\mathbb{T}^{2}))$ into
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$. Moreover, there exists a positive
constant $C_{p,T}$ such that

$\displaystyle\sup_{t\in[0,T]}\sup_{x\in{\mathbb{T}^{2}}}|(J\boldsymbol{\phi})(t,x)|\leq
C_{p,T}\bigg{(}\int_{0}^{T}\|\boldsymbol{\phi}(s)\|_{\mathbb{L}_{\sigma}^{p}}^{\gamma}\mathrm{d}s\bigg{)}^{\frac{1}{\gamma}}.$
(2.14)

###  2.3. The Biot-Savart law

Let us discuss the Biot-Savart law, which provides an explicit expression of
the velocity field $\boldsymbol{u}^{\epsilon}$ in terms of the vorticity
$\xi^{\epsilon}$, as described in [36, Section 2.4.1] and [37, Section 1.2].
Taking curl on both side of the relation
$\xi^{\epsilon}=\nabla^{\perp}\cdot\boldsymbol{u}^{\epsilon}$, we find

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
-\Delta\boldsymbol{u}^{\epsilon}&=\nabla^{\perp}\xi^{\epsilon},\\\
\nabla\cdot\boldsymbol{u}^{\epsilon}&=0,\\\ \boldsymbol{u}^{\epsilon}&\text{
is periodic}.\end{aligned}}\right.$ (2.15)

The above system allows us to express velocity field
$\boldsymbol{u}^{\epsilon}$ in terms of vorticity $\xi^{\epsilon}$. In terms
of the Fourier series, if
$\xi^{\epsilon}(x)=\frac{1}{2\pi}\sum\limits_{\eta\in\mathbb{Z}_{0}^{2}}\xi_{\eta}^{\epsilon}e^{i\eta\cdot
x}$, then

$\displaystyle\boldsymbol{u}^{\epsilon}(x)=-\frac{i}{2\pi}\sum_{\eta\in\mathbb{Z}_{0}^{2}}\frac{\eta^{\perp}}{|\eta|^{2}}\xi_{\eta}^{\epsilon}e^{i\eta\cdot
x}.$ (2.16)

It is clear from the above relation that the velocity field
$\boldsymbol{u}^{\epsilon}$ has one order more of regularity in view of
vorticity field $\xi^{\epsilon}$, that is, if
$\xi^{\epsilon}\in\mathrm{H}^{a-1,p}$ then
$\boldsymbol{u}^{\epsilon}\in\mathbb{H}^{a,p}$. In particular, the norms
$\|\boldsymbol{u}^{\epsilon}\|_{\mathbb{H}^{a,p}}$ and
$\|\xi^{\epsilon}\|_{\mathrm{H}^{a-1,p}}$ are equivalent.

From [37, Section 1.2], we know that the Biot-Savart law allows us to write
the velocity field $\boldsymbol{u}^{\epsilon}$ in terms of the vorticity
$\xi^{\epsilon}$ as

$\displaystyle\boldsymbol{u}^{\epsilon}(x)=(\boldsymbol{k}*\xi^{\epsilon})(x)=\int_{\mathbb{T}^{2}}\boldsymbol{k}(x-y)\xi^{\epsilon}(y)\mathrm{d}y,$
(2.17)

where $\boldsymbol{k}$ is the Biot-Savart kernel given by

$\displaystyle\boldsymbol{k}=\nabla^{\perp}G=\mathopen{}\mathclose{{}\left(-\frac{\partial
G}{\partial x_{2}},\frac{\partial G}{\partial x_{1}}}\right),$ (2.18)

and $G$ is the Green function of $-\Delta$ on the torus with mean zero.

Let us recall some basic properties of the Biot-Savart kernel from [6].

######  Lemma 2.3 ([6, Lemma 2.17]).

For every $p\in[1,2)$, the map $\boldsymbol{k}$ defined in (2.18) is an
$\mathbb{L}^{p}(\mathbb{T}^{2})$ divergence free (in the distributional sense)
vector field.

We have some useful estimates which play a crucial role. Using (2.16), the
Sobolev embedding
$\mathbb{H}^{1,p}\subset\mathbb{L}_{\sigma}^{\infty}(\mathbb{T}^{2})$ for
$p>2$, and the equivalency of the norms
$\|\boldsymbol{u}^{\epsilon}\|_{\mathbb{H}^{1,p}}$ and
$\|\xi^{\epsilon}\|_{\mathrm{L}^{p}}$, we obtain for any $p>2$

$\displaystyle\|\boldsymbol{k}*\xi^{\epsilon}\|_{\mathbb{L}_{\sigma}^{\infty}}=\|\boldsymbol{u}^{\epsilon}\|_{\mathbb{L}_{\sigma}^{\infty}}\leq
C_{p}\|\boldsymbol{u}^{\epsilon}\|_{\mathbb{H}^{1,p}}\leq
C_{p}\|\xi^{\epsilon}\|_{\mathrm{L}^{p}},$ (2.19)

for some positive constant $C_{p}$.

Using (2.17), Lemma 2.3, Young’s inequality for convolution (with exponent
$p\geq 1,\alpha\in[1,2)$, and $\beta\geq 1$, and
$1+\frac{1}{p}=\frac{1}{\alpha}+\frac{1}{\beta}$), we obtain

$\displaystyle\|\boldsymbol{k}*\xi^{\epsilon}\|_{\mathbb{L}_{\sigma}^{p}}=\|\boldsymbol{u}^{\epsilon}\|_{\mathbb{L}_{\sigma}^{p}}\leq\|\boldsymbol{k}\|_{\mathbb{L}_{\sigma}^{\alpha}}\|\xi^{\epsilon}\|_{\mathrm{L}^{\beta}}.$
(2.20)

###  2.4. A short note on stochastic forcing term

Let us discuss the stochastic term appearing in the system (1.2). Let
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ be a given
stochastic basis. For any given $T>0$, let
$\mathrm{Q}:\mathring{\mathrm{L}}_{\\#}^{2}(\mathbb{T}^{2})\to\mathring{\mathrm{L}}_{\\#}^{2}(\mathbb{T}^{2})$
be a positive symmetric bounded linear operator. Set
$\mathrm{L}_{\mathrm{Q}}^{2}(\mathbb{T}^{2})$ as the completion of the space
of all square integrable, zero mean-value, periodic functions
$\phi:{\mathbb{T}^{2}}\to\mathbb{R}$ with respect to the scalar product
$(\phi,\psi)_{\mathrm{L}_{\mathrm{Q}}^{2}}=(\mathrm{Q}\phi,\psi)_{\mathrm{L}^{2}}$,
for all $\phi,\psi\in\mathrm{L}_{\mathrm{Q}}^{2}(\mathbb{T}^{2})$.

Let
$\mathcal{H}_{T}=\mathrm{L}^{2}(0,T;\mathrm{L}_{\mathrm{Q}}^{2}(\mathbb{T}^{2}))$
denote the real separable Hilbert space, with the scalar product

$\displaystyle(g_{1},g_{2})_{\mathcal{H}_{T}}=\int_{0}^{T}(g_{1}(t),g_{2}(t))_{\mathrm{L}_{\mathrm{Q}}^{2}}\mathrm{d}t=\int_{0}^{T}(\mathrm{Q}g_{1}(t),g_{2}(t))_{\mathrm{L}^{2}}\mathrm{d}t.$

Let us now consider the isonormal Gaussian process
$\mathrm{W}=\\{\mathrm{W}(h);h\in\mathcal{H}_{T}\\}$ (for more details see
[41]). The map $h\mapsto\mathrm{W}(h)$ provides a linear isometry from
$\mathcal{H}_{T}$ onto $\mathcal{H}$, which is a closed subset of
$\mathrm{L}^{2}(\Omega,\mathscr{F},\mathbb{P})$, consisted of zero-mean
Gaussian random variables and the isometry can be represented as

$\displaystyle\mathbb{E}[\mathrm{W}(h)\mathrm{W}(g)]=(h,g)_{\mathcal{H}_{T}}.$

The stochastic term appearing in (1.2) should be understand in the following
sense, for $h\in\mathcal{H}_{T}$, we set

$\displaystyle\mathrm{W}(h)=\int_{0}^{T}\int_{\mathbb{T}^{2}}h(s,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),$
(2.21)

namely, the above random variable is a zero-mean Gaussian random variable with
the covariance $\mathbb{E}[(\mathrm{W}(h))^{2}]=\|h\|_{\mathcal{H}_{T}}^{2}$.

The stochastic term above, expressed through linear isometry $\mathrm{W}$, can
be interpreted in the framework of Walsh (as detailed in [51] or [16]). One
can express $\mathrm{W}(h)$ as

$\displaystyle\mathrm{W}(h)=\sum_{i}\int_{0}^{T}(h(s),\tilde{e}_{i})_{\mathrm{L}_{\mathrm{Q}}^{2}}\mathrm{d}\beta_{s}(\tilde{e}_{i}),$
(2.22)

where $\\{\tilde{e}_{i}\\}$ is a complete orthonormal basis of
$\mathrm{L}_{\mathrm{Q}}^{2}(\mathbb{T}^{2})$ and
$\beta_{s}(\tilde{e}_{i})=\mathrm{W}(\chi_{[0,s]}\tilde{e}_{i})$, hence
$\\{\beta(\tilde{e}_{i})\\}$ is a sequence of independent standard
1-dimensional Brownian motions on the probability space
$(\Omega,\mathscr{F},\mathbb{P})$ adapted to the filtration
$\\{\mathscr{F}_{t}\\}_{t\geq 0}$. Setting
$\mathrm{M}_{t}(A):=\mathrm{W}(\chi_{[0,t]}\chi_{A})$ for all $t\in[0,T]$ and
$A\in\mathcal{B}_{b}(\mathbb{R}^{2})$, we construct a martingale measure with
the covariance operator $\mathrm{Q}$ and (2.22) coincides with the stochastic
integral in the sense of Walsh ([51]). Moreover, the isonormal Gaussian
process $\mathrm{W}$ can be associated to a $\mathrm{Q}$-Wiener process
$\mathscr{W}_{t}$ on $\mathring{\mathrm{L}}_{\\#}^{2}(\mathbb{T}^{2})$ (see
[16]) in the following way

$\displaystyle(\mathscr{W}_{t},h)_{\mathrm{L}^{2}}=\mathrm{W}(\chi_{[0,t]}h),\
\ \text{ for all }\ \ h\in\mathring{\mathrm{L}}_{\\#}^{2}(\mathbb{T}^{2}),$
(2.23)

and (2.21) coincides with the integral with respect to $\mathscr{W}$, in a
sense made precise in [17, Section 3.4]. The stochastic convolution term
appearing on the right-hand side of (1.1.1) is meaningful in the ways
described above. The stochastic forcing term is periodic with zero-mean in the
space variable. It is not unexpected that we need the covariance operator
$\mathrm{Q}$ to have some regularizing effect when working in a two-
dimensional space, as stated in [16, Pages 144-146]. To address this issue,
one can consider using the covariance operator $\mathrm{Q}$ with an
appropriate regularizing effect. For example, one can choose a covariance
operator of the form

$\displaystyle\mathrm{Q}=(-\Delta)^{-a},\ \ \text{ for some}\ \ a>0.$ (2.24)

Thus, we have $\mathrm{Q}e_{\eta}=\frac{1}{|\eta|^{2a}}e_{\eta},\ \text{for
all}\ \eta\in\mathbb{Z}_{0}^{2},$ and $\\{\tilde{e}_{\eta}\\}$ is a complete
orthonormal basis of $\mathrm{L}^{2}_{\mathrm{Q}}(\mathbb{T}^{2})$ defined as
$\tilde{e}_{\eta}(x)=\frac{1}{\sqrt{2}\pi}|\eta|^{a}\cos(\eta\cdot x)$ and
$\tilde{e}_{-\eta}(x)=\frac{1}{\sqrt{2}\pi}|\eta|^{a}\sin(\eta\cdot x)$, for
$\eta\in\mathbb{Z}_{+}^{2}$. From [23, Eqn (2.24)], we know that the
stochastic term
$\int_{0}^{t}\int_{\mathbb{T}^{2}}\mathrm{G}(t-s,x,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y)$
is well-defined for $a>0$, which is equivalent to have
$G(t-\cdot,x,\cdot)\in\mathcal{H}_{t}$, for every $t>0$. It can justified as

$\displaystyle\|G(t-\cdot,x,\cdot)\|_{\mathcal{H}_{t}}^{2}\leq\frac{1}{8\pi^{2}}\sum_{\eta\in\mathbb{Z}_{0}^{2}}\frac{1}{|\eta|^{2+2a}},$

the above series is convergent if and only if $a>0$. Thus, the hypothesis
$a>0$ is sufficient for space-time continuity of the stochastic convolution’s
trajectories (see [15, Theorem 2.13]). The noise formulation of our work is
similar to [23], where the author considered two-dimensional SNSE (1.1) in the
vorticity form (1.2) with $\epsilon=1$, and established the existence and
uniqueness of the solution $\xi$ of (1.2) in the spaces
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})),$ for $p>2$ and
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$ whenever the initial data
$\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$ and
$\xi\in\mathrm{C}(\mathbb{T}^{2})$, respectively.

###  2.5. Solvability results of the system (1.2)

The main focus of this subsection is to recall the existence and uniqueness of
the solution to the system (1.2). We use the Biot-Savart law, which gives the
relation $\boldsymbol{u}^{\epsilon}=\boldsymbol{k}*\xi^{\epsilon}$. Set
$\boldsymbol{q}(\xi^{\epsilon})=\xi^{\epsilon}(\boldsymbol{k}*\xi^{\epsilon})$,
that is,

$\displaystyle[\boldsymbol{q}(\xi^{\epsilon})](x)=\xi^{\epsilon}(x)\int_{\mathbb{T}^{2}}\boldsymbol{k}(x-y)\xi^{\epsilon}(y)\mathrm{d}y.$
(2.25)

Using Hölder’s inequality and (2.19), we obtain for any $p>2$

$\displaystyle\|\boldsymbol{q}(\xi^{\epsilon})\|_{\mathbb{L}_{\sigma}^{p}}\leq\|\xi^{\epsilon}\|_{\mathrm{L}^{p}}\|\boldsymbol{k}*\xi^{\epsilon}\|_{\mathbb{L}_{\sigma}^{\infty}}\leq
C_{p}\|\xi^{\epsilon}\|_{\mathrm{L}^{p}}^{2},$ (2.26)

which implies that
$\boldsymbol{q}:\mathrm{L}^{p}(\mathbb{T}^{2})\to\mathbb{L}_{\sigma}^{p}(\mathbb{T}^{2})$
for any $p>2$. We can write the system (1.2) in an equivalent form, where the
velocity field $\boldsymbol{u}^{\epsilon}$ would not appear. That is,

$\displaystyle\xi^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}(\xi^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),\
\ \mathbb{P}\text{-a.s.}$ (2.27)

for all $t\in[0,T]$. Let us provide the definition of weak solution (in
analytic sense) to the system (1.2).

