# Well-posedness and uniform large deviation principle for stochastic Burgers-
Huxley equation perturbed by a multiplicative noise 111 1,3Department of
Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar
Highway, Roorkee, Uttarakhand 247667, INDIA.
2Theoretical Statistics and Mathematics Unit, Indian Statistical Institute
Banglore Centre-ISI Banglore, 8th Mile Mysore Road, Bangalore, Karnataka
560059, India. e-mail: Manil T. Mohan<EMAIL_ADDRESS><EMAIL_ADDRESS>e-mail: Ankit Kumar<EMAIL_ADDRESS>e-mail:
Vivek Kumar<EMAIL_ADDRESS><EMAIL_ADDRESS>*Corresponding author. Keywords: Stochastic Burgers-Huxley equations, mild
solution, large deviations. Mathematics Subject Classification (2020): Primary
60H15, 60F10; Secondary 37L55, 60H20.

Ankit Kumar1, Vivek Kumar2 and Manil T. Mohan3*

###### Abstract.

In this work, we focus on the global solvability and uniform large deviations
for the solutions of stochastic generalized Burgers-Huxley (SGBH) equation
perturbed by a small multiplicative white in time and colored in space noise.
The SGBH equation has the nonlinearity of polynomial order and noise
considered in this work is infinite dimensional with a coefficient having
linear growth. First, we prove the existence of a unique local mild solution
in the sense of Walsh to SGBH equation with the help of a truncation argument
and contraction mapping principle. Then the global solvability results are
established by using uniform bounds of the local mild solution, stopping time
arguments, tightness properties and Skorokhod’s representation theorem. By
using the uniform Laplace principle, we obtain the large deviation principle
(LDP) for the law of solutions to SGBH equation by using variational
representation methods. Further, we derive the uniform large deviation
principle (ULDP) for the law of solutions in two different topologies by using
a weak convergence method. First, in the
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ topology where the uniformity is
over $\mathrm{L}^{p}([0,1])$-bounded sets of initial conditions, and secondly
in the $\mathrm{C}([0,T]\times[0,1])$ topology with uniformity being over
bounded subsets in the $\mathrm{C}([0,1])$-norm. Finally, we consider SGBH
equation perturbed by a space-time white noise with bounded noise coefficient
and establish the ULDP for the laws of solutions. The results obtained in this
work hold true for stochastic Burgers’ as well as Burgers-Huxley equations.

##  1\. Introduction

Burgers’ equation ([4]) is one of the celebrated fundamental partial
differential equations and is the simplest model for analyzing combined effect
of nonlinear convection and diffusion. Burgers’ equation appears in various
areas of applied mathematics such as fluid mechanics, traffic flow, nonlinear
acoustics, gas dynamics, etc. The Burgers-Huxley equation is a mathematical
model that describes the dynamics of a non-linear, one-dimensional wave system
that is subject to both deterministic and stochastic influences. It is a
combination of the Burgers equation [4], which describes the nonlinear
behavior of a fluid flow, and the Huxley equation [39], which describes the
dynamics of an action potential in a nerve or muscle cell. The Burgers-Huxley
equation is a good example of convection-diffusion-reaction equations. In the
stochastic version of the equation, a random noise is added to the system to
account for the influence of external factors such as thermal fluctuations or
other sources of randomness.

###  1.1. The model

In this work, we consider the following generalized version of Burgers-Huxley
equation driven by a random forcing indexed by $\epsilon\in(0,1]$:

$\displaystyle\frac{\partial u_{\epsilon}(t,x)}{\partial t}$
$\displaystyle=\nu\frac{\partial^{2}u_{\epsilon}(t,x)}{\partial x^{2}}-\alpha
u_{\epsilon}^{\delta}(t,x)\frac{\partial u_{\epsilon}(t,x)}{\partial x}+\beta
u_{\epsilon}(t,x)(1-u_{\epsilon}^{\delta}(t,x))(u_{\epsilon}^{\delta}(t,x)-\gamma)$
$\displaystyle\quad+\sqrt{\epsilon}g(t,x,u_{\epsilon}(t,x))\mathcal{W}(t,x),~{}~{}~{}~{}~{}t>0,~{}x\in(0,1),$
(1.1)

where $\nu>0$ and $\alpha>0$ are the viscosity and convection coefficients,
respectively, $\beta>0,\ \delta\geq 1$, $\gamma\geq 1$ are parameters and
$\epsilon\in(0,1]$ is the intensity of the noise, with the Dirichlet boundary
condition and initial data

$\displaystyle u_{\epsilon}(t,0)=u_{\epsilon}(t,1)=0,\ \ t\geq 0,\ \ \text{
and }\ \ u_{\epsilon}(0,x)=u^{0}(x),$ (1.2)

respectively. The noise coefficient $g(\cdot,\cdot,\cdot)$ has linear growth
and it satisfies a Lipschitz condition in the third variable (see Hypothesis
1.1 below).

In the deterministic case, for $\beta=0$ and $\delta=1$, the equation (1.1)
represents the classical viscous Burgers equation and for $\delta=1$, it is
called the Burgers-Huxley equation. The equation is commonly used in the study
of a wide range of physical and biological systems, including fluid dynamics,
neural networks, and pattern formation. For $\beta=0$ and $\alpha=1$, the
equation gives the following generalized Burgers equations

$\displaystyle\frac{\partial u_{\epsilon}(t,x)}{\partial
t}=\nu\frac{\partial^{2}u_{\epsilon}(t,x)}{\partial x^{2}}-\alpha
u_{\epsilon}^{\delta}(t,x)\frac{\partial u_{\epsilon}(t,x)}{\partial
x}+\sqrt{\epsilon}g(t,x,u_{\epsilon}(t,x))\mathcal{W}(t,x).$ (1.3)

which is very famous as a toy model to study the turbulence in fluids and the
model has been explored a lot in the literature, for instance see [7, 8, 15,
16, 22, 24], etc. and references therein. For $\alpha=0$, the equation (1.1)
gives rise to the following equation:

$\displaystyle\frac{\partial u_{\epsilon}(t,x)}{\partial t}$
$\displaystyle=\nu\frac{\partial^{2}u_{\epsilon}(t,x)}{\partial x^{2}}+\beta
u_{\epsilon}(t,x)+\beta
u_{\epsilon}(t,x)(t-u_{\epsilon}^{\delta}(t,x))(u_{\epsilon}^{\delta}(t,x)-\gamma)$
$\displaystyle\quad+\sqrt{\epsilon}g(t,x,u_{\epsilon}(t,x))\mathcal{W}(t,x),\
\ \text{ for }\ \ (t,x)\in(0,T)\times(0,1).$ (1.4)

The equation (1.1) is known as the stochastic Huxley equation and it models
the propagation of nerve pulse in nerve fibers and wall motion in liquid
crystals (see [39]).

###  1.2. A short note on the noise $\mathcal{W}(\cdot,\cdot)$

Let us now characterize the noise $\mathcal{W}(\cdot,\cdot)$ appearing in
(1.1) on the probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\mathbb{P})$ (see [25]).
Consider a sequence $\\{\xi_{j}\\}_{j\geq 1}$ of independent standard Gaussian
random variables on this basis (see Lemma II.2.3, [23]). If
$\\{\mathrm{m}_{j}(t)\\}_{j\in\mathbb{N}}$ is an orthonormal basis in
$\mathrm{L}^{2}([0,T]),$ then

$\beta(t):=\sum_{j\in\mathbb{N}}\mathopen{}\mathclose{{}\left(\int_{0}^{t}\mathrm{m}_{j}(s)ds}\right)\xi_{j}$

is a standard Brownian motion with covariance
$\mathbb{E}[\beta(t)\beta(s)]=(t\wedge s)$. Consider a sequence
$\\{\beta_{j}(t)\\}_{j\in\mathbb{N}}$ of independent standard Brownian motions
and an orthonormal basis $\\{\varphi_{j}(x)\\}_{j\in\mathbb{N}}$ in the space
$\mathrm{L}^{2}([0,1])$. Let us define

$h_{j}(x):=\int_{0}^{x}\varphi_{j}(y)dy.$

Then the process

$\mathrm{W}(t,x):=\sum_{j\in\mathbb{N}}h_{j}(x)\beta_{j}(t)$

is Gaussian with the mean zero and covariance
$\mathbb{E}[\mathrm{W}(t,x)\mathrm{W}(s,y)]=(t\wedge s)(x\wedge y).$ The
process $\mathrm{W}$ is known as the Brownian sheet. The formal term-by-term
differentiation of the series in $\beta(t)$ suggests a representation

$\displaystyle\dot{\beta}(t)=\sum_{k\in\mathbb{N}}\mathrm{m}_{k}(t)\xi_{k},$

so that

$\displaystyle\frac{\partial^{2}\mathrm{W}}{\partial t\partial
x}(t,x)=\dot{\mathrm{W}}(t,x)=\sum_{j\in\mathbb{N}}\varphi_{j}(x)\dot{\beta}_{j}(t),$
(1.5)

and we call the process $\dot{\mathrm{W}},$ the (Gaussian) space-time white
noise. Even though the series diverges, it defines a random generalized
function on $\mathrm{L}^{2}([0,T]\times[0,1])$:

$\displaystyle\dot{\mathrm{W}}(f)=\sum_{j\in\mathbb{N}}\int_{0}^{T}\mathopen{}\mathclose{{}\left(\int_{0}^{1}f(t,x)\varphi_{j}(x)\mathrm{d}x}\right)\mathrm{d}\beta_{j}(t)=\int_{0}^{T}\int_{0}^{1}f(t,x)\mathrm{W}(\mathrm{d}t,\mathrm{d}x).$

To construct a noise, that is, white in time and colored in space, take a
sequence of non-negative numbers $\\{q_{j}\\}_{j\in\mathbb{N}}$ and define

$\displaystyle\frac{\partial^{2}\mathrm{W}^{\mathrm{Q}}}{\partial t\partial
x}(t,x)=\dot{\mathrm{W}}^{\mathrm{Q}}(t,x)=\mathcal{W}(t,x)=\sum_{j\in\mathbb{N}}q_{j}\varphi_{j}(x)\dot{\beta}_{j}(t),$
(1.6)

and we call the process $\dot{\mathrm{W}}^{\mathrm{Q}},$ the (Gaussian) white
in time and colored in space noise. Whenever we consider
$\dot{\mathrm{W}}^{\mathrm{Q}}$ in the work, we assume that

$\displaystyle\sum\limits_{j\in\mathbb{N}}q_{j}^{2}<\infty.$ (1.7)

Let the family $\\{\varphi_{j}(x)\\}_{j\in\mathbb{N}}$ be the orthonormal
basis of $\mathrm{L}^{2}([0,1])$ consisting of the eigenfunctions of the
operator $-\frac{\partial^{2}}{\partial x^{2}}$ corresponding to the
eigenvalues $\\{\lambda_{j}\\}_{j\in\mathbb{N}}$. In fact, for $j=1,2,\ldots$

$\lambda_{j}=j^{2}\pi^{2},\ \varphi_{j}(x)=\sqrt{2}\sin(j\pi x).\ $

Then an example of a family $\\{q_{j}\\}_{j\in\mathbb{N}}$ satisfying (1.7) is
$\\{\lambda_{j}^{-\eta}\\}_{j\in\mathbb{N}}$ for $\eta>\frac{1}{4}$, since
$\sum\limits_{j\in\mathbb{N}}\frac{1}{j^{4\eta}}<\infty$ provided
$\eta>\frac{1}{4}$.

###  1.3. Assumptions and solution concept

We first discuss the function spaces used in this work and then provide the
assumptions satisfied by $g(\cdot,\cdot,\cdot)$.

####  1.3.1. Function spaces

Let us denote the space of all infinite times differentiable functions having
compact support in $[0,1]$ by $\mathrm{C}_{0}^{\infty}([0,1])$. The Lebesgue
spaces are denoted by $\mathrm{L}^{p}([0,1])$, for $p\geq 1$, and the norm in
the space $\mathrm{L}^{p}([0,1])$ is denoted by $\|\cdot\|_{\mathrm{L}^{p}}$.
For $p=2$, the inner product in the space $\mathrm{L}^{2}([0,1])$ is
represented by $(\cdot,\cdot)$. Let us denote the Hilbertian Sobolev spaces by
$\mathrm{H}^{k}([0,1])$, for $k\in\mathbb{N}$ and $\mathrm{H}_{0}^{1}([0,1])$
represent the closure of $\mathrm{C}_{0}^{\infty}([0,1])$ in
$\mathrm{H}^{1}$-norm. Since we are working in a bounded domain, by the
Poincaré inequality
($\sqrt{\lambda_{1}}\|u\|_{\mathrm{L}^{2}}^{2}\leq\|\partial_{x}u\|_{\mathrm{L}^{2}}^{2}$),
we have the norm
$(\|\cdot\|_{\mathrm{L}^{2}}^{2}+\|\partial_{x}\cdot\|_{\mathrm{L}^{2}}^{2})^{\frac{1}{2}}$
is equivalent to the seminorm $\|\partial_{x}\cdot\|_{\mathrm{L}^{2}}$ and
hence $\|\partial_{x}\cdot\|_{\mathrm{L}^{2}}$ defines a norm in
$\mathrm{H}_{0}^{1}([0,1])$. Moreover, we obtain the Gelfand triplet
$\mathrm{H}_{0}^{1}([0,1])\hookrightarrow\mathrm{L}^{2}([0,1])\hookrightarrow\mathrm{H}^{-1}([0,1])$,
where $\mathrm{H}^{-1}([0,1])$ stands for the dual of
$\mathrm{H}_{0}^{1}([0,1])$. For the bounded domain $[0,1]$, one has the
compact embedding
$\mathrm{H}_{0}^{1}([0,1])\hookrightarrow\mathrm{L}^{2}([0,1])$. The duality
paring between $\mathrm{H}_{0}^{1}([0,1])$ and its dual
$\mathrm{H}^{-1}([0,1])$ as well as between $\mathrm{L}^{p}([0,1])$ and its
dual $\mathrm{L}^{\frac{p}{p-1}}([0,1])$ is denoted by
$\langle\cdot,\cdot\rangle$. In one dimension, the embedding
$\mathrm{H}^{s}([0,1])\hookrightarrow\mathrm{L}^{p}([0,1])$ is compact for any
$s>\frac{1}{2}-\frac{1}{p}$, for $p>2$.

####  1.3.2. Hypotheses and mild solution

Let us introduce the linear growth and Lipschitz conditions on the noise
coefficient.

######  Hypothesis 1.1.

The function $g:[0,T]\times[0,1]\times\mathbb{R}\to\mathbb{R}$ is a measurable
function, satisfying the following conditions:

$\displaystyle|g(t,x,r)|\leq K(1+|r|),\ \ \text{ and }\ \
|g(t,x,r)-g(t,x,s)|\leq L|r-s|,$ (1.8)

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

The fundamental solution $G(t,x,y)$ of the heat equation in the interval
$[0,1]$ with the Dirichlet boundary conditions is given by

$\displaystyle G(t,x,y)=\frac{1}{\sqrt{4\pi
t}}\sum_{m=-\infty}^{\infty}\bigg{[}e^{-\frac{(y-x-2m)^{2}}{4t}}-e^{-\frac{(y+x-2m)^{2}}{4t}}\bigg{]},$
(1.9)

for all $t\in(0,T]$ and $x,y\in[0,1]$.

Next, we provide the definition of mild solution in the sense of Walsh ([38])
to the problem (1.1)-(1.2).

######  Definition 1.2 (Mild solution).

An $\mathrm{L}^{p}([0,1])$-valued and $\mathscr{F}_{t}$-adapted stochastic
process $u_{\epsilon}:[0,\infty)\times[0,1]\times\Omega\to\mathbb{R}$ with
$\mathbb{P}$-a.s. continuous trajectories on $t\in[0,T]$, is called a mild
solution to the system (1.1)-(1.2), if for any $T>0$,
$u_{\epsilon}(t):=u_{\epsilon}(t,\cdot,\cdot)$ satisfies the following
integral equation:

$\displaystyle u_{\epsilon}(t)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial y}(t-s,x,y)\mathfrak{p}(u_{\epsilon}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{\epsilon}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y),\
\ \mathbb{P}\text{-a.s.},$ (1.10)

for all $t\in[0,T]$, where $\mathfrak{p}(u)=u^{\delta+1},\
\mathfrak{c}(u)=u(1-u^{\delta})(u^{\delta}-\gamma)$ and $G(\cdot,\cdot,\cdot)$
is the fundamental solution of the heat equation in $[0,1]$ with the Dirichlet
boundary conditions.

If $u_{\epsilon}(\cdot)$ is a mild solution to the system (1.1)-(1.2) in the
sense of Definition 1.2, then by using Proposition 3.7, [18], one can show
that $u_{\epsilon}(\cdot)$ has a continuous modification:

$\displaystyle\int_{0}^{1}u_{\epsilon}(t,y)\phi(y)\mathrm{d}y$
$\displaystyle=\int_{0}^{1}u^{0}(y)\phi(y)\mathrm{d}y+\nu\int_{0}^{t}\int_{0}^{1}u_{\epsilon}(s,y)\phi^{\prime\prime}(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\mathfrak{p}(u_{\epsilon}(s,y))\phi^{\prime}(y)\mathrm{d}y\mathrm{d}s+\beta\int_{0}^{t}\int_{0}^{1}\mathfrak{c}(u_{\epsilon}(s,y))\phi(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}g(s,y,u_{\epsilon}(s,y))\phi(y)\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y),$
(1.11)

for every test function $\phi\in\mathrm{C}^{2}([0,1]),\ \phi(x)=0,\
x\in\\{0,1\\}$ and all $t\in[0,T]$. The existence of a unique mild solution to
the system (1.1)-(1.2) will be established in Section 2.