######  Definition 2.4.

We say that an ${\mathrm{L}}^{p}$-valued ($p>2$) continuous
$\mathscr{F}_{t}$-adapted stochastic process $\xi^{\epsilon}$ is a weak
solution to the system (1.2), if it solves the following, for every
$t\in[0,T],\ \phi\in\mathrm{H}^{b}$ with $b>2,$ we have

$\displaystyle\int_{\mathbb{T}^{2}}\xi^{\epsilon}(t,y)\phi(y)\mathrm{d}y$
$\displaystyle-\int_{0}^{t}\int_{\mathbb{T}^{2}}\xi^{\epsilon}(s,y)\Delta\phi(y)\mathrm{d}x\mathrm{d}s-\int_{0}^{t}\int_{\mathbb{T}^{2}}\boldsymbol{q}(\xi^{\epsilon}(s,\cdot))(y)\cdot\nabla\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle=\int_{\mathbb{T}^{2}}\xi_{0}(y)\phi(y)\mathrm{d}y+\sqrt{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{2}}\phi(y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),\
\ \mathbb{P}\text{-a.s.}$ (2.28)

The third term appearing in the left hand side of (2.4) is well-defined, which
can be justified as follows: for $s>1$

$\displaystyle\bigg{|}\int_{\mathbb{T}^{2}}\boldsymbol{q}(\xi^{\epsilon}(s,\cdot))(y)\cdot\nabla\phi(y)\mathrm{d}y\bigg{|}$
$\displaystyle\leq\|\nabla\phi\|_{\mathrm{L}^{\infty}}\|\boldsymbol{q}(\xi^{\epsilon}(s,\cdot))\|_{\mathbb{L}_{\sigma}^{1}}$
$\displaystyle\leq
C\|\nabla\phi\|_{\mathrm{H}^{s}}\|\boldsymbol{k}*\xi^{\epsilon}(s,\cdot)\|_{\mathbb{L}_{\sigma}^{2}}\|\xi^{\epsilon}(s,\cdot)\|_{\mathrm{L}^{2}}$
$\displaystyle\leq
C\|\phi\|_{\mathrm{H}^{s+1}}\|\xi^{\epsilon}(s,\cdot)\|_{\mathrm{L}^{2}}^{2},$
(2.29)

where we have used (2.20), Hölder’s and Sobolev’s inequalities.

By the equivalency of solution concepts given in [25, Proposition 3.5], we
consider the solution to the system (1.2) according to Walsh (see [51]) being
provided by the integral equation (1.1.1).

Let us recall the the existence and uniqueness result for the solution to the
system (1.2) from [23].

######  Proposition 2.5 ([23, Theorem 1]).

Let us assume that $\xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$, for $p>2$,
and $a>0$ in (2.24). Then, there exists a unique $\mathrm{L}^{p}$-valued
$\mathscr{F}_{t}$-adapted continuous process $\xi^{\epsilon}$ satisfying the
integral equation (1.1.1). Moreover, if
$\xi_{0}\in\mathrm{C}({\mathbb{T}^{2}})$, then the solution admits a
modification which is a space-time continuous process.

##  3\. Uniform Large Deviation Principle: Additive Noise

In this section, we first recall the basics of large deviation theory. Then,
we discuss the concept of uniform large deviation principle (ULDP) and
establish it for the law of the solution of the system (1.2).

###  3.1. Large deviation principle

Let $\mathcal{E}$ denote a Polish space (complete separable metric space) with
the Borel $\sigma$-field $\mathcal{B}(\mathcal{E})$. Let
$\\{\mu^{\epsilon}\\}_{\epsilon>0}$ be a family of probability measures on
$(\mathcal{E},\mathcal{B}(\mathcal{E}))$. We provide some basic definitions
related to large deviation principle.

######  Definition 3.1 (Rate function).

A function $\mathrm{I}:\mathcal{E}\to[0,\infty]$ is called a rate function on
$\mathcal{E}$ if it is a lower semicontinuous in $\mathcal{E}$. A rate
function $\mathrm{I}(\cdot)$ on $\mathcal{E}$ is said to be a good rate
function on $\mathcal{E}$ if for every $M\in[0,\infty)$, the level set
$K:=\\{x\in\mathcal{E}:\mathrm{I}(x)\leq M\\}$ is a compact subset of
$\mathcal{E}$.

######  Definition 3.2 (Large deviation principle).

Let $\mathrm{I}:\mathcal{E}\to[0,\infty]$ be a rate function on a Polish space
$\mathcal{E}$. The sequence $\\{\mu^{\epsilon}\\}_{\epsilon>0}$ satisfies the
LDP on the space $\mathcal{E}$ with the rate function $\mathrm{I}(\cdot)$, if
the following hold:

  1. (1)

For any closed subset $\mathrm{F}\subset\mathcal{E}$,

$\displaystyle\limsup_{\epsilon\to
0}\epsilon\log\mu^{\epsilon}(\mathrm{F})\leq-\inf_{x\in\mathrm{F}}\mathrm{I}(x).$

  2. (2)

For any open subset $\mathrm{O}\subset\mathcal{E}$,

$\displaystyle\liminf_{\epsilon\to
0}\epsilon\log\mu^{\epsilon}(\mathrm{O})\geq-\inf_{x\in\mathrm{O}}\mathrm{I}(x).$

Let us now discuss the LDP for $\mathcal{E}$-valued random variables. Let
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ be a
complete probability space satisfying the usual conditions.

Given $\epsilon>0$, let
$\mathscr{G}^{\epsilon}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times\mathbb{T}^{2})\to\mathcal{E}$,
(where $\mathcal{E}_{0}$ is an another Polish space) be a measurable map and
set

$\displaystyle\mathrm{Y}^{\epsilon}=\mathscr{G}^{\epsilon}(\mathrm{W}),\ \
\text{ for all }\epsilon>0.$ (3.1)

Then, $\\{\mathrm{Y}^{\epsilon}\\}_{\epsilon>0}$ is a family of
$\mathcal{E}$-valued random variables. Let $\\{\mu^{\epsilon}\\}_{\epsilon>0}$
be the the law of $\\{\mathrm{Y}^{\epsilon}\\}_{\epsilon>0}$ on the space
$\mathcal{E}$. Then, the family $\\{\mathrm{Y}^{\epsilon}\\}_{\epsilon>0}$ is
said to satisfy the LDP on the space $\mathcal{E}$ if the family of measures
$\\{\mu^{\epsilon}\\}_{\epsilon>0}$ satisfies the LDP on the space
$\mathcal{E}$.

For any given $M>0$, define the following sets:

$\displaystyle\mathfrak{U}^{M}$
$\displaystyle:=\bigg{\\{}\phi\in\mathrm{L}^{2}([0,T]\times\mathbb{T}^{2}):\int_{0}^{T}\int_{\mathbb{T}^{2}}|v(s,y)|^{2}\mathrm{d}y\mathrm{d}s\leq
M\bigg{\\}},$ (3.2) $\displaystyle\mathscr{P}_{2}$
$\displaystyle:=\bigg{\\{}v:\int_{0}^{T}\int_{\mathbb{T}^{2}}|v(s,y)|^{2}\mathrm{d}y\mathrm{d}s<\infty,\
\mathbb{P}\text{-a.s.}\bigg{\\}},$ (3.3) $\displaystyle\mathscr{P}_{2}^{M}$
$\displaystyle:=\bigg{\\{}v\in\mathscr{P}_{2}:\phi(\omega)\in\mathfrak{U}^{M},\
\mathbb{P}\text{-a.s.}\bigg{\\}}.$ (3.4)

Note that $\mathfrak{U}_{M}$ is a compact metric space equipped with the weak
topology of $\mathrm{L}^{2}([0,T]\times\mathbb{T}^{2})$.

For every $\psi\in\mathcal{E}$, we define a map
$\mathrm{I}(\cdot):\mathcal{E}\to[0,\infty]$ by

$\displaystyle\mathrm{I}(\psi):=\inf_{\mathopen{}\mathclose{{}\left\\{v\in\mathrm{L}^{2}([0,T]\times\mathbb{T}^{2}):\psi=\mathscr{G}^{0}\mathopen{}\mathclose{{}\left(\int_{0}^{\cdot}\int_{\mathbb{T}^{2}}v(s,y)\mathrm{d}y\mathrm{d}s}\right)}\right\\}}\bigg{\\{}\frac{1}{2}$
$\displaystyle\int_{0}^{T}\int_{\mathbb{T}^{2}}|v(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}},$
(3.5)

with the convention that the infimum over the empty set is infinite. We
consider the following linear stochastic equation:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\mathrm{d}\zeta^{\epsilon}(t,x)+\mathrm{A}\zeta^{\epsilon}(t,x)\mathrm{d}t&=\sqrt{\epsilon}\mathrm{W}(\mathrm{d}t,\mathrm{d}x),\
\ (t,x)\in(0,T)\times\mathbb{T}^{2},\\\
\zeta^{\epsilon}(0,x)&=0.\end{aligned}}\right.$ (3.6)

The solution
$\zeta^{\epsilon}(t,x)=\sqrt{\epsilon}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),\
(t,x)\in(0,T]\times{\mathbb{T}^{2}}$ is a stochastic convolution that is
Gaussian and independent of the initial condition $\xi_{0}$. From [16, Theorem
12.11], we know that the LDP holds in
$\mathrm{C}([0,T];\mathrm{L}^{2}(\mathbb{T}^{2}))$ for the laws of solutions
of the linear stochastic system (3.6). Note that
$\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$ is a Banach space continuously,
and as a Borel set, embedded into $\mathrm{L}^{2}(\mathbb{T}^{2})$. Then also
the spaces $\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ and
$\mathrm{L}^{2}(0,T;\mathrm{L}^{p}(\mathbb{T}^{2}))$ are continuously, and as
a Borel set, embedded into
$\mathrm{L}^{2}(0,T;\mathrm{L}^{2}(\mathbb{T}^{2}))$. Taking into account of
[16, Theorem 12.16], one can obtain the LDP holds in
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for $p>2$. Using similar
arguments one can conclude that the LDP holds in
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$.

###  3.2. Uniform large deviation principle

In this subsection, we recall the definition ULDP, to be more specific,
Freidlin-Wentzell uniform large deviation principle (FWULDP), see [24] from
the work [43], and a uniform contraction principle for proving ULDP in a
separable Banach space from [52].

Let $(\mathcal{E},d)$ be a Polish space and $\mathcal{E}_{0}$ be a set. For
any $\xi_{0}\in\mathcal{E}_{0}$, define a rate function,
$\mathrm{I}_{\xi_{0}}:\mathcal{E}\to[0,\infty]$ (cf. [9, 43, 46]). Let
$\Lambda_{\xi_{0}}(s_{0})$ denote the level set

$\displaystyle\Lambda_{\xi_{0}}(s_{0}):=\big{\\{}f\in\mathcal{E}:\mathrm{I}_{\xi_{0}}(\psi)\leq
s_{0}\big{\\}},$

for $s_{0}\in[0,\infty]$. Furthermore, let $\mathscr{T}$ be the collection of
subsets of $\mathcal{E}_{0}$ and $b(\epsilon)$ be a function which converges
to $0$ as $\epsilon\to 0$.

If $g:=g(t,x)$ is a random field and $\mathcal{E}$ is a function space (will
be fixed later), then $g$-a.s. in $\mathcal{E}$ means that $g$ has a
stochastic version in $\mathcal{E}$, $\mathbb{P}$-a.s. For any metric space
$(\mathcal{E},d)$, the distance between an element $h\in\mathcal{E}$ and a set
$\mathrm{B}\subset\mathcal{E}$ is defined by

$\displaystyle\mathrm{dist}_{\mathcal{E}}(g,\mathrm{B}):=\inf_{h\in\mathrm{B}}d(g,h).$

######  Definition 3.3 (FWULDP).

A family of $\mathcal{E}$-valued random variables
$\\{\mathrm{Y}_{\xi_{0}}^{\epsilon}\\}_{\epsilon>0}$ is said to satisfy a
Freidlin-Wentzell uniform large deviation principle with the speed
$b(\epsilon)$ and the rate function $\mathrm{I}_{\xi_{0}}(\cdot)$, uniformly
over $\mathscr{T}$ if the following hold:

  1. (1)

For any $\mathrm{E}\in\mathscr{T},\ t_{0}\geq 0,$ and $\delta>0$

$\displaystyle\liminf_{\epsilon\to
0}\inf_{\xi_{0}\in\mathrm{E}}\inf_{\psi\in\Lambda_{\xi_{0}}(t_{0})}\bigg{\\{}b(\epsilon)\log\mu^{\epsilon}\big{(}d(\mathrm{Y}_{\xi_{0}}^{\epsilon},\psi)<\delta\big{)}+\mathrm{I}_{\xi_{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

For any $\mathrm{E}\in\mathscr{T},\ t_{0}\geq 0,$ and $\delta>0$

$\displaystyle\limsup_{\epsilon\to
0}\sup_{\xi_{0}\in\mathrm{E}}\sup_{s\in[0,t_{0}]}\bigg{\\{}b(\epsilon)\log\mu^{\epsilon}\big{(}\mathrm{dist}_{\mathcal{E}}(\mathrm{Y}_{\xi_{0}}^{\epsilon},\Lambda_{\xi_{0}}(s))\geq\delta\big{)}+s\bigg{\\}}\leq
0.$

######  Proposition 3.4 ([52, Uniform contraction principle, Theorem 2.5]).

Let $\mathscr{T}$ be a nonempty set and $\mathrm{X}_{1},\mathrm{X}_{2}$ be any
two separable Banach spaces. Further assume that the family
$\\{\nu^{\epsilon}\\}_{\epsilon>0}$ of probability measures on
$(\mathrm{X}_{1},\mathcal{B}(\mathrm{X}_{1}))$, satisfies an LDP with the good
rate function $\mathrm{I}:\mathrm{X}_{1}\to[0,\infty]$. For any given
$\xi_{0}\in\mathscr{T}$, let
$\mathrm{S}_{\xi_{0}}:\mathrm{X}_{1}\to\mathrm{X}_{2}$ be a locally Lipschitz
continuous mapping in the sense that for every $R>0$, there exists a positive
constant $L_{R}$, such that for all
${\xi_{0}}\in\mathscr{T},x_{1},x_{2}\in\mathrm{X}_{1}$ with
$\|x_{1}\|_{\mathrm{X_{1}}}\leq R$ and $\|x_{1}\|_{\mathrm{X_{2}}}\leq R$,

$\displaystyle\|\mathrm{S}_{\xi_{0}}(x_{1})-\mathrm{S}_{\xi_{0}}(x_{2})\|_{\mathrm{X_{2}}}\leq
L_{R}\|x_{1}-x_{2}\|_{\mathrm{X_{1}}}.$

For any given ${\xi_{0}}\in\mathscr{T}$ and $\epsilon>0$, let
$\mu^{\epsilon}_{\xi_{0}}=\nu^{\epsilon}\circ(\mathrm{S}_{\xi_{0}})^{-1}$.
Then, the family $\\{\mu^{\epsilon}_{\xi_{0}}\\}_{\epsilon>0}$ of measures
satisfies the LDP on the space $\mathrm{X}_{2}$ uniformly in
${\xi_{0}}\in\mathscr{T}$ with the rate function $\mathrm{J}_{\xi_{0}}$
defined as

$\displaystyle\mathrm{J}_{\xi_{0}}(y)=\inf\\{\mathrm{I}(x):\
x\in(\mathrm{S}_{\xi_{0}})^{-1}(y)\\},\ \ \text{ for all }\ \
y\in\mathrm{X_{2}}.$

Let us now move to the proof of ULDP for the system (1.2), with respect to the
bounded initial conditions $\xi_{0}$ in $\mathrm{L}^{p}(\mathbb{T}^{2})$-norm,
for $p>2$ with the help of uniform contraction principle (Proposition 3.4).

A solution to the following linear stochastic equation:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\mathrm{d}\zeta(t,x)+\mathrm{A}\zeta(t,x)\mathrm{d}t&=\mathrm{W}(\mathrm{d}t,\mathrm{d}x),\
\ (t,x)\in(0,T]\times{\mathbb{T}^{2}},\\\ \zeta(0,x)&=0,\end{aligned}}\right.$
(3.7)

is defined by the variation of constant formula:

$\displaystyle\zeta(t,x)=\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\mathrm{W}(\mathrm{d}s,\mathrm{d}y),\
\ (t,x)\in(0,T]\times{\mathbb{T}^{2}}.$ (3.8)

Furthermore, we have the following result:

######  Lemma 3.5 ([23, Lemma 7]).

For $a>0$ in (2.24) and $p>2$, we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\zeta(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}<\infty.$
(3.9)

Moreover, $\zeta(\cdot,\cdot)$ has a continuous modification on
$[0,T]\times{\mathbb{T}^{2}}$.

Let us set $\beta=\xi-\zeta$. Since the additive noise is independent of the
unknown, we can see that $\beta$ satisfies the following equation:

$\displaystyle\frac{\partial}{\partial
t}\beta(t)=\Delta\beta(t)-\nabla\cdot\boldsymbol{q}(\beta(t)+\zeta(t)),\
t\in[0,T],$ (3.10)

whose mild solution is given by

$\displaystyle\beta(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}((\beta+\zeta)(s,\cdot))(y)\mathrm{d}y\mathrm{d}s,$
(3.11)

where $\boldsymbol{q}$ is defined in (2.25). Let us recall the following
result from [23].

######  Lemma 3.6 ([23, Lemma 11]).

Let us assume that $\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$,
$a>0$ in (2.24). Then, for a constant $C_{p}>0$, we have

$\displaystyle\sup_{t\in[0,T]}\|\beta(t)\|_{\mathrm{L}^{p}}^{p}\leq\big{(}\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}+C_{1}(\zeta)\big{)}e^{C_{2}(\zeta)},$

where the constants

$C_{1}(\zeta)=C_{p}T\sup\limits_{t\in[0,T]}\|\zeta(t)\|_{\mathrm{L}^{p}}^{2p}\
\text{ and }\
\mathrm{C}_{2}(\zeta)=C_{p}T\bigg{(}1+\sup\limits_{t\in[0,T]}\|\zeta(t)\|_{\mathrm{L}^{p}}^{2}\bigg{)}.$

Next, we prove the local Lipschitz continuity of solutions of (3.10) with
respect to $\zeta\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})),$ for
$p>2$.

######  Lemma 3.7.

For any $T>0$, $R_{1}>0$ and $R_{2}>0$, there exists a positive constant
$\widehat{C}=\big{(}C_{p}(C_{1}+R_{2}^{p})Te^{C_{p}T(1+C_{1}^{2/p}+R_{2}^{2})}\big{)}^{\frac{1}{p}}$
such that the solution of (3.10) satisfies

$\displaystyle\|\beta_{\zeta_{1}}-\beta_{\zeta_{2}}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq\widehat{C}\|\zeta_{1}-\zeta_{2}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})},$
(3.12)

for all $\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$ with
$\|\xi_{0}\|_{\mathrm{L}^{p}}\leq R_{1}$ and
$\zeta_{1},\zeta_{2}\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ with
$\|\zeta_{1}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq R_{2}$
and $\|\zeta_{2}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq
R_{2}$.

###### Proof.

From Lemma 3.6, for every $R_{1}>0$ and $R_{2}>0$, there exists
$C_{1}=C_{1}(R_{1},R_{2},T)=(R_{1}^{p}+C_{p}TR_{2}^{2p})e^{C_{p}T(1+R_{2}^{2})}>0$,
such that for all $\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$ with
$\|\xi_{0}\|_{\mathrm{L}^{p}}\leq R_{1}$ and
$\zeta\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ with
$\|\zeta\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq R_{2}$,

$\displaystyle\sup_{t\in[0,T]}\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}\leq
C_{1},\ \mathbb{P}\text{-a.s.}$ (3.13)

Let $\zeta_{1}$ and $\zeta_{2}$ be any two functions in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ such that
$\|\zeta_{1}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\\\ \leq
R_{2}$ and
$\|\zeta_{2}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq R_{2}$.
We denote the corresponding solutions of (3.10) by $\beta_{\zeta_{i}}$, for
$i=1,2$. Then, $\beta_{\zeta}=\beta_{\zeta_{1}}-\beta_{\zeta_{2}}$, where
$\zeta=\zeta_{1}-\zeta_{2}$, satisfies the following system for $t\in[0,T]$:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial}{\partial
t}\beta_{\zeta}(t)&=\Delta\beta_{\zeta}(t)-\big{[}\nabla\cdot\boldsymbol{q}(\beta_{\zeta_{1}}(t)+\zeta_{1}(t))-\nabla\cdot\boldsymbol{q}(\beta_{\zeta_{2}}(t)+\zeta_{2}(t))\big{]},\\\
\beta_{\zeta}(0)&=0.\end{aligned}}\right.$ (3.14)

Taking inner product with $|\beta_{\zeta}(\cdot)|^{p-2}\beta_{\zeta}(\cdot)$
to the first equation of the above system, we find

$\displaystyle\frac{1}{p}\frac{\mathrm{d}}{\mathrm{d}t}\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}+(p-1)\||\beta_{\zeta}(t)|^{\frac{p-2}{2}}\nabla\beta_{\zeta}(t)\|_{\mathrm{L}^{2}}^{2}$
$\displaystyle\quad=(p-1)\big{(}|\beta_{\zeta}(t)|^{p-2}\nabla\beta_{\zeta}(t),\boldsymbol{q}(\beta_{\zeta_{1}}(t)+\zeta_{1}(t))-\boldsymbol{q}(\beta_{\zeta_{2}}(t)+\zeta_{2}(t))\big{)},$
(3.15)

for a.e. $t\in[0,T]$. Let us take

$\displaystyle\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\beta_{\zeta_{1}}+\zeta_{1}):=\big{(}|\beta_{\zeta}|^{p-2}\boldsymbol{k}*(\beta_{\zeta_{1}}+\zeta_{1})\cdot\nabla\beta_{\zeta},\beta_{\zeta_{1}}+\zeta_{1}\big{)}.$
(3.16)

Then, by fixing $|\beta_{\zeta}|^{p-2}$, we have

$\displaystyle\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\beta_{\zeta_{1}}+\zeta_{1})-\mathrm{b}(\beta_{\zeta_{2}}+\zeta_{2},\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2})$
$\displaystyle=\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\beta_{\zeta_{1}}+\zeta_{1}-(\beta_{\zeta_{2}}+\zeta_{2}))$
$\displaystyle\quad+b((\beta_{\zeta_{1}}+\zeta_{1})-(\beta_{\zeta_{2}}+\zeta_{2}),\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2})$
$\displaystyle=\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\beta_{\zeta})+\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\zeta)+\mathrm{b}(\beta_{\zeta}+\zeta,\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2})$
$\displaystyle=\mathrm{b}(\beta_{\zeta_{1}}+\zeta_{1},\beta_{\zeta},\zeta)+\mathrm{b}(\beta_{\zeta},\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2})+\mathrm{b}(\zeta,\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2})$
$\displaystyle=\big{(}|\beta_{\zeta}|^{p-2}\boldsymbol{k}*(\beta_{\zeta_{1}}+\zeta_{1})\cdot\nabla\beta_{\zeta},\zeta\big{)}+\big{(}|\beta_{\zeta}|^{p-2}\boldsymbol{k}*\beta_{\zeta}\cdot\nabla\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2}\big{)}$
$\displaystyle\quad+\big{(}|\beta_{\zeta}|^{p-2}\boldsymbol{k}*\zeta\cdot\nabla\beta_{\zeta},\beta_{\zeta_{2}}+\zeta_{2}\big{)},$
(3.17)

where we use the fact that
$\mathrm{b}(\boldsymbol{u},\beta_{\zeta},\beta_{\zeta})=\int_{\mathbb{T}^{2}}(\boldsymbol{u}\cdot\nabla\beta_{\zeta})|\beta_{\zeta}|^{p-2}\beta_{\zeta}\mathrm{d}x=0$,
provided $\nabla\cdot\boldsymbol{u}=0$ (cf. [1, Lemma 2.2]). Using (3.16) and
(3.2) in (3.2), we obtain for a.e. $t\in[0,T]$

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}+p(p-1)\||\beta_{\zeta}(t)|^{\frac{p-2}{2}}\nabla\beta_{\zeta}(t)\|_{\mathrm{L}^{2}}^{2}$
$\displaystyle=p(p-1)\big{\\{}\big{(}\boldsymbol{k}*(\beta_{\zeta_{1}}(t)+\zeta_{1}(t))\cdot\nabla\beta_{\zeta}(t)|\beta_{\zeta}(t)|^{p-2},\zeta(t)\big{)}$
$\displaystyle\quad+\big{(}\boldsymbol{k}*\beta_{\zeta}(t)\cdot\nabla\beta_{\zeta}|\beta_{\zeta}(t)|^{p-2},\beta_{\zeta_{2}}(t)+\zeta_{2}(t)\big{)}$
$\displaystyle\quad+\big{(}\boldsymbol{k}*\zeta(t)\cdot\nabla\beta_{\zeta}(t)(\zeta)|\beta_{\zeta}(t)|^{p-2},\beta_{\zeta_{2}(t)}+\zeta_{2}(t)\big{)}\big{\\}}$
$\displaystyle=I_{1}+I_{2}+I_{3}.$ (3.18)

Let us consider the term $I_{1}$ in the right hand side of the above equality,
and estimate it using (2.19), Hölder’s and Young’s inequalities as

$\displaystyle|I_{1}|$ $\displaystyle\leq
C_{p}\||\beta_{\zeta}|^{\frac{p-2}{2}}\nabla\beta_{\zeta}\|_{\mathrm{L}^{2}}\||\beta_{\zeta}|^{\frac{p-2}{2}}\zeta\|_{\mathrm{L}^{2}}\|\boldsymbol{k}*(\beta_{\zeta_{1}}+\zeta_{1})\|_{\mathbb{L}_{\sigma}^{\infty}}$
$\displaystyle\leq\frac{1}{4}\||\beta_{\zeta}|^{\frac{p-2}{2}}\nabla\beta_{\zeta}\|_{\mathrm{L}^{2}}^{2}+C_{p}\|\beta_{\zeta}\|_{\mathrm{L}^{p}}^{p-2}\|\zeta\|_{\mathrm{L}^{p}}^{2}\|\beta_{\zeta_{1}}+\zeta_{1}\|_{\mathrm{L}^{p}}^{2}$
$\displaystyle\leq\frac{1}{4}\||\beta_{\zeta}|^{\frac{p-2}{2}}\nabla\beta_{\zeta}\|_{\mathrm{L}^{2}}^{2}+C_{p}\|\beta_{\zeta}\|_{\mathrm{L}^{p}}^{p}+C_{p}\|\beta_{\zeta_{1}}+\zeta_{1}\|_{\mathrm{L}^{p}}^{p}\|\zeta\|_{\mathrm{L}^{p}}^{p},$
(3.19)

for $p>2$. Similarly, we can estimate the terms $I_{2}$ and $I_{3}$ from (3.2)
as

$\displaystyle|I_{2}|$
$\displaystyle\leq\frac{1}{4}\||\beta_{\zeta}|^{\frac{p-2}{2}}\nabla\beta_{\zeta}\|_{\mathrm{L}^{2}}^{2}+C_{p}\|\beta_{\zeta_{2}}+\zeta_{2}\|_{\mathrm{L}^{p}}^{2}\|\beta_{\zeta}\|_{\mathrm{L}^{p}}^{p},$
(3.20) $\displaystyle|I_{3}|$
$\displaystyle\leq\frac{1}{4}\||\beta_{\zeta}|^{\frac{p-2}{2}}\nabla\beta_{\zeta}\|_{\mathrm{L}^{2}}^{2}+C_{p}\|\beta_{\zeta}\|_{\mathrm{L}^{p}}^{p}+C_{p}\|\beta_{\zeta_{2}}+\zeta_{2}\|_{\mathrm{L}^{p}}^{p}\|\zeta\|_{\mathrm{L}^{p}}^{p},$
(3.21)

for $p>2$.

Combining (3.16)-(3.21), we obtain

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}+\frac{p(p-1)}{4}\||\beta_{\zeta}(t)|^{\frac{p-2}{2}}\nabla\beta_{\zeta}(t)\|_{\mathrm{L}^{2}}^{2}$
$\displaystyle\leq
C_{p}\big{(}1+\|\beta_{\zeta_{2}}(t)+\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{2}\big{)}\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}$
$\displaystyle\quad+C_{p}\big{(}\|\beta_{\zeta_{1}}(t)+\zeta_{1}(t)\|_{\mathrm{L}^{p}}^{p}+\|\beta_{\zeta_{2}}(t)+\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{p}\big{)}\|\zeta(t)\|_{\mathrm{L}^{p}}^{p}.$
(3.22)

Applying Gronwall’s inequality in (3.2) and using the fact that
$\|\zeta_{1}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq R_{2}$
and $\|\zeta_{2}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))}\leq
R_{2}$, along with (3.13), we obtain

$\displaystyle\|\beta_{\zeta}(t)\|_{\mathrm{L}^{p}}^{p}$
$\displaystyle\leq\bigg{(}C_{p}\int_{0}^{T}\big{(}\|\beta_{\zeta_{1}}(t)+\zeta_{1}(t)\|_{\mathrm{L}^{p}}^{p}+\|\beta_{\zeta_{2}}(t)+\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{p}\big{)}\|\zeta(t)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}t\bigg{)}$
$\displaystyle\qquad\times\exp\bigg{\\{}C_{p}\int_{0}^{T}\big{(}1+\|\beta_{\zeta_{2}}(t)+\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{2}\big{)}\mathrm{d}s\bigg{\\}}$
$\displaystyle\leq
C_{p}T\sup_{t\in[0,T]}\big{\\{}\big{(}\|\beta_{\zeta_{1}}(t)\|_{\mathrm{L}^{p}}^{p}+\|\zeta_{1}(t)\|_{\mathrm{L}^{p}}^{p}+\|\beta_{\zeta_{2}}(t)\|_{\mathrm{L}^{p}}^{p}+\|\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{p}\big{)}\|\zeta(t)\|_{\mathrm{L}^{p}}^{p}\big{\\}}$
$\displaystyle\qquad\times\exp{\bigg{\\{}C_{p}T\bigg{(}1+\sup\limits_{t\in[0,T]}\big{\\{}\|\beta_{\zeta_{2}}(t)\|_{\mathrm{L}^{p}}^{2}+\|\zeta_{2}(t)\|_{\mathrm{L}^{p}}^{2}\big{\\}}\bigg{)}\bigg{\\}}}$
$\displaystyle\leq\big{(}C_{p}(C_{1}+R_{2}^{p})Te^{C_{p}T(1+C_{1}^{2/p}+R_{2}^{2})}\big{)}\sup_{t\in[0,T]}\|\zeta(t)\|_{\mathrm{L}^{p}}^{p},$
(3.23)

where the constant
$C_{1}=(R_{1}^{p}+C_{p}TR_{2}^{2p})e^{C_{p}T(1+R_{2}^{2})}$. ∎

Let us now establish the ULDP for the laws of solutions to the system (1.2).
For any given $T>0$ and the initial data
$\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, define a map
$\mathrm{S}_{\xi_{0}}(\cdot)$ from
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ to
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ such that

$\displaystyle\mathrm{S}_{\xi_{0}}(\zeta)=\beta_{\zeta},\ \ \text{ for all }\
\ \zeta\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})),\text{ for }p>2,$
(3.24)

where $\beta_{\zeta}$ is the solution to the system (3.10). For given
$\phi\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ with $p>2$, we
define

$\displaystyle\mathrm{J}_{\xi_{0}}(\phi):=\inf\big{\\{}\mathrm{I}(\psi):\
\psi\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})),\
\psi+\mathrm{S}_{\xi_{0}}(\psi)=\phi,\ \phi(0)=\xi_{0}\big{\\}},$ (3.25)

where $\mathrm{I}(\cdot)$ is the rate function defined in (3.5).