###  1.4. Literature Survey

The SGBH equation (1.1) covers both the models given by equations (1.3) and
(1.1). The global solvability results of deterministic generalized Burgers-
Huxley (GBH) equation and numerical studies have been carried out in [13, 19,
28], etc. and references therein. For the global solvability results of the
stochastic counterpart, one can see [20, 21, 26, 27], etc. In the works [20,
21, 26], the noise coefficient $g(\cdot,\cdot,\cdot)$ is taken as a bounded
function, while in [27] Burgers-Huxley equation is considered with the
$\mathrm{Q}$-Wiener process having noise coefficient as a Hilbert-Schmidt
operator which satisfy a linear growth and Lipschitz conditions.

The large deviation theory is one of the most popular research topics in
probability theory and it explores the limiting behavior of the probabilities
of rare events in the terms of a rate function. Though the topic of large
deviations is much older, the modern theories were mainly developed by Donsker
and Vardhan [10, 36], and Freidlin and Wentzell [12]. In this work, we are
following the uniform large deviation principle (ULDP) introduced by Freidlin
and Wentzell which is known as the Freidlin-Wentzell uniform large deviation
principle (FWULDP) in the honor of their names. It is one of the most popular
definitions of ULDP. Specifically, the work [12] describes the asymptotic
behavior of probability of large deviations of the law of the solution to a
family of small noise finite-dimensional SDEs, away from its law of large
number dynamics. In order to prove ULDP, one can use different types of
uniform Lapalce principles (ULP) with additional assumptions available in the
literature. In the work [3], the authors proved FWULDP for diverse families of
infinite-dimensional stochastic dynamical systems by using certain variational
representations. They developed the weak convergence method for proving the
ULDP by using the ULP. In order to obtain the ULDP for the law of the solution
to certain SPDEs, they have provided a sufficient condition, and to verify the
condition, one needs some basic qualitative properties of the solution like
existence, uniqueness, and tightness of the controlled analogues of the
original SPDEs. Adopting the approach established in [3], the authors in [2,
5, 6, 29, 30, 31, 34], etc. obtained several results in this direction for
different types of SPDEs. The author in the work [29] established the
equivalences between four different kinds of definitions of ULDP and ULP.

One of the main concerns in the method provided by [3] is that it requires the
convergence of initial conditions and therefore one cannot use this method
directly to prove ULDP for SPDEs having non-compact sets of the initial
conditions. For several applications, for example, in the case of the study of
exit times of a stochastic process $``u"$ from its domain, the large
deviations of $``u"$ must be uniform over the initial conditions lie in the
bounded subsets of the space that are not necessarily compact [30]. Recently,
in the work [29], the author introduced a new definition called the
equicontinuous uniform Laplace principle (EULP) of ULP and showed that the
definition of EULP is equivalent to the definition of FWULDP without any
compactness assumptions. In order to prove EULP, he came with a sufficient
condition under which the law of the solution follows the ULDP with uniformity
being over initial conditions that belong to bounded but not necessarily
compact sets. In the work [31], the authors adopted the methodology from [29]
and established the ULDP for the law of the solution of the stochastic Burgers
type equations. Here, they considered the nonlinearity of polynomial growth of
any order and the driving noise as a finite-dimensional Wiener process. Later,
the author in [33] proved the ULDP for the law of the solutions to a class of
semilinear SPDEs driven by Brownian sheet with bounded noise coefficient. In
both works [31] and [33], the uniformity is considered with respect to the
initial conditions that are bounded and do not necessarily belongs to a
compact set.

###  1.5. Objectives and novelties

In this work, we are mainly considering the SGBH system (1.1)-(1.2) consisting
of a multiplicative noise which is white in time and colored in space with the
coefficient $g(\cdot,\cdot,\cdot)$ having a linear growth with respect to the
unknown $u_{\epsilon}$ (Hypothesis 1.1). To the best of our knowledge, this
type of noise for this model and the following results on this model have not
been explored yet.

  * •

First we prove the existence of a unique local mild solution to the system
(1.1)-(1.2) in the sense of Walsh (Definition 1.2) by using a truncated system
and fixed point arguments. Then by using the uniform energy estimates of the
local solution, stopping time arguments, tightness properties and Skorokhod’s
representation theorem, we prove the global existence of the solution.

  * •

Next, we move to the LDP part, where we discuss two different types of LDP
depending on the availability of the initial conditions.

    1. (1)

The first LDP result is based on a variational representation method developed
in [3] and to use it, we need the initial data over a compact subset.

    2. (2)

If we are given the non-compact sets of initial data, then it is not enough to
establish ULP as mentioned in [3], and hence for this situation, we borrow
ideas from the works [29, 31]. We verify the sufficient condition for EULP and
hence ULDP in two different topologies: the
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ topology where the uniformity is
over $\mathrm{L}^{p}([0,1])$-bounded sets of initial conditions (for
$p\geq\max\\{6,2\delta+1\\}$), and secondly in the
$\mathrm{C}([0,T]\times[0,1])$ topology with uniformity being over bounded
subsets in the $\mathrm{C}([0,1])$-norm. To prove the ULDP in
$\mathrm{C}([0,T]\times[0,1])$, we use the embedding
$\mathrm{C}([0,T]\times[0,1])\hookrightarrow\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$
and Vardhan’s contraction principle [35].

  * •

Finally, we consider SGBH equation perturbed by a space-time white noise with
bounded noise coefficient and establish the ULDP for the laws of solutions in
the $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1])),$ for $p\geq 2\delta+1$ and
$\mathrm{C}([0,T]\times[0,1])$ topologies.

The well-posedess and LDP results obtained in this work hold true for
stochastic Burgers as well as Burgers-Huxley equations. The main difference
between the works [31, 33, 34] is that none of them considers the infinite
dimensional multiplicative time white and space colored noise with coefficient
having linear growth.

The main difficulties in proving our results are the presence of a polynomial
type of nonlinearity and the infinite-dimensional of noise with unbounded
noise coefficient. The polynomial nonlinearites are handled by using Green’s
function estimates (see Appendix A). We overcame the difficulty of a proper
Burkholder-Davis-Gundy (BDG) inequality in $\mathrm{L}^{p}$-spaces by using
the results from [37, 40]. Recently, the authors in [37, 40] obtained BDG
inequality in UMD Banach spaces, which helped us to handle several maximal
inequalities. Precisely, the authors in [37] showed that if $\mathrm{M}$ is an
$\mathrm{L}^{p}$-bounded martingale, $1<p<\infty$, with $\mathrm{M}_{0}=0$,
that takes values in a UMD Banach space $(\mathrm{X},\|\cdot\|_{\mathrm{X}})$
over a measure space $(S,\Sigma,\mu)$, then for all $t\geq 0$,

$\displaystyle\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{0\leq s\leq
t}\|\mathrm{M}_{s}(\sigma)\|_{\mathrm{X}}^{p}}\right]\leq
C_{p,\mathrm{X}}\mathbb{E}\mathopen{}\mathclose{{}\left[\|[\mathrm{M}(\sigma)]_{t}^{\frac{1}{2}}\|_{\mathrm{X}}^{p}}\right],$
(1.12)

where the quadratic variation $[\mathrm{M}(\sigma)]_{t}$ is considered
pointwise in $\sigma\in S$. As $\mathrm{L}^{p}([0,1]),$ $1<p<\infty$ is a UMD
Banach space, we can use the above BDG inequality in the sequel. We also point
out that in the case of space-time white noise with bounded noise coefficient,
we are able to prove the ULDP in the $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$
topology for $p\geq 2\delta+1$, as we are not using Lemma 2.3 for the
tightness of stochastic convolution.

###  1.6. Organization of the paper

The existence and uniqueness of local as well as global mild solution in the
sense of Walsh of the system (1.1)-(1.2) is discussed in Section 2 (Theorem
2.1, Lemmas 2.2 and 2.3 and Theorem 2.4). In Section 3, we start with the
basics of LDP and state a sufficient Condition 3.3 and recall a general result
for LDP (Theorem 3.4). Then, we move to the proof of the existence and
uniqueness of solutions to the stochastic controlled integral equation (3.1)
and skeleton integral equation (3.1) (Theorems 3.5 and 3.6, respectively).
Later, we prove our main theorem (Theorem 3.7) of this section, that is, LDP
for the laws of solutions to the system (1.1)-(1.2). In Section 4, we first
recall some useful definitions and results from [29] of ULDP and the
sufficient Condition 4.4 which ensures EULP and hence ULDP by Theorem 4.3.
Then, we state and prove several intermediate results (Proposition 4.9,
Corollary 4.10) which help us to obtain our main results (Theorems 4.7 and
4.8) of the section. We obtain the proof of Theorem 4.7 by using uniform
convergence in probability (Theorem 4.11) and Theorem 4.6. Similarly, we prove
Theorem 4.11 by combining convergence in probability in supremum norm (Theorem
4.13) along with Theorem 4.6. In Section 5, we establish the ULDP for the law
of the solution of the system (1.1)-(1.2) with space-time white noise with a
bounded noise coefficient by adopting an approach given in [31]. We wind up
the article by recalling some useful results from [15, 18] in Appendix A.

##  2\.  Solvability Results

In this section, we discuss the solvability of the system (1.1)-(1.2). In
order achieve it, we first establish a local solvability result (Theorem 2.1)
using the truncation defined in (2.1) and a fixed point argument. Then, we
establish the uniform energy estimate (Lemma 2.2) and tightness of stocahstic
convolution term (Lemma 2.3) followed by global solvability results (Theorem
2.4).