Let us move to the main result for this subsection, that is, ULDP for the
system (1.2) in $\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for $p>2$.

######  Theorem 3.8.

For any given $T>0$ and the initial data
$\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$. Let $\xi^{\epsilon}$ be
the solution of (1.2), and $\mu^{\epsilon}_{\xi_{0}}$ be the law of
$\xi^{\epsilon}$ in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for $p>2$. Then, the
family of measures $\\{\mu^{\epsilon}_{\xi_{0}}\\}_{\epsilon>0}$ satisfies an
LDP in $\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$ , for $p>2$ with the
rate function $\mathrm{J}_{\xi_{0}}(\cdot)$ defined in (3.25) uniformly with
respect to the initial conditions $\xi_{0}$ belongs to a bounded subsets of
$\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$.

###### Proof.

For the given initial data $\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, with
$p>2$, let $\mathrm{S}_{\xi_{0}}(\cdot)$ and $\mathrm{J}_{\xi_{0}}(\cdot)$ be
the mappings defined in (3.24) and (3.25), respectively. From Lemma 3.7, we
infer that $\zeta+\mathrm{S}_{\xi_{0}}(\zeta)$ is locally Lipschitz continuous
for $\zeta\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, with $p>2$,
uniformly over the initial conditions $\xi_{0}$ lying in a bounded subset in
$\mathrm{L}^{p}(\mathbb{T}^{2})$-norm.

Let $\zeta^{\epsilon}$ be the solution to the system (3.6) and
$\nu^{\epsilon}$ be the law of $\zeta^{\epsilon}$. Then, we have

$\displaystyle\xi^{\epsilon}=\beta_{\zeta^{\epsilon}}^{\epsilon}+\zeta^{\epsilon}=(\mathrm{I}+\mathrm{S}_{\xi_{0}})(\zeta^{\epsilon}).$

Since, $\mu^{\epsilon}_{\xi_{0}}$ is the law of $\xi^{\epsilon}$, we have
$\mu^{\epsilon}_{\xi_{0}}=\nu^{\epsilon}\circ(\mathrm{I}+\mathrm{S}_{\xi_{0}})^{-1}$.
From [16, Theorem 12.16], we know that $\\{\nu^{\epsilon}\\}_{\epsilon>0}$
satisfies the LDP on the space
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for $p>2$, with the rate
function $\mathrm{I}(\cdot)$ defined in (3.5). By the uniform contraction
principle (Proposition 3.4), we obtain that the family of measures
$\\{\mu^{\epsilon}_{\xi_{0}}\\}_{\epsilon>0}$ satisfies the LDP on the space
$\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for $p>2$ uniformly over
the initial conditions $\xi_{0}$ lying in bounded subsets in
$\mathrm{L}^{p}(\mathbb{T}^{2})$-norm, with the rate function defined by

$\displaystyle\mathrm{J}_{\xi_{0}}(\phi)$
$\displaystyle=\inf\big{\\{}\mathrm{I}(\psi):\psi\in(\mathrm{I}+\mathrm{S}_{\xi_{0}})^{-1}(\phi)\big{\\}}$
$\displaystyle=\inf\big{\\{}\mathrm{I}(\psi):\psi\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2})),\psi+\mathrm{S}_{\xi_{0}}(\psi)=\phi\big{\\}},$

and hence the proof completes. ∎

Now, our aim is to prove ULDP for the law of the solutions to the system (1.2)
in $\mathrm{C}([0,T]\times\mathbb{T}^{2})$. For that we only need to prove the
following result in $\mathrm{C}([0,T]\times\mathbb{T}^{2})$.

######  Lemma 3.9.

For any $T>0,\ R_{1}>0$ and $R_{2}>0$, the solution of (3.10) satisfies:

$\displaystyle\|\beta_{\zeta_{1}}-\beta_{\zeta_{2}}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\leq
C\mathopen{}\mathclose{{}\left(T,R_{1},R_{2}}\right)\|\zeta_{1}-\zeta_{2}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})},$
(3.26)

for all $\xi\in\mathrm{C}(\mathbb{T}^{2})$, with
$\|\xi_{0}\|_{\mathrm{C}(\mathbb{T}^{2})}\leq R_{1}$ and
$\zeta_{1},\zeta_{2}\in\mathrm{C}([0,T]\times\mathbb{T}^{2})$ with

$\displaystyle\|\zeta_{1}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\leq R_{2}\
\text{ and }\ \|\zeta_{2}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\leq
R_{2}.$ (3.27)

###### Proof.

From [23, Theorem 1], we know that
$\xi\in\mathrm{C}([0,T]\times\mathbb{T}^{2})$, whenever the initial data
$\xi_{0}\in\mathrm{C}(\mathbb{T}^{2})$ and [23, Lemma 7] implies that $\zeta$
has a continuous modification on $[0,T]\times{\mathbb{T}^{2}}$, and hence
$\beta\in\mathrm{C}([0,T]\times\mathbb{T}^{2})$. Therefore, there exists a
constant $\widetilde{C}>0$ such that

$\displaystyle\|\beta_{\zeta}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\leq\widetilde{C}(R_{1},R_{2},T),\
\mathbb{P}\text{-a.s.}$ (3.28)

Let $\zeta_{1}$ and $\zeta_{2}$ be any two functions in the space
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$ such that (3.27) be satisfied. We
denote the corresponding solutions of (3.2) by $\beta_{\zeta_{i}}$, for
$i=1,2$. Then, for $\beta_{\zeta}=\beta_{\zeta_{1}}-\beta_{\zeta_{2}}$, where
$\zeta=\zeta_{1}-\zeta_{2}$, we have

$\displaystyle|\beta_{\zeta_{1}}(t,x)-\beta_{\zeta_{2}}(t,x)|$
$\displaystyle=\bigg{|}\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\big{(}\boldsymbol{q}((\beta_{\zeta_{1}}+\zeta_{1})(s,\cdot))(y)-\boldsymbol{q}((\beta_{\zeta_{2}}+\zeta_{2})(s,\cdot))(y)\big{)}\mathrm{d}y\mathrm{d}s\bigg{|}$
$\displaystyle\leq\bigg{|}\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot(\boldsymbol{k}*(\beta_{\zeta_{1}}+\zeta_{1})(s,y))(\beta_{\zeta}+\zeta)(s,y)\mathrm{d}y\mathrm{d}s\bigg{|}$
$\displaystyle\quad+\bigg{|}\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot(\boldsymbol{k}*(\beta_{\zeta}+\zeta)(s,y))(\beta_{\zeta_{2}}+\zeta_{2})\mathrm{d}y\mathrm{d}s\bigg{|}$
$\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{|}\nabla_{y}G(t-s,x,y)\big{|}\big{|}(\beta_{\zeta_{1}}+\zeta_{1})(s,y)\big{|}\big{|}(\beta_{\zeta}+\zeta)(s,y)\big{|}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+C\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{|}\nabla_{y}G(t-s,x,y)\big{|}\big{|}(\beta_{\zeta}+\zeta)(s,y)\big{|}\big{|}(\beta_{\zeta_{2}}+\zeta_{2})(s,y)\big{|}\mathrm{d}y\mathrm{d}s$
$\displaystyle\leq
C\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{|}\nabla_{y}G(t-s,x,y)\big{|}\big{\\{}\big{|}(\beta_{\zeta_{1}}+\zeta_{1})(s,y)\big{|}+\big{|}(\beta_{\zeta_{2}}+\zeta_{2})(s,y)\big{|}\big{\\}}\big{|}\beta_{\zeta}(s,y)\big{|}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+C\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{|}\nabla_{y}G(t-s,x,y)\big{|}\big{\\{}\big{|}(\beta_{\zeta_{1}}+\zeta_{1})(s,y)\big{|}+(\beta_{\zeta_{2}}+\zeta_{2})(s,y)\big{|}\big{|}\zeta(s,y)\big{|}\mathrm{d}y\mathrm{d}s.$

Applying Gronwall’s inequality in above inequality, and using (2.9) and
(3.28), we obtain

$\displaystyle\sup_{(t,x)\in[0,T]\times\mathbb{T}^{2}}|\beta_{\zeta_{1}}(t,x)-\beta_{\zeta_{2}}(t,x)|$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left(T^{\frac{1}{2}}\big{\\{}\|\beta_{\zeta_{1}}+\zeta_{1}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}+\|\beta_{\zeta_{2}}+\zeta_{2}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\big{\\}}\|\zeta\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}}\right)$
$\displaystyle\qquad\times\exp\mathopen{}\mathclose{{}\left\\{T^{\frac{1}{2}}\big{\\{}\|\beta_{\zeta_{1}}+\zeta_{1}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}+\|\beta_{\zeta_{2}}+\zeta_{2}\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}\big{\\}}}\right\\}$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left(T^{\frac{1}{2}}\big{(}\widetilde{C}+R_{2}\big{)}\|\zeta\|_{\mathrm{C}([0,T]\times\mathbb{T}^{2})}}\right)\times
e^{2T^{\frac{1}{2}}(\widetilde{C}+R_{2})},$ (3.29)

and (3.26) follows. ∎

Let us now establish the ULDP for the laws of solutions to the system (1.2).
For any given $T>0$ and the initial data
$\xi_{0}\in\mathrm{C}(\mathbb{T}^{2})$, define a map
$\mathrm{S}_{\xi_{0}}(\cdot)$ from $\mathrm{C}([0,T]\times\mathbb{T}^{2})$ to
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$ such that

$\displaystyle\mathrm{S}_{\xi_{0}}(\zeta)=\beta_{\zeta},\ \ \text{ for all }\
\ \zeta\in\mathrm{C}([0,T]\times\mathbb{T}^{2}),$ (3.30)

where $\beta_{\zeta}$ is the solution to the system (3.2). For given
$\phi\in\mathrm{C}([0,T]\times\mathbb{T}^{2})$, we define

$\displaystyle\mathrm{J}_{\xi_{0}}(\phi):=\inf\big{\\{}\mathrm{I}(\psi):\
\psi\in\mathrm{C}([0,T]\times\mathbb{T}^{2}),\
\psi+\mathrm{S}_{\xi_{0}}(\psi)=\phi,\ \phi(0)=\xi_{0}\big{\\}},$ (3.31)

where $\mathrm{I}(\cdot)$ is the rate function defined in (3.5).

######  Theorem 3.10.

For any given $T>0$ and the initial data
$\xi_{0}\in\mathrm{C}(\mathbb{T}^{2})$. Let $\xi^{\epsilon}$ be the solution
of (1.2), and $\mu^{\epsilon}_{\xi_{0}}$ be the law of $\xi^{\epsilon}$ in the
space $\mathrm{C}([0,T]\times\mathbb{T}^{2})$. Then, the family of measures
$\\{\mu^{\epsilon}_{\xi_{0}}\\}_{\epsilon>0}$ satisfies an LDP in
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$ with the rate function
$\mathrm{J}_{\xi_{0}}(\cdot)$ defined in (3.31) uniformly with respect to the
initial conditions $\xi_{0}$ belongs to bounded subsets of
$\mathrm{C}(\mathbb{T}^{2})$.

###### Proof.

Since the proof of this theorem is based on Theorem 3.8 with (3.30) and (3.31)
as substitutes for (3.24) and (3.25), respectively, we are not supplying any
additional proof. ∎

##  4\. Finite dimensional noise

In this section, we consider the system (1.4) and establish the well-posedness
results. If $\xi^{\epsilon}$ is a solution of the integral equation (1.1.2) in
the sense of Walsh ([51]), then by using [29, Proposition 3.7], one can prove
that $\xi^{\epsilon}$ has a continuous modification:

$\displaystyle\int_{\mathbb{T}^{2}}\xi^{\epsilon}(t,y)\phi(y)\mathrm{d}y$
$\displaystyle=\int_{\mathbb{T}^{2}}\xi_{0}(y)\phi(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\xi^{\epsilon}(t,y)\Delta\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad-\int_{0}^{t}\int_{\mathbb{T}^{2}}\boldsymbol{q}(\xi^{\epsilon}(s,\cdot))(y)\cdot\nabla\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}\sigma_{j}(s,y,\xi^{\epsilon}(s,y))\phi(y)\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s),\
\ \mathbb{P}\text{-a.s.},$ (4.1)

for every test function $\phi\in\mathrm{H}^{b}$ with $b>2$, and all time
$t\in[0,T]$.

Let us introduce the assumptions on the noise coefficient $\sigma_{j},$ for
$j=1,\ldots,n$:

######  Hypothesis 4.1.

The function
$\sigma_{j}:[0,T]\times\mathbb{R}^{2}\times\mathbb{R}\to\mathbb{R}$ is a
measurable function, satisfying the following conditions:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} |\sigma_{j}(t,x,r)|&\leq
K(1+|r|),\\\ |\sigma_{j}(t,x,r)-\sigma_{j}(t,x,s)|&\leq
L|r-s|,\end{aligned}}\right.$ (4.2)

for $j=1,\ldots,n$, for all $t\in[0,T]$, $x\in{\mathbb{T}^{2}}$ and
$r,s\in\mathbb{R}$, where $K,L$ are some positive constants.

Under the above assumption, we state our main result of this section.

######  Theorem 4.2.

Let $\xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$, for $p>2$ be given and
Hypothesis 4.1 hold. Then, there exists a unique $\mathrm{L}^{p}$-valued
$\mathscr{F}_{t}$-adapted continuous process $\xi^{\epsilon}$ of the integral
equation (1.1.2). Moreover, if the initial data
$\xi_{0}\in\mathrm{C}({\mathbb{T}^{2}})$, then the solution $\xi^{\epsilon}$
admits a modification which is a space-time continuous process.

###  4.1. Existence and uniqueness results

In this subsection, we prove Theorem 4.2 by establishing several auxiliary
results. In order to establish a solution to the system (1.4), we use a
truncation method. Let $\Pi_{R}(\cdot)$ be a $\mathrm{C}^{1}(\mathbb{R})$
function such that

$\Pi_{R}(r)=\mathopen{}\mathclose{{}\left\\{\begin{aligned} &1,\ \ \text{ if
}\ \ |r|\leq R,\\\ &0,\ \ \text{ if }\ \ |r|\geq R+1,\end{aligned}}\right.$
(4.3)

and $|\Pi_{R}^{\prime}(r)|\leq 2$ for all $r\in\mathbb{R}$. Also, by the mean
value theorem we have $|\Pi_{R}(r_{1})-\Pi_{R}(r_{2})|\leq 2|r_{1}-r_{2}|$,
for all $r_{1},r_{2}\in\mathbb{R}$. Let us introduce the truncated system for
$\xi_{R}^{\epsilon}(t,x)=\nabla^{\perp}\cdot\boldsymbol{u}_{R}^{\epsilon}(t,x),\
(t,x)\in[0,T]\times{\mathbb{T}^{2}}$ as

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{\partial\xi_{R}^{\epsilon}}{\partial
t}(t,x)&+\boldsymbol{u}_{R}^{\epsilon}(t,x)\cdot\nabla\xi_{R}^{\epsilon}(t,x)\Pi_{R}(\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}})-\Delta\xi_{R}^{\epsilon}(t,x)\\\
&=\sqrt{\epsilon}\sum_{j=1}^{n}\sigma_{j,R}(t,x,\xi_{R}^{\epsilon}(t,x))\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{W}^{j}(t),\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\
\nabla\cdot\boldsymbol{u}_{R}^{\epsilon}(t,x)&=0,\
(t,x)\in[0,T]\times{\mathbb{T}^{2}},\\\ \xi_{R}^{\epsilon}(0,x)&=\xi_{0}(x),\
x\in{\mathbb{T}^{2}}.\end{aligned}}\right.$ (4.4)

We write the mild formulation of the truncated system (4.4) in sense of Walsh
(see [51]) as

$\displaystyle\xi_{R}^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}_{R}(\xi_{R}^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,R}(t,y,\xi_{R}^{\epsilon}(t,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(t),\
\ \mathbb{P}\text{-a.s.,}$ (4.5)

for all $t\in[0,T]$, where
$\boldsymbol{q}_{R}(\xi^{\epsilon}(\cdot))(\cdot)=\Pi_{R}(\|\xi^{\epsilon}(\cdot)\|_{\mathrm{L}^{p}})\boldsymbol{q}(\xi^{\epsilon}(\cdot))(\cdot)$
and
$\sigma_{j,R}(\cdot,\cdot,\xi^{\epsilon}(\cdot,\cdot))=\Pi_{R}(\|\xi^{\epsilon}(\cdot)\|_{\mathrm{L}^{p}})\sigma_{j}(\cdot,\cdot,\xi^{\epsilon}(\cdot,\cdot)).$

Set

$\displaystyle\mathcal{A}_{1}\xi_{R}^{\epsilon}(t,x)$
$\displaystyle:=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y,$
$\displaystyle\mathcal{A}_{2}\xi_{R}^{\epsilon}(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}_{R}(\xi_{R}^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s,$
$\displaystyle\mathcal{A}_{3}\xi_{R}^{\epsilon}(t,x)$
$\displaystyle:=\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,R}(s,y,\xi_{R}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s),$
$\displaystyle\mathcal{A}\xi_{R}^{\epsilon}(t,x)$
$\displaystyle:=\mathcal{A}_{1}\xi_{R}^{\epsilon}(t,x)+\mathcal{A}_{2}\xi_{R}^{\epsilon}(t,x)+\sqrt{\epsilon}\mathcal{A}_{3}\xi_{R}^{\epsilon}(t,x).$
(4.6)

Let us define a Banach space $\mathscr{H}$ consisting of the adapted processes
$u:\Omega\times[0,T]\to\mathrm{L}^{p}(\mathbb{T}^{2})$ such that

$\displaystyle\|u\|_{\mathscr{H}}^{p}:=\sup_{t\in[0,T]}e^{-\lambda
pt}\mathbb{E}\big{[}\|u(t)\|_{\mathrm{L}^{p}}^{p}\big{]},$

where $\lambda>0$ is defined in (4.16) below.

For $\xi^{\epsilon}\in\mathrm{L}^{p}(\mathbb{T}^{2})$ for $p>2$, we obtain
that the function $\boldsymbol{q}_{R}$ is defined from
$\mathrm{L}^{p}(\mathbb{T}^{2})$ to $\mathbb{L}_{\sigma}^{p}(\mathbb{T}^{2})$
for any $p>2$. Let us recall a result from [23].

######  Lemma 4.3 ([23, Lemma 9]).

Fix $R\geq 1$ and $p>2$. Then, there exist positive constants $C_{p}$ and
$C_{R,p}$ such that the following hold:

$\displaystyle\|\boldsymbol{q}_{R}(\xi)\|_{\mathbb{L}_{\sigma}^{p}}$
$\displaystyle\leq C_{p}(R+1)^{2},$ (4.7)
$\displaystyle\|\boldsymbol{q}_{R}(\xi)-\boldsymbol{q}_{R}(\tilde{\xi})\|_{\mathbb{L}_{\sigma}^{p}}$
$\displaystyle\leq C_{R,p}\|\xi-\tilde{\xi}\|_{\mathrm{L}^{p}},$ (4.8)

for all $\xi,\tilde{\xi}\in\mathrm{L}^{p}(\mathbb{T}^{2})$.

######  Proposition 4.4.

Let us assume that $\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, for $p>2$, and
$R\geq 1$. Then, there exists a unique $\mathrm{L}^{p}$-valued
$\mathscr{F}_{t}$-adapted continuous process $\xi_{R}^{\epsilon}$ satisfying
the truncated integral equation (4.1) such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C(R,T).$ (4.9)

###### Proof.

The proof of this proposition is based on a fixed point argument. The proof is
divided into two parts. In the first part, we establish that the operator
$\mathcal{A}$ is well-defined in the Banach space $\mathscr{H}$. In the second
part, we show that $\mathcal{A}$ is a contraction map and then the existence
and uniqueness of truncated integral equation (4.1) follows from fixed point
arguments.

Claim 1. The operator $\mathcal{A}$ is well-defined on the Banach space
$\mathscr{H}$.

From (4.1), we have

$\displaystyle\|\mathcal{A}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\leq
C_{p}\mathopen{}\mathclose{{}\left(\|\mathcal{A}_{1}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}+\|\mathcal{A}_{2}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}+\|\mathcal{A}_{3}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}}\right).$
(4.10)

Using (2.10) and Young’s inequality, we obtain

$\displaystyle\|\mathcal{A}_{1}\xi_{R}^{\epsilon}\|_{\mathscr{H}}$
$\displaystyle=\sup_{t\in[0,T]}e^{-\lambda
t}\|\mathcal{A}_{1}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}$
$\displaystyle\leq\sup_{t\in[0,T]}e^{-\lambda
t}\big{\\{}\|G(t,0,\cdot)\|_{\mathrm{L}^{1}}\|\xi_{0}\|_{\mathrm{L}^{p}}\big{\\}}<\infty.$
(4.11)

Applying (1) for $\beta=1$ and $\alpha=p$, we obtain

$\displaystyle\|\mathcal{A}_{2}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\leq\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\boldsymbol{q}_{R}(\xi_{R}^{\epsilon}(s,\cdot))\|_{\mathbb{L}_{\sigma}^{p}}\mathrm{d}s\leq
C_{p}(R+1)^{2}t^{\frac{1}{2}},$

which implies

$\displaystyle\|\mathcal{A}_{2}\xi_{R}^{\epsilon}\|_{\mathscr{H}}^{p}=\sup_{t\in[0,T]}e^{-\lambda
t}\|\mathcal{A}_{2}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}<\infty.$
(4.12)

Now, we consider the term
$\mathbb{E}\big{[}\|\mathcal{A}_{3}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\big{]}$,
and estimate it using Hypothesis 4.1, Fubini’s theorem, Burkholder-Davis-
Gundy’s (BDG) (see [35, Theorem 1.1.7]), [29, Proposition 3.5], Hölder’s,
Minkowski’s and Young’s inequalities as

$\displaystyle\mathbb{E}\bigg{[}\|\mathcal{A}_{3}\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle=\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\int_{\mathbb{T}^{2}}\bigg{|}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,R}(s,x,\xi_{R}^{\epsilon}(t,x))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)\bigg{|}^{p}\mathrm{d}x\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\int_{\mathbb{T}^{2}}\bigg{|}\sum_{j=1}^{n}\int_{0}^{t}\bigg{|}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,R}(s,y,\xi_{R}^{\epsilon}(s,y))\mathrm{d}y\bigg{|}^{2}\mathrm{d}s\bigg{|}^{\frac{p}{2}}\mathrm{d}x\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\bigg{|}\int_{\mathbb{T}^{2}}\bigg{|}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,R}(s,,y,\xi_{R}^{\epsilon}(s,y))\mathrm{d}y\bigg{|}^{p}\mathrm{d}x\bigg{|}^{\frac{2}{p}}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\big{\|}G(t-s,\cdot,\cdot)*\sigma_{j,R}(s,\cdot,\xi_{R}^{\epsilon}(s,\cdot))\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\big{\|}G(t-s,\cdot,\cdot)\|_{\mathrm{L}^{1}}^{2}\|\sigma_{j,R}(s,,\cdot,\xi_{R}^{\epsilon}(s,\cdot))\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\|\sigma_{j,R}(s,,\cdot,\xi_{R}^{\epsilon}(s,\cdot))\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}(tn)^{\frac{p}{2}-1}\int_{0}^{t}\mathbb{E}\mathopen{}\mathclose{{}\left[\big{(}1+\Pi_{R}^{p}(\|\xi^{\epsilon}_{R}(s)\|_{\mathrm{L}^{p}})\|\xi_{R}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p})}\right]\mathrm{d}s$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}K^{\frac{p}{2}-1}T^{\frac{p}{2}}\big{(}1+(R+1)^{p}\big{)}.$
(4.13)

Combining (4.10)-(4.1), the claim follows.

Claim 2. $\mathcal{A}$ is a contraction. Let
$\xi_{R,1}^{\epsilon},\xi_{R,2}^{\epsilon}\in\mathscr{H}$. Without loss of
generality, we can assume that
$\|\xi_{R,1}^{\epsilon}\|_{\mathrm{L}^{p}}\leq\|\xi_{R,2}^{\epsilon}\|_{\mathrm{L}^{p}}$.
To end this, we need to establish that $\mathcal{A}$ is a contraction on the
Banach space $\mathscr{H}$. Let us start with the term
$\|\mathcal{A}_{2}\xi_{R,1}^{\epsilon}(\cdot)-\mathcal{A}_{2}\xi_{R,2}^{\epsilon}(\cdot)\|_{\mathrm{L}^{p}}$
and estimate it using (1) with $\alpha=p,\ \beta=1$ to obtain

$\displaystyle\|\mathcal{A}_{2}\xi_{R,1}^{\epsilon}(t)-\mathcal{A}_{2}\xi_{R,2}^{\epsilon}(t)\|_{\mathrm{L}^{p}}$
$\displaystyle=\bigg{\|}\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\big{(}\boldsymbol{q}_{R}(\xi_{R,1}^{\epsilon}(s,\cdot)-\boldsymbol{q}_{R}(\xi_{R,2}^{\epsilon}(s,\cdot))\big{)}(y)\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\boldsymbol{q}_{R}(\xi_{R,1}^{\epsilon}(s,\cdot)-\boldsymbol{q}_{R}(\xi_{R,2}^{\epsilon}(s,\cdot))\|_{\mathbb{L}_{\sigma}^{p}}\mathrm{d}s,$

for all $t\in[0,T]$. From the above inequality and Lemma 4.3 (see (4.8)), we
obtain for all $t\in[0,T]$

$\displaystyle\|\mathcal{A}_{2}\xi_{R,1}^{\epsilon}-\mathcal{A}_{2}\xi_{R,2}^{\epsilon}\|_{\mathscr{H}}^{p}$
$\displaystyle\leq C_{R,p}\sup_{t\in[0,T]}e^{-\lambda
pt}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s\bigg{)}^{p}\bigg{]}$
$\displaystyle\leq
C_{R,p}T^{\frac{p-1}{2p}}\sup_{t\in[0,T]}\bigg{(}\int_{0}^{t}(t-s)^{-\frac{1}{2}}e^{-\lambda(t-s)}e^{-\lambda
s}\mathbb{E}\big{[}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\mathrm{d}s\bigg{)}$
$\displaystyle\leq
C_{R,p}T^{\frac{p-1}{2p}}\mathopen{}\mathclose{{}\left(\frac{\Gamma(\frac{1}{2})}{\lambda^{\frac{1}{2}}}}\right)\|\xi_{R,1}^{\epsilon}-\xi_{R,2}^{\epsilon}\|_{\mathscr{H}}^{p}=C_{R,p}T^{\frac{p-1}{2p}}\mathopen{}\mathclose{{}\left(\frac{\sqrt{\pi}}{\lambda^{\frac{1}{2}}}}\right)\|\xi_{R,1}^{\epsilon}-\xi_{R,2}^{\epsilon}\|_{\mathscr{H}}^{p},$
(4.14)

for $p>2$, where $\Gamma(\cdot)$ denotes the Gamma function.

Using similar arguments as in (4.1), Hypothesis 4.1 (Lipschitz continuity of
$\sigma_{j}$) and Minkowski’s inequality, we find

$\displaystyle\mathbb{E}\big{[}\|\mathcal{A}_{3}\xi_{R,1}^{\epsilon}(t)-\mathcal{A}_{3}\xi_{R,2}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\big{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\big{\|}G(t-s,\cdot,\cdot)*(\sigma_{j,R}(s,\cdot,\xi_{R,1}^{\epsilon}(s,\cdot))-\sigma_{j,R}(s,\cdot,\xi_{R,2}^{\epsilon}(s,\cdot)))\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\big{\|}G(t-s,\cdot,\cdot)\|_{\mathrm{L}^{1}}^{2}\|\sigma_{j,R}(s,\cdot,\xi_{R,1}^{\epsilon}(s,\cdot))-\sigma_{j,R}(s,\cdot,\xi_{R,2}^{\epsilon}(s,\cdot)\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\sum_{j=1}^{n}\int_{0}^{t}\|\sigma_{j,R}(s,\cdot,\xi_{R,1}^{\epsilon}(s,\cdot))-\sigma_{j,R}(s,\cdot,\xi_{R,2}^{\epsilon}(s,\cdot)\big{\|}_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C_{\epsilon,n,p,L}\mathbb{E}\bigg{[}\int_{0}^{t}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}$
$\displaystyle\leq
C_{\epsilon,n,p,L}\mathopen{}\mathclose{{}\left(\bigg{\\{}\mathbb{E}\bigg{[}\int_{0}^{t}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{]}^{\frac{p}{2}}\bigg{\\}}^{\frac{2}{p}}}\right)^{\frac{p}{2}}$
$\displaystyle\leq
C_{\epsilon,n,p,L}\mathopen{}\mathclose{{}\left(\int_{0}^{t}\bigg{\\{}\mathbb{E}\big{[}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}}^{\frac{2}{p}}\mathrm{d}s}\right)^{\frac{p}{2}}$
$\displaystyle\leq C_{\epsilon,n,p,L}\bigg{(}\frac{1}{2\lambda}(e^{2\lambda
t}-1)\bigg{)}^{\frac{p}{2}}\sup_{s\in[0,t]}\bigg{\\{}e^{-\lambda
ps}\mathbb{E}\big{[}\|\xi_{R,1}^{\epsilon}(s)-\xi_{R,2}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}},$

for $p\geq 2$. From the above inequality, we obtain

$\displaystyle\|\mathcal{A}_{3}\xi_{R,1}^{\epsilon}-\mathcal{A}_{3}\xi_{R,2}^{\epsilon}\|_{\mathscr{H}}^{p}\leq
C_{\epsilon,n,p,L}\bigg{(}\frac{1}{2\lambda}(1-e^{-2\lambda
t})\bigg{)}^{\frac{p}{2}}\|\xi_{R,1}^{\epsilon}-\xi_{R,2}^{\epsilon}\|_{\mathscr{H}}^{p}.$
(4.15)

Combining (4.1)-(4.15), and choosing $\lambda>0$ sufficiently large so that

$\displaystyle
C_{R,p}T^{\frac{p-1}{2p}}\mathopen{}\mathclose{{}\left(\frac{\sqrt{\pi}}{\lambda^{\frac{1}{2}}}}\right)+C_{\epsilon,n,p,L}\bigg{(}\frac{1}{2\lambda}(1-e^{-2\lambda
T})\bigg{)}^{\frac{p}{2}}<1,$ (4.16)

for $p>2$. For the above value of $\lambda$, the mapping $\mathcal{A}$ is a
contraction on the Banach space $\mathscr{H}$.