###  2.1. Local solution

In order to obtain a mild solution of the system (1.1)-(1.2), we use a
truncation technique. Let $\Pi_{R}(\cdot)$ be a $\mathrm{C}^{1}(\mathbb{R})$
function such that

$\Pi_{R}(r)=\mathopen{}\mathclose{{}\left\\{\begin{aligned} 1,\ \ &\text{ if
}\ \ |r|\leq R,\\\ 0,\ \ &\text{ if }\ \ |r|\geq R+1,\end{aligned}}\right.$
(2.1)

and $\big{|}\frac{\mathrm{d}\Pi_{R}}{\mathrm{d}r}(r)\big{|}\leq 2$, for all
$r\in\mathbb{R}$. Note that by the mean value theorem, one can show that
$|\Pi_{R}(r_{1})-\Pi_{R}(r_{2})|\leq 2|r_{1}-r_{2}|$, for all
$r_{1},r_{2}\in\mathbb{R}$. Let us now introduce the following truncated
system:

$\displaystyle\frac{\partial u_{\epsilon,R}(t,x)}{\partial t}$
$\displaystyle=\nu\frac{\partial^{2}u_{\epsilon,R}(t,x)}{\partial
x^{2}}+\frac{\alpha}{\delta+1}\frac{\partial}{\partial
x}\Pi_{R}(\|u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(t,x))+\beta\Pi_{R}(\|u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(t,x))$
$\displaystyle\quad+\sqrt{\epsilon}\Pi_{R}(\|u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}})g(t,x,u_{\epsilon,R}(t,x))\frac{\partial^{2}\mathrm{W}^{\mathrm{Q}}(t,x)}{\partial
x\partial t},$ (2.2)

for $(t,x)\in(0,T)\times(0,1)$, with the same boundary and initial conditions
given in (1.2). Let us write down the mild form of the above truncated
equation

$\displaystyle u_{\epsilon,R}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,u_{\epsilon,R}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y),\
\ \mathbb{P}\text{-a.s.}$ (2.3)

Set

$\displaystyle\mathscr{A}_{1}u_{\epsilon,R}(t,x)$
$\displaystyle:=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y,$
$\displaystyle\mathscr{A}_{2}u_{\epsilon,R}(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{0}^{1}\frac{\partial G}{\partial
y}(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s,$
$\displaystyle\mathscr{A}_{3}u_{\epsilon,R}(t,x)$
$\displaystyle:=\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s,$
$\displaystyle\mathscr{A}_{4}u_{\epsilon,R}(t,x)$
$\displaystyle:=\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,u_{\epsilon,R}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y),$
$\displaystyle\mathscr{A}u_{\epsilon,R}(t,x)$
$\displaystyle:=\mathscr{A}_{1}u_{\epsilon,R}(t,x)+\frac{\alpha}{\delta+1}\mathscr{A}_{2}u_{\epsilon,R}(t,x)+\beta\mathscr{A}_{3}u_{\epsilon,R}(t,x)+\mathscr{A}_{4}u_{\epsilon,R}(t,x).$
(2.4)

Let us define a Banach space $\mathcal{H}$ consisting of the adapted processes
$u:\Omega\times[0,T]\to\mathrm{L}^{p}([0,1])$ such that

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

where $\lambda$ will be fixed later.

######  Theorem 2.1.

Let us assume that $u^{0}\in\mathrm{L}^{p}([0,1])$, for $p\geq 2\delta+1$.
Then, there exists a unique $\mathrm{L}^{p}([0,1])$-valued
$\mathscr{F}_{t}$-adapted continuous process $u_{\epsilon,R}(\cdot,\cdot)$
satisfying the truncated integral equation (2.1) such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C(R,T).$ (2.5)

###### Proof.

The proof of this theorem is based on fixed point arguments. We divide the
proof into two steps. In the first step, we show that the operator
$\mathscr{A}$ is well defined in the space $\mathcal{H}$. In the second step,
we prove that $\mathscr{A}$ is a contraction map and then the existence and
uniqueness of the truncated integral equation (2.1) follows from the
contraction mapping principle.

Step 1. For $p\geq 2\delta+1$,
$\mathbb{E}\bigg{[}\sup\limits_{t\in[0,T]}\|\mathscr{A}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C(R,T)$.

Using (A.1) and Young’s inequality, we find

$\displaystyle\mathbb{E}\big{[}\|\mathscr{A}_{1}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\big{]}=\bigg{\|}\int_{0}^{1}G(t,\cdot,y)u^{0}(y)\mathrm{d}y\bigg{\|}_{\mathrm{L}^{p}}^{p}\leq
C\|u^{0}\|_{\mathrm{L}^{p}}^{p}.$ (2.6)

Using similar arguments as in Proposition 3.3 [20], we obtain the following
estimates:

$\displaystyle\mathbb{E}\bigg{[}\|\mathscr{A}_{2}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C(R+1)^{p(\delta+1)}\int_{0}^{t}(t-s)^{-1+\frac{p-\delta}{2p}}\mathrm{d}s\leq
C(R+1)^{p(\delta+1)}t^{\frac{p-\delta}{2p}},$ (2.7)
$\displaystyle\mathbb{E}\bigg{[}\|\mathscr{A}_{3}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C\int_{0}^{t}\bigg{\\{}(R+1)^{p(\delta+1)}(1+\gamma)(t-s)^{-\frac{1}{2}+\frac{p-\delta}{2p}}+(R+1)^{p}\gamma$
$\displaystyle\qquad+(R+1)^{p(2\delta+1)}(t-s)^{-\frac{1}{2}+\frac{p-2\delta}{2p}}\bigg{\\}}\mathrm{d}s$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left((R+1)^{p(\delta+1)}(1+\gamma)t^{\frac{1}{2}+\frac{p-\delta}{2p}}+(R+1)^{p}\gamma
t+(R+1)^{p(2\delta+1)}t^{\frac{1}{2}+\frac{p-2\delta}{2p}}}\right),$ (2.8)

for $p\geq 2\delta+1$. Similarly, we consider the term
$\mathbb{E}\big{[}\|\mathscr{A}_{4}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\big{]}$,
and estimate it using Hypothesis 1.1, Fubini’s theorem, BDG (cf. (1.12)),
Hölder’s, Minkowski’s inequalities, (A.1), (A.7) and Young’s inequality for
convolution as

$\displaystyle\mathbb{E}\bigg{[}\|\mathscr{A}_{4}u_{\epsilon,R}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle=\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{\|}\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,u_{\epsilon,R}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle=\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{\|}\sum_{j\in\mathbb{N}}\int_{0}^{t}\big{(}G(t-s,\cdot,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,\cdot,u_{\epsilon,R}(s,y)),q_{j}\varphi_{j}(y)\big{)}\mathrm{d}\beta_{j}(s)\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{\|}\bigg{(}\sum_{j\in\mathbb{N}}\int_{0}^{t}\big{|}\big{(}G(t-s,\cdot,y)\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,u_{\epsilon,R}(s,y)),q_{j}\varphi_{j}(y)\big{)}\big{|}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{\|}\bigg{(}\sum_{j\in\mathbb{N}}\int_{0}^{1}q_{j}^{2}\varphi_{j}^{2}(y)\mathrm{d}y$
$\displaystyle\qquad\qquad\times\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)\Pi_{R}^{2}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g^{2}(s,y,u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s\bigg{)}\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{\|}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)\Pi_{R}^{2}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g^{2}(s,y,u_{\epsilon,R}(s,y))\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{\frac{p}{2}}}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-1}\big{\|}e^{-\frac{|\cdot|^{2}}{a_{1}(t-s)}}*\big{|}\Pi_{R}^{2}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g^{2}(s,\cdot,u_{\epsilon,R}(s,\cdot))\big{|}\big{\|}_{\mathrm{L}^{\frac{p}{2}}}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
CK\epsilon^{\frac{p}{2}}t^{\frac{p-2}{4}}\mathbb{E}\bigg{[}\int_{0}^{t}(t-s)^{-\frac{1}{2}}\Pi_{R}^{p}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\big{(}1+\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{)}\mathrm{d}s\bigg{]}$
$\displaystyle\leq
CK\epsilon^{\frac{p}{2}}\big{(}1+R^{p}\big{)}t^{\frac{p}{4}},$ (2.9)

where we have used the fact that $\\{\varphi_{j}(x)\\}_{j\in\mathbb{N}}$ is an
orthonormal family of $\mathrm{L}^{2}([0,1])$ and
$\sum\limits_{j\in\mathbb{N}}q_{j}^{2}<\infty$. Combining (2.6)-(2.1), the
claim follows.