Therefore an application of the Banach fixed point theorem yields the
existence of a unique fixed point for the operator $\mathcal{A}$ and this
gives the existence of a unique solution of the truncated integral equation
and the estimate (4.9) holds from Claim 1. ∎

By Proposition 4.4, we obtain the existence of unique local solution of the
integral equation (1.1.2), that is, up to a stopping time
$\tau_{R}:=\inf\\{t\geq 0:\|\xi^{\epsilon}(t)\|_{\mathrm{L}^{p}}\geq
R\\}\wedge T$. To prove that the solution we achieved is a global one, we must
demonstrate that a uniform bound holds. Consider the following integral
equation:

$\displaystyle\xi_{m}^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0,m}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}_{m}(\xi_{m}^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)$
$\displaystyle=:I_{1,m}+I_{2,m}+I_{3,m},$ (4.17)

where we have taken the sequence of bounded Borel measurable functions
$\boldsymbol{q}_{m}(t,x,r)=\boldsymbol{q}(t,x,r)$ and
$\sigma_{j,m}(t,x,r)=\sigma_{j}(t,x,r)$, for $|r|\leq m$, and
$\boldsymbol{q}_{m}(t,x,r)=\sigma_{j,m}(t,x,r)=0$, for $|r|\geq m+1$ such that
they are globally Lipschitz in $r\in\mathbb{R}$ with constants independents of
$m$.

The following result provides a uniform energy estimate for
$\xi_{m}^{\epsilon}$.

######  Lemma 4.5.

For each $m\in\mathbb{N}$, $p>2$, there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{m}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}$
$\displaystyle+2p(p-1)\int_{0}^{T}\||\xi_{m}^{\epsilon}(s)|^{\frac{p-2}{2}}\nabla\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{]}$
$\displaystyle\leq\big{(}2\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}+C_{\epsilon,p,T}\big{)}e^{C_{\epsilon,p,T}}.$
(4.18)

###### Proof.

Applying Itô’s formula to the function $|\cdot|^{p}$ for the process
$\xi_{m}^{\epsilon}(\cdot,x)$ for $x\in{\mathbb{T}^{2}}$ and then taking
integration over the spatial domain ${\mathbb{T}^{2}}$ (cf. [25]), we deduce
for all $t\in[0,T]$

$\displaystyle\|\xi_{m}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}$
$\displaystyle=\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}-p(p-1)\int_{0}^{t}\||\xi_{m}^{\epsilon}(s)|^{\frac{p-2}{2}}\nabla\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s$
$\displaystyle\quad+p(p-1)\int_{0}^{t}\big{(}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\nabla\xi_{m}^{\epsilon}(s,y),\boldsymbol{q}_{m}(\xi_{m}^{\epsilon}(s,y))\big{)}\mathrm{d}s$
$\displaystyle\quad+p\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\xi_{m}^{\epsilon}(s,y)\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)$
$\displaystyle\quad+\frac{\epsilon}{2}p(p-1)\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\sigma_{j,m}^{2}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle=:\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}-p(p-1)I_{1}+p(p-1)I_{2}+I_{3}+I_{4}.$
(4.19)

Consider the term $I_{2}$ and using
$\boldsymbol{q}(\xi_{m}^{\epsilon})=\xi^{\epsilon}_{m}(\boldsymbol{k}*\xi_{m}^{\epsilon})$,
$\boldsymbol{u}_{m}^{\epsilon}=\boldsymbol{k}*\xi_{m}^{\epsilon}$ and [1,
Lemma 2.2] as

$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{t}\big{(}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\xi_{m}^{\epsilon}(s,y)\nabla\xi_{m}^{\epsilon}(s,y),\boldsymbol{u}_{m}^{\epsilon}(s,y)\big{)}\mathrm{d}s=0,$
(4.20)

since $\boldsymbol{u}_{m}^{\epsilon}$ is a divergence free velocity field.

Now, we consider the term
$\mathbb{E}\bigg{[}\sup\limits_{t\in[0,T]}|I_{3}|\bigg{]}$ and estimate it
using BDG (see [35, Theorem 1.1.7]), Hölder’s and Young’s inequalities as

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}|I_{3}|\bigg{]}$
$\displaystyle=p\sqrt{\epsilon}\mathbb{E}\bigg{[}\sup_{s\in[0,T]}\bigg{|}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\xi_{m}^{\epsilon}(s,y)$
$\displaystyle\qquad\times\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)\bigg{|}\bigg{]}$
$\displaystyle\leq
C_{\epsilon,p}\mathbb{E}\bigg{[}\int_{0}^{T}\bigg{(}\int_{\mathbb{T}^{2}}|\xi_{m}^{\epsilon}(s,y)|^{p-2}\xi_{m}^{\epsilon}(s,y)\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\bigg{)}^{2}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq
C_{\epsilon,p}\mathbb{E}\bigg{[}\int_{0}^{T}\bigg{(}\int_{\mathbb{T}^{2}}\big{(}1+|\xi_{m}^{\epsilon}(s,y)|^{p}\big{)}\mathrm{d}y\bigg{)}^{2}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq
C_{\epsilon,p}\mathbb{E}\bigg{[}T+\int_{0}^{T}\|\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{2p}\mathrm{d}s\bigg{]}^{\frac{1}{2}}$
$\displaystyle\leq\frac{1}{2}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{m}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}+C_{\epsilon,p}\int_{0}^{T}\mathbb{E}\big{[}\|\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\mathrm{d}s+C_{\epsilon,p,T}.$
(4.21)

Using Hypothesis 4.1, we estimate the term $I_{4}$ as

$\displaystyle I_{4}\leq
C_{\epsilon,p}\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{(}1+|\xi_{m}^{\epsilon}(s,y)|^{p}\big{)}\mathrm{d}y\mathrm{d}s\leq
C_{\epsilon,p}\int_{0}^{t}\|\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s+C_{\epsilon,p,T}.$
(4.22)

Combining (4.1)-(4.22), we find

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{m}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle+2p(p-1)\mathbb{E}\bigg{[}\int_{0}^{T}\||\xi_{m}^{\epsilon}(s)|^{\frac{p-2}{2}}\nabla\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{]}$
$\displaystyle\leq
2\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}+C_{\epsilon,p}\int_{0}^{t}\mathbb{E}\big{[}\|\xi_{m}^{\epsilon}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\mathrm{d}s+C_{\epsilon,p,T}.$
(4.23)

Applying Gronwall’s inequality in (4.1), we obtain the required uniform energy
estimate (4.5). ∎

Let us go back to the integral equation (1.1.2), and establish the existence
and uniqueness result stated in Theorem 4.2.

###### Proof of Theorem 4.2.

Uniqueness. We employ a stopping time argument to prove this. Let us assume
that $\xi_{1}^{\epsilon}$ and $\xi_{2}^{\epsilon}$ be any two solutions of the
integral equation (1.1.2). Then, $\xi_{1}^{\epsilon}$ and $\xi_{2}^{\epsilon}$
are the solutions of the truncated integral equation (4.1). Let us define the
following stopping times

$\displaystyle\tau_{R}^{1}:=\inf\\{t\geq
0:\|\xi_{1}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\geq R\\}\wedge T\ \ \text{ and }\
\ \tau_{R}^{2}:=\inf\\{t\geq 0:\|\xi_{2}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\geq
R\\}\wedge T.$

By Proposition 4.4, we obtain $\xi_{1}^{\epsilon}(t)=\xi_{2}^{\epsilon}(t),\
t\in[0,\tau_{R}^{1}\wedge\tau_{R}^{2}]$. By the uniform energy estimate (4.5)
for $\xi_{i}^{\epsilon}$, $i=1,2$, and an application of Markov’s inequality
yields (see Remark 4.6 below)

$\displaystyle\mathbb{P}\big{(}\tau_{R}^{1}\wedge\tau_{R}^{2}<T\big{)}\to 0,\
\ \text{ as }\ \ R\to\infty$

and the uniqueness follows.

Existence. Let us now show the existence of the solution $\xi^{\epsilon}$ to
the integral equation (1.1.2) in the interval $[0,T]$. The proof of this part
is based on Skorokhod’s representation theorem (see [21, Theorem 3]) and [26,
Lemma 1.1]. By Lemma 2.2, we obtain that the sequence of processes

$\displaystyle I_{2,m}$
$\displaystyle:=\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}_{m}(\xi_{m}^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s,\
\ t\in[0,T]$ (4.24)

is tight in $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for $p>2$,
uniformly in $m$. By [29, Corollary 3.6], the sequence of processes

$\displaystyle I_{3,m}$
$\displaystyle:=\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s),\
\ t\in[0,T]$ (4.25)

is also tight in $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for
$p>2$. Note that the sequence

$\displaystyle I_{1,m}$
$\displaystyle:=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0,m}(y)\mathrm{d}y,\ \
t\in[0,T]$ (4.26)

is tight in $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$. Therefore,
one can conclude that $\xi_{m}^{\epsilon}(t)=\sum\limits_{i=1}^{3}I_{i,m},\
t\in[0,T]$ is tight in $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$,
for $p>2$, uniformly in $m$.

By Skorokhod’s representation theorem, for a given pair of subsequences
$\\{\xi_{m}^{\epsilon}\\}$ and $\\{\xi_{l}^{\epsilon}\\}$, there exist
subsequences $m_{i}$ and $l_{i}$ (denoted by $\xi_{m_{i}}^{\epsilon}$, and
$\xi_{l_{i}}^{\epsilon}$) and a sequence of random variables
$z_{i}^{\epsilon}:=(\widetilde{\xi}_{i}^{\epsilon},\bar{\xi}_{i}^{\epsilon},\bar{\mathrm{W}}^{j}_{i})$,
$i=1,2,\ldots,$ in
$\mathcal{I}=\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))\times\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))\times\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$,
for $p>2$ in some probability space
$(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}\\}_{t\geq
0},\bar{\mathbb{P}})$ such that the sequence $z_{i}$ converges to
$z^{\epsilon}=(\widetilde{\xi}^{\epsilon},\bar{\xi}^{\epsilon},\bar{\mathrm{W}}^{j}),\
\bar{\mathbb{P}}$-a.s., in $\mathcal{I}$ as $i\to\infty$ and the laws of
$z^{\epsilon}_{i}$ and
$(\xi_{m_{i}}^{\epsilon},\xi_{l_{i}}^{\epsilon},\mathrm{W}^{j})$ coincide.
Here, the two random fields $\bar{\mathrm{W}}^{j}$ and
$\bar{\mathrm{W}}^{j}_{i}$ are Wiener process defined on different stochastic
basis $(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}\\}_{t\geq
0},\bar{\mathbb{P}})$ and
$(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}^{i}\\}_{t\geq
0},\bar{\mathbb{P}})$, respectively, where $\bar{\mathscr{F}}_{t}$ and
$\bar{\mathscr{F}}_{t}^{i}$ are the completion of the $\sigma$-fields
generated by $z^{\epsilon}(t,x)$ and $z_{i}^{\epsilon}(t,x)$ for all
$(t,x)\in[0,T]\times{\mathbb{T}^{2}}$, respectively. Using [26, Lemma 1.1],
our aim is to show that $z_{i}^{\epsilon}$ converges to $z^{\epsilon}$ weakly.
From [29, Proposition 3.7], we have the equivalency between weak and mild
solutions. Letting $i\to\infty$, in the weak formulation (4) with
$\widetilde{\xi}_{i}^{\epsilon}$ in place of $\xi^{\epsilon}$, for every
smooth function $\phi\in\mathrm{H}^{b}$ with $b>2$, we have

$\displaystyle\int_{\mathbb{T}^{2}}\widetilde{\xi}^{\epsilon}(t,y)\phi(y)\mathrm{d}y$
$\displaystyle=\int_{\mathbb{T}^{2}}\xi_{0}(y)\phi(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\widetilde{\xi}^{\epsilon}(t,y)\Delta\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad-\int_{0}^{t}\int_{\mathbb{T}^{2}}\boldsymbol{q}(\widetilde{\xi}^{\epsilon}(s,\cdot))(y)\cdot\nabla\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}\sigma_{j}(s,y,\widetilde{\xi}^{\epsilon}(s,y))\phi(y)\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s),\
\ \mathbb{P}\text{-a.s.},$ (4.27)

for all $t\in[0,T]$ on the probability space
$(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}^{i}\\}_{t\geq
0},\bar{\mathbb{P}})$. Similarly, (4.1) hols for $\bar{\xi}^{\epsilon}$ also.
Then by the uniqueness of the solution we deduce that
$\widetilde{\xi}^{\epsilon}=\bar{\xi}^{\epsilon}$. Using [26, Lemma 1.1], we
conclude that $\xi^{\epsilon}_{m}$ converges in
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for $p>2$, in
probability to some random element
$\xi^{\epsilon}\in\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for
$p>2$.