Step 2. $\mathscr{A}$ is a contraction. Let
$u_{\epsilon,R},v_{\epsilon,R}\in\mathcal{H}$. With no loss of generality, we
can assume that
$\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}\leq\|v_{\epsilon,R}\|_{\mathrm{L}^{p}}$.
Using Taylor’s formula, Hölder’s inequality, the embedding
$\mathrm{L}^{q}([0,1])\hookrightarrow\mathrm{L}^{p}([0,1])$, for $q\geq p$,
mean value theorem, and the definition of mapping $\Pi_{R}(\cdot)$ given in
(2.1), we have

$\displaystyle\|\Pi_{R}(\|u_{\epsilon,R}\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R})-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\mathfrak{c}(v_{\epsilon,R})\|_{\mathrm{L}^{1}}$
$\displaystyle\leq\|(\Pi_{R}((\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}))-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}}))\mathfrak{c}(u_{\epsilon,R})\|_{\mathrm{L}^{1}}+\|\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})(\mathfrak{c}(u_{\epsilon,R})-\mathfrak{c}(v_{\epsilon,R}))\|_{\mathrm{L}^{1}}$
$\displaystyle\leq\|(\Pi_{R}((\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}))-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}}))((1+\gamma)u_{\epsilon,R}^{\delta+1}-\gamma
u_{\epsilon,R}-u_{\epsilon,R}^{2\delta+1})\|_{\mathrm{L}^{1}}$
$\displaystyle\quad+\|\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}}((1+\gamma)(u_{\epsilon,R}^{\delta+1}-v_{\epsilon,R}^{\delta+1})-\gamma(u_{\epsilon,R}-v_{\epsilon,R})-(u_{\epsilon,R}^{2\delta+1}-v_{\epsilon,R}^{2\delta+1}))\|_{\mathrm{L}^{1}}$
$\displaystyle\leq\big{|}\Pi_{R}((\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}))-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{|}\mathopen{}\mathclose{{}\left\\{(1+\gamma)\|u_{\epsilon,R}\|_{\mathrm{L}^{\delta+1}}^{\delta+1}+\gamma\|u_{\epsilon,R}\|_{\mathrm{L}^{1}}+\|u_{\epsilon,R}\|_{\mathrm{L}^{2\delta+1}}^{2\delta+1}}\right\\}$
$\displaystyle\quad+\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{\\{}(1+\gamma)(1+\delta)\|(u_{\epsilon,R}-v_{\epsilon,R})(\theta
u_{\epsilon,R}+(1-\theta)v_{\epsilon,R}))^{\delta}\|_{\mathrm{L}^{1}}$
$\displaystyle\qquad+\gamma\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{1}}+(1+2\delta)\|(u_{\epsilon,R}-v_{\epsilon,R})(\theta
u_{\epsilon,R}+(1-\theta)v_{\epsilon,R}))^{2\delta}\|_{\mathrm{L}^{1}}\big{\\}}$
$\displaystyle\leq\big{|}\Pi_{R}((\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}))-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{|}\mathopen{}\mathclose{{}\left\\{(1+\gamma)\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}^{\delta+1}+\gamma\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}+\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}^{2\delta+1}}\right\\}$
$\displaystyle\quad+\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{\\{}(1+\gamma)(1+\delta)\|\theta
u_{\epsilon,R}+(1-\theta)v_{\epsilon,R})\|_{\mathrm{L}^{\frac{p\delta}{p-1}}}^{\delta}$
$\displaystyle\qquad+\gamma\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}}+(1+2\delta)\|\theta
u_{\epsilon,R}+(1-\theta)v_{\epsilon,R})\|_{\mathrm{L}^{\frac{2p\delta}{p-1}}}^{2\delta}\big{\\}}\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}}$
$\displaystyle\leq\big{|}\Pi_{R}((\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}))-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{|}\mathopen{}\mathclose{{}\left\\{(1+\gamma)\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}^{\delta+1}+\gamma\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}+\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}^{2\delta+1}}\right\\}$
$\displaystyle\quad+\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\big{\\{}(1+\gamma)(1+\delta)(\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}+\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})^{\delta}$
$\displaystyle\quad+\gamma+(1+2\delta)(\|u_{\epsilon,R}\|_{\mathrm{L}^{p}}+\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})^{2\delta}\big{\\}}\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
2\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}}\mathopen{}\mathclose{{}\left\\{(1+\gamma)(R+1)^{\delta+1}+\gamma
R+1+(R+1)^{2\delta+1}}\right\\}$
$\displaystyle\quad+\big{\\{}2^{\delta}(1+\gamma)(1+\delta)(R+1)^{\delta}+\gamma+2^{2\delta}(1+2\delta)(R+1)^{2\delta}\big{\\}}\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(\delta,\gamma,R)\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}},$ (2.10)

for $p\geq 2\delta+1$. Using similar calculations as in the above inequality
yields

$\displaystyle\|\Pi_{R}(\|u_{\epsilon,R}\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R})-\Pi_{R}(\|v_{\epsilon,R}\|_{\mathrm{L}^{p}})\mathfrak{p}(v_{\epsilon,R})\|_{\mathrm{L}^{1}}\leq
C(\delta,R)\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathrm{L}^{p}},$ (2.11)

for $p\geq\delta+1$. Using (2.11), (A.2), (A.7), Minkowski’s, Hölder’s and
Young’s inequalities, we find

$\displaystyle\|\mathscr{A}_{2}(u_{\epsilon,R}(t))-\mathscr{A}_{2}(v_{\epsilon,R}(t))\|_{\mathrm{L}^{p}}$
$\displaystyle\leq\bigg{\|}\int_{0}^{t}\int_{0}^{1}\frac{\partial G}{\partial
y}(t-s,\cdot,y)\big{(}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,y))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(v_{\epsilon,R}(s,y))\big{)}\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-1}\big{\|}e^{\frac{-|\cdot|^{2}}{a_{2}(t-s)}}*\big{|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,\cdot))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(v_{\epsilon,R}(s,\cdot))\big{|}\big{\|}_{\mathrm{L}^{p}}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-1}\|e^{\frac{-|\cdot|^{2}}{a_{2}(t-s)}}\|_{\mathrm{L}^{p}}\big{\|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,\cdot))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(v_{\epsilon,R}(s,\cdot))\big{\|}_{\mathrm{L}^{1}}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-1+\frac{1}{2p}}\big{\|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(u_{\epsilon,R}(s,\cdot))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{p}(v_{\epsilon,R}(s,\cdot))\big{\|}_{\mathrm{L}^{1}}\mathrm{d}s$
$\displaystyle\leq
C(\delta,R)\int_{0}^{t}(t-s)^{-1+\frac{1}{2p}}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s.$
(2.12)

From above inequality, we deduce for all $t\in[0,T]$

$\displaystyle\|\mathscr{A}_{2}(u_{\epsilon,R})-\mathscr{A}_{2}(v_{\epsilon,R})\|_{\mathcal{H}}^{p}$
$\displaystyle\leq C(\delta,R)\sup_{t\in[0,T]}e^{-\lambda
pt}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{2p}}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s\bigg{)}^{p}\bigg{]}$
$\displaystyle\leq
C(\delta,R,p)T^{\frac{p-1}{2p}}\sup_{t\in[0,T]}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{2p}}e^{-\lambda(t-s)}e^{-\lambda
s}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\mathrm{d}s\bigg{)}$
$\displaystyle\leq
C(\delta,R,p,T)\mathopen{}\mathclose{{}\left(\frac{\Gamma(\frac{1}{2p})}{\lambda^{\frac{1}{2p}}}}\right)\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathcal{H}}^{p},$
(2.13)

where we have used Hölder’s inequality and $\Gamma(\cdot)$ denotes the gamma
function.

Using (2.11), (A.2), (A.7), Minkowski’s, Hölder’s and Young’s inequalities, we
find

$\displaystyle\|\mathscr{A}_{3}(u_{\epsilon,R}(t))-\mathscr{A}_{3}(v_{\epsilon,R}(t))\|_{\mathrm{L}^{p}}$
$\displaystyle\leq\bigg{\|}\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)\big{(}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(s,y))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(v_{\epsilon,R}(s,y))\big{)}\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-\frac{1}{2}}\big{\|}e^{\frac{-|\cdot|^{2}}{a_{1}(t-s)}}*\big{|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(s,\cdot))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(v_{\epsilon,R}(s,\cdot))\big{|}\big{\|}_{\mathrm{L}^{p}}\mathrm{d}s$
$\displaystyle\leq\int_{0}^{t}(t-s)^{-1}\|e^{\frac{-|\cdot|^{2}}{a_{2}(t-s)}}\|_{\mathrm{L}^{p}}\big{\|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(u_{\epsilon,R}(s,\cdot))-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})\mathfrak{c}(v_{\epsilon,R}(s,\cdot))\big{\|}_{\mathrm{L}^{1}}\mathrm{d}s$
$\displaystyle\leq
C(\delta,\gamma,R)\int_{0}^{t}(t-s)^{-\frac{1}{2}+\frac{1}{2p}}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s,$
(2.14)

for all $t\in[0,T]$. From the above inequality, we find

$\displaystyle\|\mathscr{A}_{3}(u_{\epsilon,R})-\mathscr{A}_{3}(v_{\epsilon,R})\|_{\mathcal{H}}^{p}$
$\displaystyle\leq
C(\delta,R,p)T^{\frac{p^{2}-1}{2p}}\sup_{t\in[0,T]}\bigg{(}\int_{0}^{t}(t-s)^{-\frac{1}{2}+\frac{1}{2p}}e^{-\lambda(t-s)}e^{-\lambda
s}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\mathrm{d}s\bigg{)}$
$\displaystyle\leq
C(\delta,\gamma,R,p,T)\mathopen{}\mathclose{{}\left(\frac{\Gamma(\frac{1}{2}+\frac{1}{2p})}{\lambda^{\frac{1}{2}+\frac{1}{2p}}}}\right)\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathcal{H}}^{p}.$
(2.15)