Let us now assume that the initial data
$\xi_{0}\in\mathrm{C}({\mathbb{T}^{2}})$. Then our solution $\xi^{\epsilon}$
of the integral equation (1.1.2) is the sum of three terms. The first term
$\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y$ is continuous, using the
properties of heat kernel $G$ (see Lemma 2.1(ii)). We know that
$\xi_{0}\in\mathrm{L}^{p}(\mathbb{T}^{2})$, for any $p\in[1,\infty]$, since
$\xi_{0}\in\mathrm{C}({\mathbb{T}^{2}})$. Therefore,
$\xi\in\mathrm{C}([0,T];\mathrm{L}^{p}(\mathbb{T}^{2}))$, for all $p>2$ and
(2.20) implies
$\boldsymbol{q}(\xi)\in\mathrm{C}([0,T];\mathbb{L}_{\sigma}^{p}({\mathbb{T}^{2}}))$.
Fixing $p>4$, we conclude by Lemma 2.2 (2) that
$J\boldsymbol{q}(\xi)\in\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$. For $p>2$,
the tightness of final term
$\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j,m}(s,y,\xi_{m}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)$
in the space $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$ is immediate from [29,
Corollary 3.6]. Hence, the solution $\xi^{\epsilon}$ has a stochastic
modification which is continuous in $(t,x)\in[0,T]\times{\mathbb{T}^{2}}$.
This completes the proof of Theorem 4.2. ∎

######  Remark 4.6.

One can show the existence of solution in the following way also: We first
consider the solution $\xi_{R}^{\epsilon}$ of the truncated integral equation
(4.1) for any $R\in\mathbb{N}$. Let us define a sequence of stopping times

$\displaystyle\tau_{R}:=\inf\\{t\geq
0:\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\geq R\\}\wedge T,\ \ \text{ for
}\ \ R\in\mathbb{N}.$

From Proposition 4.4, we know that $\xi_{R}^{\epsilon}(t,\cdot)$ is the unique
solution of the integral equation (1.1.2) for any $t\in[0,\tau_{R}]$. By the
uniqueness of solution, we infer that for any $P\geq R$,
$\tau_{R}\leq\tau_{P}$ and $\xi_{P}^{\epsilon}(t)=\xi_{R}^{\epsilon}(t)$ for
$t\leq\tau_{R}$. Let us define $\xi^{\epsilon}(t)=\xi_{R}^{\epsilon}(t)$ for
$t\leq\tau_{R}$. Using the similar arguments we are able to construct a
solution $\xi^{\epsilon}(t)$ of the integral equation (1.1.2) in the random
interval $[0,\tau_{\infty}]$, where $\tau_{\infty}=\sup\limits_{R\geq
0}\tau_{R}$. Now, our aim is to establish that $\tau_{\infty}=T,\
\mathbb{P}$-a.s., which is equivalent to show that
$\lim\limits_{R\to\infty}\mathbb{P}\big{(}\tau_{R}<T\big{)}=0$.

From the uniform energy estimate (4.5), we have

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C\big{(}\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}+1\big{)},$

for $p>2$. Since $\tau_{R}\leq\tau_{\infty}$, we have

$\displaystyle\mathbb{P}\big{(}\tau_{\infty}<T\big{)}\leq\mathbb{P}\big{(}\tau_{R}<T\big{)}=\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}>R^{p}\bigg{)}.$

Applying Markov’s inequality in the above inequality, we find

$\displaystyle\mathbb{P}\big{(}\tau_{R}<T\big{)}\leq\frac{1}{R^{p}}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{R}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq\frac{C\big{(}\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}+1\big{)}}{R^{p}}.$

Passing $R\to\infty$ leads to our required result, that is, $\tau_{\infty}=T,\
\mathbb{P}$-a.s.

##  5\. Uniform large Deviation Principle: Finite-dimensional Noise

In this section, we first recall the definition of equicontinuous uniform
Laplace principle (EULP) (see [43], for ULDP, see Subsection 3.2). Later, we
state a sufficient condition under which EULP holds. We start with the
following additional information:

One should note that FWULDP depends on the choice of the Polish space
$(\mathcal{E},d)$. Recall that a family
$\mathrm{B}\subset\mathrm{C}_{b}(\mathcal{E})$ (space of bounded and
continuous functions from $\mathcal{E}$ to $\mathbb{R}$) of functions from
$\mathcal{E}\to\mathbb{R}$ is equibounded and equicontinuous if

$\displaystyle\sup_{g\in\mathrm{B}}\sup_{\psi\in\mathcal{E}}|g(\psi)|<\infty\
\ \text{ and }\ \ \lim_{\delta\to
0}\sup_{g\in\mathrm{B}}\sup_{\begin{subarray}{c}\psi_{1},\psi_{2}\in\mathcal{E},\\\
d(\psi_{1},\psi_{2})<\delta\end{subarray}}|g(\psi_{1})-g(\psi_{2})|=0,$

respectively.

######  Definition 5.1 (EULP).

Let $\mathscr{T}$ be the collection of subsets of $\mathcal{E}_{0}$ and
$b(\epsilon)$ be a function which converges to $0$ as $\epsilon\to 0$. A
family of $\mathcal{E}$-valued random variables
$\\{\mathrm{Y}_{\xi_{0}}^{\epsilon}\\}_{\epsilon>0}$ is said to satisfy an
_equicontinuous uniform Laplace principle_ with the speed $b(\epsilon)$ and
the rate function $\mathrm{I}_{\xi_{0}}(\cdot)$, uniformly over $\mathscr{T}$
if for any $\mathrm{E}\in\mathscr{T}$ and any equibounded and equicontinuous
family $\mathrm{B}\subset\mathrm{C}_{b}(\mathcal{E})$,

$\displaystyle\lim_{\epsilon\to
0}\sup_{g\in\mathrm{B}}\sup_{\xi_{0}\in\mathrm{E}}\bigg{|}b(\epsilon)\log\mathbb{E}\bigg{[}\exp\bigg{(}-\frac{g(\mathrm{Y}_{\xi_{0}}^{\epsilon})}{b(\epsilon)}\bigg{)}\bigg{]}+\inf_{h\in\mathcal{E}}\big{\\{}g(h)+\mathrm{I}_{\xi_{0}}(\psi)\big{\\}}\bigg{|}=0.$
(5.1)

An equivalency between ULDP and EULP is established in [43].

######  Proposition 5.2 ([43, Theorem 2.10]).

EULP and FWULDP are equivalent.

Let us provide a sufficient condition for a family of random variables to
satisfy EULP. Then by Proposition 5.2, we obtain FWULDP. Let $\mathscr{P}_{2}$
be the set of $\mathscr{F}_{t}$-adapted controls $v:[0,T]\to\mathbb{R}^{n}$
such that
$\mathbb{P}\big{(}\|v\|_{\mathrm{L}^{2}(0,T;\mathbb{R}^{n})}<\infty\big{)}=1$.
For $M>0$, let $\mathscr{P}_{2}^{M}\subset\mathscr{P}_{2}$ be the set of
admissible controls such that

$\displaystyle\mathscr{P}_{2}^{M}:=\big{\\{}v\in\mathscr{P}_{2}:\mathbb{P}\big{(}\|v\|_{\mathrm{L}^{2}(0,T;\mathbb{R}^{n})}\leq
M\big{)}=1\big{\\}}.$

######  Condition 5.1.

Assume that there exists a family of measurable maps
$\mathscr{G}^{\epsilon}:\mathrm{C}([0,T];\mathbb{R}^{n})\to\mathcal{E}$,
indexed by $\epsilon\in(0,1]$, $\xi_{0}\in\mathcal{E}_{0}$. Let
$\mathrm{W}(\cdot)$ be a $n$-dimensional Wiener process and
$\mathrm{Y}_{v}^{\epsilon}:=\mathscr{G}^{\epsilon}\big{(}\xi_{0},\sqrt{\epsilon}\mathrm{W}+\int_{0}^{\cdot}v(s)\mathrm{d}s\big{)}$.
Define $\mathscr{G}^{0}:\mathrm{C}([0,T];\mathbb{R}^{n})\to\mathcal{E}$, the
limiting case of $\mathscr{G}^{\epsilon}$ as $\epsilon\to 0$. Let
$\mathscr{T}$ be collection of the subset of $\mathcal{E}_{0}$ such that for
any $\mathrm{E}\in\mathscr{T},\ M>0$ and $\delta>0$,

$\displaystyle\lim_{\epsilon\to
0}\sup_{\xi_{0}\in\mathrm{E}}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\mathopen{}\mathclose{{}\left\\{d\bigg{(}\mathscr{G}^{\epsilon}\bigg{(}\xi_{0},\sqrt{\epsilon}\mathrm{W}+\int_{0}^{\cdot}v(s\mathrm{d}s\bigg{)},\mathscr{G}^{0}\bigg{(}\xi_{0},\int_{0}^{\cdot}v(s)\mathrm{d}s\bigg{)}\bigg{)}>\delta}\right\\}=0.$
(5.2)

######  Theorem 5.3 ([43, Theorem 2.13]).

Assume that Condition 5.1 holds. Then, the family
$\mathrm{Y}_{\xi_{0}}^{\epsilon}:=\mathscr{G}^{\epsilon}(\xi_{0},\sqrt{\epsilon}\mathrm{W})$
satisfies the EULP uniformly over $\mathscr{T}$ with the speed
$b(\epsilon)=\epsilon$, and the rate function
$\mathrm{I}_{\xi_{0}}:\mathcal{E}\to\mathbb{R}$,

$\displaystyle\mathrm{I}_{\xi_{0}}(h)=\inf\bigg{\\{}\frac{1}{2}\int_{0}^{T}|v(s)|_{\mathbb{R}^{n}}^{2}\mathrm{d}s:\
h:=\mathscr{G}^{0}\bigg{(}\xi_{0},\int_{0}^{\cdot}v(s)\mathrm{d}s\bigg{)}\bigg{\\}}.$
(5.3)

If we verify Condition 5.1, then by Theorem 5.3, the family
$\\{\mathrm{Y}_{\xi_{0}}^{\epsilon}\\}_{\epsilon>0}$ satisfies the EULP and by
equivalency we obtain that the family
$\\{\mathrm{Y}_{\xi_{0}}^{\epsilon}\\}_{\epsilon>0}$ satisfies ULDP. Thus, our
main aim is to verify Condition 5.1.

###  5.1. Controlled and skeleton equation

The unique mild solution to the system (1.4) is a probabilistically strong
solution, since (2.4) is satisfied and an estimate similar to (4.5) can be
established for $\xi^{\epsilon}$. Let us fix
$\mathcal{E}=\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$,
$\mathcal{E}_{0}=\mathrm{L}^{p}({\mathbb{T}^{2}})$ and the solution map of the
integral equation (1.1.2) be represented by
$\xi^{\epsilon}=\mathscr{G}^{\epsilon}(\xi_{0},\sqrt{\epsilon}\mathrm{W})$.

Let us denote the solution
$\xi_{v}^{\epsilon}(t,x):=\mathscr{G}^{\epsilon}\big{(}\xi_{0},\sqrt{\epsilon}\mathrm{W}(t)+\int_{0}^{t}v(s)\mathrm{d}s\big{)}$
of the following stochastic controlled integral (SCI) equation

$\displaystyle\xi_{v}^{\epsilon}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}(\xi_{v}^{\epsilon}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi_{v}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)$
$\displaystyle\quad+\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi_{v}^{\epsilon}(s,y))v_{j}(s)\mathrm{d}y\mathrm{d}s,\
\mathbb{P}\text{-a.s.},$ (5.4)

for all $t\in[0,T]$. It we choose $\epsilon=0$, then the above SCI equation
changes to skeleton equation (limiting case). Let us denote the solution
$\xi_{v}^{0}(t,x):=\mathscr{G}^{0}\big{(}\xi_{0},\int_{0}^{t}v(s)\mathrm{d}s\big{)}$
of the following skeleton equation:

$\displaystyle\xi_{v}^{0}(t,x)$
$\displaystyle=\int_{\mathbb{T}^{2}}G(t,x,y)\xi_{0}(y)\mathrm{d}y+\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\boldsymbol{q}(\xi_{v}^{0}(s,\cdot))(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi_{v}^{0}(s,y))v_{j}(s)\mathrm{d}y\mathrm{d}s,\
\text{ for all }\ t\in[0,T].$ (5.5)

For $\psi\in\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, we define the
rate function

$\displaystyle\mathrm{I}_{\xi_{0}}(\psi):=\inf_{\big{\\{}v\in\mathrm{L}^{2}(0,T;\mathbb{R}^{n}):\psi=\mathscr{G}^{0}\big{(}\xi_{0},\int_{0}^{\cdot}v(s)\mathrm{d}s\big{)}\big{\\}}}\bigg{\\{}\frac{1}{2}\int_{0}^{T}|v(s)|^{2}_{\mathbb{R}^{n}}\mathrm{d}s\bigg{\\}},$
(5.6)

where $\psi$ satisfies the skeleton equation (5.1).

The following results discusses the existence and uniqueness of solution to
the above SCI and skeleton equations.

######  Theorem 5.4.

Assume that Hypothesis 4.1 holds and $v\in\mathscr{P}_{2}^{M}$. Then, for any
initial data $\xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$, for $p>2$, there
exists a unique solution
$\xi_{v}^{\epsilon}\in\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$ of
the SCI equation (5.1).

###### Proof.

The main ingredient of this proof is the Girsanov’s theorem (see [16, Theorem
10.14]).

For any given $v\in\mathscr{P}_{2}^{M}$, we define

$\displaystyle\frac{\mathrm{d}\widetilde{\mathbb{P}}}{\mathrm{d}\mathbb{P}}:=\exp\bigg{\\{}-\frac{1}{\sqrt{\epsilon}}\int_{0}^{T}v(s)\mathrm{d}\mathrm{W}(s)-\frac{1}{2\epsilon}\int_{0}^{T}|v(s)|_{\mathbb{R}^{n}}^{2}\mathrm{d}s\bigg{\\}},$
(5.7)

where $\mathrm{W}(\cdot)$ is a finite-dimensional Wiener process. Then, the
stochastic process

$\displaystyle\widetilde{\mathrm{W}}(t):=\mathrm{W}(t)+\frac{1}{\sqrt{\epsilon}}\int_{0}^{t}v(s)\mathrm{d}s,\
\ t\in[0,T],$

is a finite-dimensional Wiener process under the probability measure
$\widetilde{\mathbb{P}}$. Also,

$\displaystyle\exp\bigg{\\{}-\frac{1}{\sqrt{\epsilon}}\int_{0}^{T}v(s)\mathrm{d}\mathrm{W}(s)-\frac{1}{2\epsilon}\int_{0}^{T}|v(s)|^{2}_{\mathbb{R}^{n}}\mathrm{d}s\bigg{\\}},$

is an exponential martingale and $\widetilde{\mathbb{P}}$ is another
probability measure on the probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ and
$\widetilde{\mathbb{P}}$ is equivalent to the measure $\mathbb{P}$. Using
Girsanov’s theorem, we obtain hat $\widetilde{\mathrm{W}}(\cdot)$ is a real
valued finite-dimensional Wiener process with respect to the measure
$\mathbb{P}$. By Theorem 4.2, we obtain the existence and uniqueness of the
solution to the SCI equation (5.1) with respect to the new probability measure
$\widetilde{\mathbb{P}}$. Hence, we conclude the well-posedness to the SCI
equation (5.1) with respect to the probability measure $\mathbb{P}$ also. ∎

######  Theorem 5.5.

Assume that Hypothesis 4.1 holds and $v\in\mathrm{L}^{2}(0,T;\mathbb{R}^{n})$.
Then, for any initial data $\xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$, for
$p>2$, there exists a unique solution
$\xi_{v}^{0}\in\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, to the
skeleton equation (5.1).

###### Proof.

Since $v\in\mathrm{L}^{2}(0,T;\mathbb{R}^{n})$, and

$\displaystyle\bigg{|}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi_{v}^{0}(s,y))v_{j}(s)\mathrm{d}y\mathrm{d}s\bigg{|}$
$\displaystyle\leq
C\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}\big{(}1+|\xi_{v}^{0}(s,y)|\big{)}|v_{j}(s)|\mathrm{d}y\mathrm{d}s$
$\displaystyle\leq
C\sum_{j=1}^{n}\bigg{(}\int_{0}^{t}|v_{j}(s)|^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\\{}t^{\frac{1}{2}}+\bigg{(}\int_{0}^{t}\bigg{(}\int_{\mathbb{T}^{2}}|\xi_{v}^{0}(s,y)|\mathrm{d}y\bigg{)}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\\}}$
$\displaystyle\leq
C\|v\|_{\mathrm{L}^{2}(0,T;\mathbb{R}^{n})}\bigg{\\{}T^{\frac{1}{2}}+T^{\frac{1}{4}}\bigg{(}\int_{0}^{T}\|\xi_{v}^{0}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{4}}\bigg{\\}}<\infty,$

the well-posedness of the skeleton equation (5.1) can be shown by using
similar arguments as in the proof of Theorem 4.2, with some minor modification
in the proof. ∎

######  Remark 5.6.

Let us assume that $\xi_{0}\in\mathrm{C}(\mathbb{T}^{2})$. Using the well-
posedness established in Section 4, and [29, Lemma 3.1, Corollary 3.2 and
Proposition 3.5], we conclude that the solution $\xi_{v}^{\epsilon}$ of the
integral equation (5.1) has a modification which is a space-times continuous
process. Moreover, the solution $\xi_{v}^{0}$ of the integral equation (5.1)
has a version which belongs to the space
$\mathrm{C}([0,T]\times\mathbb{T}^{2})$.