Using similar arguments as in (2.1), Hypothesis 1.1 (Lipschitz continuity of
$g$) and Minkowski’s inequality, we obtain

$\displaystyle\mathbb{E}\big{[}\|\mathscr{A}_{4}(u_{\epsilon,R}(t))-\mathscr{A}_{4}(v_{\epsilon,R}(t))\|_{\mathrm{L}^{p}}^{p}\big{]}$
$\displaystyle\leq
C\epsilon^{p}\mathbb{E}\bigg{[}\bigg{(}\bigg{\|}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)\big{(}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,u_{\epsilon,R}(s,y))$
$\displaystyle\qquad-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,y,v_{\epsilon,R}(s,y))\big{)}^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}^{p}\bigg{]}$
$\displaystyle\leq
C(\epsilon,p)\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-1}\big{\|}e^{-\frac{|\cdot|^{2}}{a_{1}(t-s)}}*\big{|}\Pi_{R}(\|u_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,\cdot,u_{\epsilon,R}(s,\cdot))$
$\displaystyle\qquad-\Pi_{R}(\|v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}})g(s,\cdot,v_{\epsilon,R}(s,\cdot))\big{|}^{2}\big{\|}_{\mathrm{L}^{\frac{p}{2}}}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
C(\epsilon,L,R,p)\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
C(\epsilon,L,R,p)\mathopen{}\mathclose{{}\left(\bigg{\\{}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}\bigg{\\}}^{\frac{2}{p}}}\right)^{\frac{p}{2}}$
$\displaystyle\leq
C(\epsilon,L,R,p)\mathopen{}\mathclose{{}\left(\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}\bigg{\\{}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}}^{\frac{2}{p}}\mathrm{d}s}\right)^{\frac{p}{2}}$
$\displaystyle\leq
C(\epsilon,L,R,p)\bigg{\\{}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}e^{2\lambda
s}\mathrm{d}s\bigg{\\}}^{\frac{p}{2}}\sup_{s\in[0,t]}\bigg{\\{}e^{-\lambda
ps}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}}$
$\displaystyle\leq C(\epsilon,L,R,p)\bigg{\\{}e^{2\lambda
t}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}e^{-2\lambda(t-s)}\mathrm{d}s\bigg{\\}}^{\frac{p}{2}}\sup_{s\in[0,t]}\bigg{\\{}e^{-\lambda
ps}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}}$
$\displaystyle\leq
C(\epsilon,L,R,p)\mathopen{}\mathclose{{}\left(\frac{e^{2\lambda
t}\Gamma\big{(}\frac{1}{p}\big{)}}{(2\lambda)^{\frac{1}{p}}}}\right)^{\frac{p}{2}}\sup_{s\in[0,t]}\bigg{\\{}e^{-\lambda
ps}\mathbb{E}\big{[}\|u_{\epsilon,R}(s)-v_{\epsilon,R}(s)\|_{\mathrm{L}^{p}}^{p}\big{]}\bigg{\\}},$
(2.16)

for $p\geq 2$. Thus, from (2.1), one can conclude that

$\displaystyle\|\mathscr{A}_{4}(u_{\epsilon,R})-\mathscr{A}_{4}(v_{\epsilon,R})\|_{\mathcal{H}}^{p}\leq
C(\epsilon,L,R,p)\mathopen{}\mathclose{{}\left(\frac{\Gamma\big{(}\frac{1}{p}\big{)}}{\lambda^{\frac{1}{p}}}}\right)^{\frac{p}{2}}\|u_{\epsilon,R}-v_{\epsilon,R}\|_{\mathcal{H}}^{p}.$
(2.17)

Combining (2.1)-(2.17), and fixing $\lambda>0$ sufficiently large such that

$\displaystyle
C(\alpha,\beta,\gamma,\delta,\epsilon,L,R,p,T)\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\frac{\Gamma(\frac{1}{2p})}{\lambda^{\frac{1}{2p}}}}\right)+\mathopen{}\mathclose{{}\left(\frac{\Gamma(\frac{1}{2}+\frac{1}{2p})}{\lambda^{\frac{1}{2}+\frac{1}{2p}}}}\right)+\mathopen{}\mathclose{{}\left(\frac{\Gamma\big{(}\frac{1}{p}\big{)}}{\lambda^{\frac{1}{p}}}}\right)^{\frac{p}{2}}}\right]<1,$
(2.18)

for $p\geq 2\delta+1$. For the above value of $\lambda$, the map $\mathscr{A}$
is a contraction on the Banach space $\mathcal{H}$. Therefore there exists a
unique fixed point for the map $\mathscr{A}$ by the contraction mapping
principle and this gives the existence of a unique solution of the truncated
integral equation (2.1) and the estimate (2.5) holds from Step 1. ∎

By the previous result we obtain the existence of a unique local mild solution
of the system (1.1)-(1.2) up to a stopping time

$\tau_{R}:=\inf\\{t\geq 0:\|u_{\epsilon}(t)\|_{\mathrm{L}^{p}}\geq R\\}\wedge
T.$

###  2.2. Uniform estimates of the solution

We now move to the proof of the existence of global solution of the integral
equation (1.2). We need uniform energy estimates for our solution. Consider
the following system:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
u_{\epsilon,m}}{\partial
t}(t,x)&=\nu\frac{\partial^{2}u_{\epsilon,m}}{\partial
x^{2}}(t,x)+\frac{\alpha}{\delta+1}\frac{\partial}{\partial
x}\mathfrak{p}_{m}(u_{\epsilon,m}(t,x))+\beta\mathfrak{c}_{m}(u_{\epsilon,m}(t,x))\\\
&\quad+\sqrt{\epsilon}g_{m}(t,x,u_{\epsilon,m}(t,x))\frac{\partial^{2}\mathrm{W}^{\mathrm{Q}}}{\partial
t\partial x},\ \ (t,x)\in(0,T)\times(0,1),\\\
u_{\epsilon,m}(t,0)&=u_{\epsilon,m}(t,1)=0,\\\
u_{\epsilon,m}(0,x)&=u^{0}_{m},\end{aligned}}\right.$ (2.19)

where we take the sequence of bounded Borel measurable functions
$\mathfrak{p}_{m}(t,x,r)=\mathfrak{p}(t,x,r)$,
$\mathfrak{c}_{m}(t,x,r)=\mathfrak{c}(t,x,r)$ and $g_{m}(t,x,r)=g(t,x,r)$, for
$|r|\leq m$ and
$\mathfrak{p}_{m}(t,x,r)=\mathfrak{c}_{m}(t,x,r)=g_{m}(t,x,r)=0$, for $|r|\geq
m+1$ such that they are globally Lipschitz in $r\in\mathbb{R}$. The functions
$\mathfrak{p}_{m}(t,x,r),\ \mathfrak{c}_{m}(t,x,r)$ and $g_{m}(t,x,r)$
inherits the properties of $\mathfrak{p}(t,x,r),\ \mathfrak{c}(t,x,r)$, and
$g(t,x,r)$, respectively whenever $|r|\leq m$. Let us consider a bounded and
smooth sequence $\\{u^{0}_{m}\\}_{m\in\mathbb{N}}$ converging to $u^{0}$ in
the space $\mathrm{L}^{p}([0,1])$. We have the following mild formulation to
the system (2.19):

$\displaystyle u_{\epsilon,m}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}_{m}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\mathfrak{p}_{m}(u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}_{m}(u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g_{m}(s,y,u_{\epsilon,m}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle=:I_{m}^{1}+\frac{\alpha}{\delta+1}I_{\epsilon,m}^{2}+\beta
I_{\epsilon,m}^{3}+\sqrt{\epsilon}I_{\epsilon,m}^{4}.$ (2.20)

Let us now prove a uniform energy estimate which plays a crucial role to
obtain the global solution to the integral equation (1.2).

######  Lemma 2.2.

For each $m\in\mathbb{N},\ p\geq 2\delta+1$, there exists a constant $C>0$
such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,m}(t)\|^{p}_{\mathrm{L}^{p}}+$
$\displaystyle\nu
p(p-1)\int_{0}^{T}\||u_{\epsilon,m}(s)|^{\frac{p-2}{2}}\partial_{x}u_{\epsilon,m}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s+p\beta\int_{0}^{T}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p+2\delta}}^{p+2\delta}\mathrm{d}s\bigg{]}$
$\displaystyle\leq C\big{(}1+\|u^{0}\|_{\mathrm{L}^{p}}^{p}\big{)}.$ (2.21)

###### Proof.

Applying Itô formula to the function $|\cdot|^{p}$ for the process
$u_{\epsilon,m}(\cdot,x)$ for $x\in(0,1)$ and then taking integration over the
spatial domain $[0,1]$ (see Lemma 4.2, [15]), we deduce for all $t\in[0,T]$

$\displaystyle\|u_{\epsilon,m}(t)\|_{\mathrm{L}^{p}}^{p}$
$\displaystyle\leq\|u^{0}_{m}\|_{\mathrm{L}^{p}}^{p}-\nu
p(p-1)\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}\mathopen{}\mathclose{{}\left|\frac{\partial}{\partial
x}u_{\epsilon,m}(s,x)}\right|^{2}\mathrm{d}x\mathrm{d}s$
$\displaystyle\quad+\frac{p(p-1)\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}\frac{\partial}{\partial
x}u_{\epsilon,m}(s,x)\mathfrak{p}_{m}(u_{\epsilon,m}(s,x))\mathrm{d}x\mathrm{d}s$
$\displaystyle\quad+p\beta\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}u_{\epsilon,m}(s,x)\mathfrak{c}_{m}(u_{\epsilon,m}(s,x))\mathrm{d}x\mathrm{d}s$
$\displaystyle\quad+p\sqrt{\epsilon}\sum_{j\in\mathbb{N}}\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}u_{\epsilon,m}(s,x)g_{m}(s,x,u_{\epsilon,m}(s,x))q_{j}\varphi_{j}(x)\mathrm{d}x\mathrm{d}\beta_{j}(s)$
$\displaystyle\quad+\frac{\epsilon}{2}p(p-1)\sum_{j\in\mathbb{N}}\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}g_{m}^{2}(s,x,u_{\epsilon,m}(s,x))q_{j}^{2}\varphi_{j}^{2}(x)\mathrm{d}x\mathrm{d}s$
$\displaystyle\leq\|u^{0}_{m}\|_{\mathrm{L}^{p}}^{p}+I_{1}(t)+I_{2}(t)+I_{3}(t)+I_{4}(t)+I_{5}(t),\
\mathbb{P}\text{-a.s.}$ (2.22)

For $I_{1}(t)$, we have

$\displaystyle I_{1}(t)=-\nu
p(p-1)\int_{0}^{t}\||u_{\epsilon,m}(s)|^{\frac{p-2}{2}}\partial_{x}u_{\epsilon,m}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s.$
(2.23)

Using the fact that $\big{(}\mathfrak{p}(u),|u|^{p-2}u\big{)}=0$ ([27]), we
obtain $I_{2}=0$.

Next, we consider the term $I_{3}(\cdot)$ and estimate it using Hölder’s and
Young’s inequalities as

$\displaystyle I_{3}(t)$
$\displaystyle:=p\beta\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}u_{\epsilon,m}(s,x)\mathfrak{c}_{m}(u_{\epsilon,m}(s,x)\mathrm{d}x\mathrm{d}s$
$\displaystyle=p\beta\int_{0}^{t}\int_{0}^{1}\big{(}(1+\gamma)|u_{\epsilon,m}(s,x)|^{p-2}(u_{\epsilon,m}(s,x))^{\delta+2}-\gamma|u_{\epsilon,m}(s,x)|^{p}$
$\displaystyle\qquad-|u_{\epsilon,m}(s,x)|^{p-2}(u_{\epsilon,m}(t,x))^{\delta+2}\big{)}\mathrm{d}x\mathrm{d}s$
$\displaystyle\leq
p\beta\int_{0}^{t}\big{(}(1+\gamma)\|u_{\epsilon,m}(s)\|^{p+\delta}_{\mathrm{L}^{p+\delta}}-\gamma\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}-\|u_{\epsilon,m}(s)\|^{p+2\delta}_{\mathrm{L}^{p+2\delta}}\big{)}\mathrm{d}s$
$\displaystyle\leq
p\beta\int_{0}^{t}\big{(}(1+\gamma)\|u_{\epsilon,m}(s)\|^{\frac{p+2\delta}{2}}_{\mathrm{L}^{p+2\delta}}\|u_{\epsilon,m}(s)\|^{\frac{p}{2}}_{\mathrm{L}^{p}}-\gamma\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}-\|u_{\epsilon,m}(s)\|^{p+2\delta}_{\mathrm{L}^{p+2\delta}}\big{)}\mathrm{d}s$
$\displaystyle\leq
p\beta\int_{0}^{t}\mathopen{}\mathclose{{}\left(\frac{1}{2}\|u_{\epsilon,m}(s)\|^{{p+2\delta}}_{\mathrm{L}^{p+2\delta}}+\frac{(1+\gamma)^{2}}{2}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}-\gamma\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}-\|u_{\epsilon,m}(s)\|^{p+2\delta}_{\mathrm{L}^{p+2\delta}}}\right)\mathrm{d}s$
$\displaystyle\leq
p\beta\int_{0}^{t}\mathopen{}\mathclose{{}\left(-\frac{1}{2}\|u_{\epsilon,m}(s)\|^{{p+2\delta}}_{\mathrm{L}^{p+2\delta}}+\frac{(1+\gamma^{2})}{2}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}}\right)\mathrm{d}s.$
(2.24)

Using Hypothesis 1.1, we obtain

$\displaystyle|I_{5}(t)|$ $\displaystyle\leq
C(\epsilon,p)\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}g_{m}^{2}(s,x,u_{\epsilon,m}(s,x))x\mathrm{d}x\mathrm{d}s$
$\displaystyle\leq
C(\epsilon,p)\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}\big{(}1+|u_{\epsilon,m}(s,x))|^{2}\big{)}\mathrm{d}x\mathrm{d}s$
$\displaystyle\leq
C(\epsilon,p,T)\mathopen{}\mathclose{{}\left(1+\int_{0}^{t}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}\mathrm{d}s}\right).$
(2.25)

In order to estimate $I_{4}(t)$ we apply BDG (see Theorem 1.1.7, [25]),
Hölder’s and Young’s inequalities to get

$\displaystyle\mathbb{E}$
$\displaystyle\bigg{[}\sup_{t\in[0,T]}\bigg{|}p\sqrt{\epsilon}\sum_{j\in\mathbb{N}}\int_{0}^{t}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p-2}u_{\epsilon,m}(s,x)g_{m}(s,y,u_{\epsilon,m}(s,x))q_{j}\varphi_{j}(x)\mathrm{d}x\mathrm{d}\beta_{j}(s)\bigg{|}\bigg{]}$
$\displaystyle\leq
C(p,\epsilon)\mathbb{E}\mathopen{}\mathclose{{}\left[\sum_{j\in\mathbb{N}}\int_{0}^{T}\mathopen{}\mathclose{{}\left(\int_{0}^{1}g_{m}(s,x,u_{\epsilon,m}(s,x))|u_{\epsilon,m}(s,x)|^{p-1}q_{j}\varphi_{j}(x)\mathrm{d}x}\right)^{2}\mathrm{d}s}\right]^{1/2}$
$\displaystyle\leq
C(p,K,\epsilon)\mathbb{E}\mathopen{}\mathclose{{}\left[\sum_{j\in\mathbb{N}}\int_{0}^{T}\mathopen{}\mathclose{{}\left(\int_{0}^{1}(1+|u_{\epsilon,m}(s,x)|^{p})q_{j}\varphi_{j}(x)\mathrm{d}x}\right)^{2}\mathrm{d}s}\right]^{1/2}$
$\displaystyle\leq C(p,K,\epsilon,,T)$
$\displaystyle\qquad+C(p,K,\epsilon)\mathbb{E}\mathopen{}\mathclose{{}\left[\int_{0}^{T}\bigg{(}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p}\mathrm{d}x\bigg{)}\bigg{(}\sum_{j\in\mathbb{N}}q_{j}^{2}\int_{0}^{1}|u_{\epsilon,m}(s,x)|^{p}\varphi_{j}^{2}(x)\mathrm{d}x\bigg{)}\mathrm{d}s}\right]^{1/2}$
$\displaystyle\leq
C(p,K,\epsilon,T)+C(p,K,\epsilon)\mathbb{E}\mathopen{}\mathclose{{}\left[\int_{0}^{T}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p}}^{2p}\mathrm{d}s}\right]^{1/2}$
$\displaystyle\leq
C(p,K,\epsilon,T)+C(p,K,\epsilon)\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{s\in[0,T]}\|u_{\epsilon,m}(s)\|^{\frac{p}{2}}_{\mathrm{L}^{p}}\mathopen{}\mathclose{{}\left(\int_{0}^{T}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s}\right)^{1/2}}\right]$
$\displaystyle\leq
C(p,K,\epsilon,T)+\frac{1}{2}\mathbb{E}\bigg{[}\sup_{s\in[0,T]}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}\bigg{]}+C(p,K,\epsilon)\int_{0}^{T}\mathbb{E}\big{[}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}\big{]}\mathrm{d}s.$
(2.26)