###  5.2. Main Theorems

In this subsection, we establish that the family of solutions $\xi^{\epsilon}$
of the integral equation (4) satisfies ULDP in two different topologies
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for $p>2$ and
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$.

######  Theorem 5.7 (ULDP in
$\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$).

Let $p>2$. The family $\\{\xi_{v}^{\epsilon}\\}_{\epsilon>0}$ satisfies the
ULDP on $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, with rate
function $\mathrm{I}_{\xi_{0}}(\cdot)$ given by (5.6), where the uniformity is
over $\mathrm{L}^{p}({\mathbb{T}^{2}})$-bounded sets of initial conditions, if
the following hold:

  1. (1)

For any $N>0$, $t_{0}\geq 0$, and $\delta>0$

$\displaystyle\liminf_{\epsilon\to 0}\inf_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\inf_{\psi\in\Lambda_{\xi_{0}}(t_{0})}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\|\xi_{v}^{\epsilon}-\psi\|_{\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))}<\delta\big{)}+\mathrm{I}_{\xi_{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

For any $N>0$, $t_{0}\geq 0$, and $\delta>0$

$\displaystyle\limsup_{\epsilon\to 0}\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{s\in[0,t_{0}]}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\mathrm{dist}_{\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))}(\xi_{v}^{\epsilon},\Lambda_{\xi_{0}(s)})>\delta\big{)}+s\bigg{\\}}\leq
0.$

######  Theorem 5.8 (ULDP in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$).

The family $\\{\xi_{v}^{\epsilon}\\}_{\epsilon>0}$ satisfies the ULDP on
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$, with rate function
$\mathrm{I}_{\xi_{0}}(\cdot)$ given by (5.6), where the uniformity is over
$\mathrm{C}({\mathbb{T}^{2}})$-bounded sets of initial conditions, if the
following hold:

  1. (1)

For any $N>0$, $t_{0}\geq 0$, and $\delta>0$

$\displaystyle\liminf_{\epsilon\to 0}\inf_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\inf_{\psi\in\Lambda_{\xi_{0}}(t_{0})}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\|\xi_{v}^{\epsilon}-\psi\|_{\mathrm{C}([0,T]\times{\mathbb{T}^{2}})}<\delta\big{)}+\mathrm{I}_{\xi_{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

For any $N>0$, $t_{0}\geq 0$, and $\delta>0$

$\displaystyle\limsup_{\epsilon\to 0}\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{s\in[0,t_{0}]}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\mathrm{dist}_{\mathrm{C}([0,T]\times{\mathbb{T}^{2}})}(\xi_{v}^{\epsilon},\Lambda_{\xi_{0}(s)})>\delta\big{)}+s\bigg{\\}}\leq
0.$

######  Proposition 5.9 (Global estimate).

For any $p>2$, there exists a constant $C>0$ depending on $p,M$, and $T$, such
that for $\epsilon\in(0,1],\ \xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$ and
$v\in\mathscr{P}_{2}^{M}$,

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}+$
$\displaystyle
2p(p-1)\mathbb{E}\bigg{[}\int_{0}^{T}\||\xi_{v}^{\epsilon}(s)|^{\frac{p-2}{2}}\nabla\xi_{v}^{\epsilon}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{]}$
$\displaystyle\leq C\big{(}1+\|\xi_{0}\|_{\mathrm{L}^{p}}^{p}\big{)}.$ (5.8)

###### Proof.

The proof of this proposition can be obtained in a similar manner as in the
proof of (4.1) in Lemma 4.5. ∎

######  Corollary 5.10.

The random field
$\sup\limits_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p},\ p>2$
is bounded in probability, that is, for any given $T>0,\ M>0$, and $N>0$,

$\displaystyle\lim_{R\to\infty}\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}>R\bigg{)}=0.$
(5.9)

###### Proof.

Applying Markov’s inequality along with the global estimate (5.9) yields the
required result (5.9). ∎

######  Corollary 5.11 (Uniform convergence in probability).

For any $T>0,\ \delta>0,\ N>0,\ M>0$ and $p>2$

$\displaystyle\lim_{\epsilon\to 0}\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\|\xi_{v}^{\epsilon}-\xi_{v}^{0}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))}>\delta\bigg{)}=0.$
(5.10)

###### Proof.

For any $v\in\mathscr{P}_{2}^{M}$, $\|\xi_{0}\|_{\mathrm{L}^{p}}\leq N$, for
$p>2$ and $\epsilon\in(0,1]$, we have

$\displaystyle\xi_{v}^{\epsilon}(t,x)$ $\displaystyle-\xi_{v}^{0}(t,x)$
$\displaystyle=\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\big{(}\boldsymbol{q}(\xi_{v}^{\epsilon}(s,\cdot))-\boldsymbol{q}(\xi_{v}^{0}(s,\cdot))(y)\big{)}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\big{(}\sigma_{j}(s,y,\xi_{v}^{\epsilon}(s,y))-\sigma_{j}(s,y,\xi_{v}^{0}(s,y))\big{)}v_{j}(s)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\sum_{j=1}^{n}\int_{0}^{t}\int_{\mathbb{T}^{2}}G(t-s,x,y)\sigma_{j}(s,y,\xi_{v}^{\epsilon}(s,y))\mathrm{d}y\mathrm{d}\mathrm{W}^{j}(s)$
$\displaystyle=:I_{1,\epsilon}+I_{2,\epsilon}+\sqrt{\epsilon}I_{3,\epsilon}.$
(5.11)

We consider $I_{1,\epsilon}$ and estimate it using (2.19) and (1) with
$\alpha=p,\ \beta=1$, to find

$\displaystyle\|I_{1,\epsilon}(t)\|_{\mathrm{L}^{p}}$
$\displaystyle=\bigg{\|}\int_{0}^{t}\int_{\mathbb{T}^{2}}\nabla_{y}G(t-s,x,y)\cdot\big{(}\boldsymbol{q}(\xi_{v}^{\epsilon}(s,\cdot))-\boldsymbol{q}(\xi_{v}^{0}(s,\cdot))(y)\big{)}\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\boldsymbol{q}(\xi_{v}^{\epsilon}(s,\cdot))-\boldsymbol{q}(\xi_{v}^{0}(s,\cdot))\|_{\mathbb{L}_{\sigma}^{p}}\mathrm{d}s$
$\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|(\boldsymbol{k}*\xi_{v}^{\epsilon}(s))(\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s))+(\boldsymbol{k}*\xi_{v}^{\epsilon}(s)$
$\displaystyle\qquad-\boldsymbol{k}*\xi_{v}^{0}(s))\xi_{v}^{0}(s)\|_{\mathbb{L}_{\sigma}^{p}}\mathrm{d}s$
$\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}}\mathopen{}\mathclose{{}\left[\|\boldsymbol{k}*\xi_{v}^{\epsilon}(s)\|_{\mathbb{L}_{\sigma}^{\infty}}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}}\right.$
$\displaystyle\qquad\mathopen{}\mathclose{{}\left.+\|\boldsymbol{k}*(\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s))\|_{\mathbb{L}_{\sigma}^{\infty}}\|\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}}\right]\mathrm{d}s$
$\displaystyle\leq
C_{p}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\big{(}\|\xi_{v}^{\epsilon}(s)\|_{\mathrm{L}^{p}}+\|\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}\big{)}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s,$
(5.12)

for $p>2$ and $t\in[0,T]$. For any $P>0$, $\epsilon\in(0,1],\
v\in\mathscr{P}_{2}^{M},\ \|\xi_{0}\|_{\mathrm{L}^{p}}\leq N$ and $T>0$, we
define

$\displaystyle\Upsilon_{P}^{\epsilon}:=\bigg{\\{}\omega\in\Omega:\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\leq
P,\ \sup_{t\in[0,T]}\|\xi_{v}^{0}(t)\|_{\mathrm{L}^{p}}\leq P\bigg{\\}}.$
(5.13)

Using Hölder’s inequality in (5.2) and (5.13), we arrive at

$\displaystyle\|I_{1,\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}\leq
C_{p,T,P}\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s,$
(5.14)

where $\chi_{A}$ is the characteristic function of the set $A$. Let us
consider the second term $I_{2,\epsilon}$ in the right hand side of (5.2) and
estimate it using [29, Lemma 3.1(i)] to obtain

$\displaystyle\|I_{2,\epsilon}(t)\|_{\mathrm{L}^{p}}\leq
C\sum_{j=1}^{n}\int_{0}^{t}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}|v_{j}(s)|\mathrm{d}s.$

Hölder’s inequality and (5.13) results to

$\displaystyle\|I_{2,\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\leq
C\sum_{j=1}^{n}\bigg{(}\int_{0}^{t}|v_{j}(s)|^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{(}\int_{0}^{t}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}$
$\displaystyle\leq
CM^{\frac{p}{2}}\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}\|\xi_{v}^{\epsilon}(s)-\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s.$
(5.15)

Combining (5.2)-(5.2) and applying Gronwall’s inequality, we deduce

$\displaystyle\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t,x)-\xi_{v}^{0}(t,x)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{p}^{\epsilon}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\sup_{t\in[0,T]}\|I_{3,\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{p}^{\epsilon}}\exp\bigg{\\{}C\int_{0}^{T}\big{[}(t-s)^{-\frac{1}{2}}+M^{\frac{p}{2}}\big{]}\mathrm{d}s\bigg{\\}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\sup_{t\in[0,T]}\|I_{3,\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{p}^{\epsilon}}.$
(5.16)

Let $\varrho>0$. By Proposition 5.9, there exists a $P>0$ large enough such
that

$\displaystyle\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\big{[}1-\mathbb{P}(\Upsilon_{P}^{\epsilon})\big{]}$
$\displaystyle\leq\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\bigg{\\{}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}>P\bigg{)}+\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{v}^{0}(t)\|_{\mathrm{L}^{p}}>P\bigg{)}\bigg{\\}}$
$\displaystyle\leq\varrho.$ (5.17)

From (5.2) and (5.2), for any $\epsilon\in(0,1],\
\xi_{0}\in\mathrm{L}^{p}({\mathbb{T}^{2}})$ and $v\in\mathscr{P}_{2}^{M}$, we
find

$\displaystyle\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)-\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}>\delta\bigg{)}$
$\displaystyle\leq\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\big{[}1-\mathbb{P}(\Upsilon_{P}^{\epsilon})\big{]}$
$\displaystyle\quad+\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)-\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\delta\bigg{)}$
$\displaystyle\leq\varrho+\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sqrt{\epsilon}\sup_{t\in[0,T]}\|I_{3,\epsilon}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\frac{\delta}{C}\bigg{)}.$
(5.18)

Using the linear growth of $\sigma_{j}$ (see Hypothesis 4.1), we have for any
$t\in[0,T]$

$\displaystyle\sup_{\epsilon\in(0,1]}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\sigma_{j}(t,\cdot,\xi_{v}^{\epsilon}(t,\cdot))\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C\bigg{(}1+\sup_{\epsilon\in(0,1]}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|\xi_{v}^{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\bigg{)}<\infty.$
(5.19)

Using [29, Corollary 3.6], we obtain for any $\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N$ and $v\in\mathscr{P}_{2}^{M}$ the family

$\displaystyle\\{I_{3,\epsilon}:\epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))$, for $p>2$.
By Prokhorov’s Theorem (see [3, Section 5]), we find that for any sequence
$\epsilon_{m}\to 0,\ v_{m}\in\mathscr{P}_{2}^{M}$ and
$\|\xi_{0,m}\|_{\mathrm{L}^{p}}\leq N$, there exists a subsequence (denoted by
$(\epsilon_{m},v_{m},\xi_{0,m})$) such that $I_{3,\epsilon_{m}}(t,x)$
converges in distribution. Now, we multiply by the subsequence
$\sqrt{\epsilon_{m}}\downarrow 0$, then it is obvious that

$\displaystyle\sqrt{\epsilon_{m}}\sup_{t\in[0,T]}\|I_{3,\epsilon_{m}}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon_{m}}}\Rightarrow
0\ \ \text{ as }\ \ m\to\infty,$

where “$\Rightarrow$” stands for convergence in distribution. Thus, the
original sequence

$\sqrt{\epsilon}\sup\limits_{t\in[0,T]}\|I_{3,\epsilon}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}\Rightarrow
0\ \text{ as }\ \epsilon\to 0.$

As we can see that the sequential limit is zero, which is a constant,
therefore the sequence converges in probability also, that is,

$\displaystyle\lim_{\epsilon\to
0}\mathbb{P}\bigg{(}\sqrt{\epsilon}\sup_{t\in[0,T]}\|I_{3,\epsilon}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\frac{\delta}{C}\bigg{)}=0.$

Due to arbitrary choice of $\epsilon_{m}\downarrow 0,\
v_{m}\in\mathscr{P}_{2}^{M}$, and $\|\xi_{0,m}\|_{\mathrm{L}^{p}}\leq N$, we
deduce

$\displaystyle\lim_{\epsilon\to 0}\sup_{\|\xi_{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{v\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\|\xi_{v}^{\epsilon}-\xi_{v}^{0}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}({\mathbb{T}^{2}}))}>\delta\bigg{)}\leq\varrho.$
(5.20)

The choice of $\varrho$ is arbitrary, hence (5.10) follows. ∎

###### Proof of Theorem 5.7.

Combining Theorem 5.3 and Corollary 5.11, we obtain the required result. ∎

Let us now move to the proof of Theorem 5.8, that is, ULDP in
$\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$ topology. For that, we need the
following intermediate result.

######  Lemma 5.12.

For any $\|\xi_{0}\|_{\mathrm{C}({\mathbb{T}^{2}})}\leq N$, the family

$\displaystyle\\{\xi_{v}^{\epsilon}-\xi_{v}^{0}:v\in\mathscr{P}_{2}^{M},\epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T]\times{\mathbb{T}^{2}})$.

###### Proof.

From (5.2), we have

$\displaystyle\xi_{v}^{\epsilon}(t,x)-$
$\displaystyle\xi_{v}^{0}(t,x)=:I_{1,\epsilon}+I_{2,\epsilon}+\sqrt{\epsilon}I_{3,\epsilon}.$

Using similar arguments as in (5.2), we obtain

$\displaystyle\|I_{1,\epsilon}(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C_{p}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\big{(}\|\xi_{v}^{\epsilon}(s)\|_{\mathrm{L}^{p}}+\|\xi_{v}^{0}(s)\|_{\mathrm{L}^{p}}\big{)}^{2}\mathrm{d}s.$
(5.21)