Combining the estimates for $I_{1}(t)$-$I_{5}(t)$, we deduce

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,m}(t)\|^{p}_{\mathrm{L}^{p}}$
$\displaystyle+2\nu
p(p-1)\int_{0}^{T}\||u_{\epsilon,m}(s)|^{\frac{p-2}{2}}\partial_{x}u_{\epsilon,m}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s+p\beta\int_{0}^{T}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p+2\delta}}^{p+2\delta}\mathrm{d}s\bigg{]}$
$\displaystyle\leq
2\|u^{0}_{m}\|_{\mathrm{L}^{p}}^{p}+C(\epsilon,\alpha,\beta,\gamma,\delta,K,T,p)\int_{0}^{T}\mathbb{E}\big{[}\|u_{\epsilon,m}(s)\|^{p}_{\mathrm{L}^{p}}\big{]}\mathrm{d}s.$

Applying Gronwall’s inequality in the above inequality, we conclude

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,m}(t)\|^{p}_{\mathrm{L}^{p}}+2\nu
p(p-1)$
$\displaystyle\int_{0}^{T}\||u_{\epsilon,m}(s)|^{\frac{p-2}{2}}\partial_{x}u_{\epsilon,m}(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s+p\beta\int_{0}^{T}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p+2\delta}}^{p+2\delta}\mathrm{d}s\bigg{]}$
$\displaystyle\leq C\big{(}1+\|u^{0}_{m}\|_{\mathrm{L}^{p}}^{p}\big{)},$

which completes the proof. ∎

The next task is to prove the tightness of $u_{\epsilon,m}$. The tightness of
the terms $I_{\epsilon,m}^{2}$ and $I_{\epsilon,m}^{3}$, of (2.2) hold from
Lemmas A.1, A.2 and A.3 in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p\geq 2\delta+1$. Let us
establish the tightness of the final term $I_{\epsilon,m}^{4}$ in the right
hand side of (2.2).

######  Lemma 2.3.

Assume that the initial data $u^{0}\in\mathrm{L}^{p}([0,1])$, for $p>6$, then
the stochastic convolution term $I_{\epsilon,m}^{4}$ is uniformly tight in
$\mathrm{C}([0,T]\times[0,1])$.

###### Proof.

The proof of this lemma is based the Arzela-Ascoli theorem and Aldous’
tightness criterion. First, we prove that the stochastic term
$I_{\epsilon,m}^{4}$ is uniformly bounded. We need the following fact about
Green’s function $G$ given in (1.9) (see [38])

$G(t,x,y)=\widetilde{p}(t,x,y)+H(t,x,y),$

where $H(t,x,y)$ is a smooth function in $(t,x,y)$ and $\widetilde{p}$ is the
heat kernel defined as

$\widetilde{p}(t,x,y)=\frac{1}{\sqrt{4\pi
t}}\exp\mathopen{}\mathclose{{}\left(\frac{-(x-y)^{2}}{4t}}\right).$

Using BDG inequality (Theorem 1.1.7, [25]), Hölder’s inequality (with
conjugate exponents $1\leq p_{1}<\frac{3}{2}$ and $1\leq
q_{1}\leq\frac{p}{2}$), Lemma 2.2 and Equation (3.13) in [38], we find for any
$t\in[0,T]$

$\displaystyle\mathbb{E}\bigg{[}\bigg{|}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,m}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)\bigg{|}^{p}\bigg{]}$
$\displaystyle\leq
C\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,x,y)g^{2}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
C\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2p_{1}}(t-s,x,y)\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{p}{2p_{1}}}\mathbb{E}\bigg{[}\bigg{(}\int_{0}^{t}\int_{0}^{1}g^{2q_{1}}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{p}{2q_{1}}}\bigg{]}$
$\displaystyle\leq
C(K)t^{\frac{(3-2q_{1})p}{4q_{1}}+\frac{p}{2q_{1}}-1}\mathbb{E}\bigg{[}\int_{0}^{t}\int_{0}^{1}g^{p}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{]}$
$\displaystyle\leq
C(p,K,T)\bigg{(}1+\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,m}(s)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\bigg{)}$
$\displaystyle\leq C(p,K,T,u^{0}),$ (2.27)

for all $m\in\mathbb{N}$. Note that $I_{\epsilon,m}^{4}(x,t)$ can be written
as

$\displaystyle I_{\epsilon,m}^{4}(x,t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{1}\tilde{p}(t-s,x,y)g_{m}(s,y,u_{\epsilon,m}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}H(t-s,x,y)g_{m}(s,y,u_{\epsilon,m}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle=:I_{\epsilon,m}^{4,1}(x,t)+I_{\epsilon,m}^{4,2}(x,t),$ (2.28)

for $(x,t)\in[0,1]\times[0,T]$. The second term on the right hand side of
(2.2) is a convolution of $\mathrm{W}^{\mathrm{Q}}$ with a smooth function
$H$. It can also shown to be smooth and hence we consider the first term only.
It can be easily seen that

$\displaystyle\big{\\{}\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t+k)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}\big{\\}}^{\frac{1}{p}}$
$\displaystyle\leq
C(p)\big{\\{}\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t+k)-I_{\epsilon,m}^{4,1}(x,t+k)\big{|}^{p}\big{]}\big{\\}}^{\frac{1}{p}}$
$\displaystyle\quad+C(p)\big{\\{}\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t+k)-I_{\epsilon,m}^{4,1}(x+h,t)\big{|}^{p}\big{]}\big{\\}}^{\frac{1}{p}}.$
(2.29)

Let us consider the term
$\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}$,
and estimate it using BDG inequality (Theorem 1.1.7, [25]), Hölder’s
inequality (with conjugate exponents $\frac{p}{2}$ and $q:=\frac{p}{p-2}$),
(2.1) and Lemma 2.2, as

$\displaystyle\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}$
$\displaystyle\leq
C(p)\mathbb{E}\bigg{[}\bigg{|}\int_{0}^{t}\int_{0}^{1}\big{(}\widetilde{p}(t-s,x+h,y)-\widetilde{p}(t-s,x,y)\big{)}^{2}g^{2}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{|}^{\frac{p}{2}}\bigg{]}$
$\displaystyle\leq
C(p)\mathbb{E}\bigg{[}\int_{0}^{t}\int_{0}^{1}g^{p}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{]}$
$\displaystyle\qquad\times\bigg{[}\int_{0}^{t}\int_{0}^{1}\big{|}\widetilde{p}(t-s,x+h,y)-\widetilde{p}(t-s,x,y)\big{|}^{2q}\mathrm{d}y\mathrm{d}s\bigg{]}^{\frac{p}{2q}}$
$\displaystyle\leq
C(p,K,T,u^{0})\bigg{[}\int_{0}^{t}\int_{0}^{1}\frac{1}{(4\pi(t-s))^{q}}\big{|}e^{\frac{-(y-x-h)^{2}}{4(t-s)}}-e^{-\frac{(y-x)^{2}}{4(t-s)}}\big{|}^{2q}\mathrm{d}y\mathrm{d}s\bigg{]}^{\frac{p}{2q}}.$
(2.30)

Fix $y-x=hz$ and $t-s=h^{2}\ell$, in the above inequality, to find

$\displaystyle\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}$
$\displaystyle\leq
C(p,K,T,u^{0})\bigg{[}h^{3-2q}\int_{0}^{t}\int_{0}^{\infty}\ell^{-q}\big{|}e^{-\frac{(z-1)^{2}}{4\ell}}-e^{-\frac{z^{2}}{4\ell}}\big{|}^{2q}\mathrm{d}z\mathrm{d}\ell\bigg{]}^{\frac{p}{2q}}$
$\displaystyle\leq C(p,K,T,u^{0})h^{\frac{p}{2}-3},$ (2.31)

provided $q<\frac{3}{2}$, which implies $p>6$. Thus, we conclude that

$\displaystyle\big{\\{}\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t+k)-I_{\epsilon,m}^{4,1}(x,t+k)\big{|}^{p}\big{]}\big{\\}}^{\frac{1}{p}}\leq
C(p,K,T,u^{0})h^{\frac{1}{2}-\frac{3}{p}},\ \ \text{ for }\ \ p>6.$ (2.32)

Let us now consider the second term of the inequality (2.2). Applying BDG
inequality (Theorem 1.1.7, [25]), we find

$\displaystyle\big{\\{}\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x,t+k)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}\big{\\}}^{\frac{1}{p}}$
$\displaystyle\leq
C(p)\bigg{\\{}\mathbb{E}\bigg{[}\bigg{|}\int_{0}^{t}\int_{0}^{1}\big{(}\widetilde{p}(t+k-s,x,y)-\widetilde{p}(t-s,x,y)\big{)}^{2}g^{2}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{|}^{\frac{p}{2}}\bigg{]}\bigg{\\}}^{\frac{1}{p}}$
$\displaystyle\quad+C(p)\bigg{\\{}\mathbb{E}\bigg{[}\bigg{|}\int_{t}^{t+k}\int_{0}^{1}\widetilde{p}^{2}(t+k-s,x,y)g^{2}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{|}^{\frac{p}{2}}\bigg{]}\bigg{\\}}^{\frac{1}{p}}$
$\displaystyle=:J_{1}+J_{2}.$ (2.33)

We consider the term $J_{1}$ of above inequality, and estimate it using
Hölder’s inequality and similar arguments as in (2.2) by putting $t-s=k\ell$,
$y-x=\sqrt{k}z$, as

$\displaystyle J_{1}$ $\displaystyle\leq
C(p)\bigg{\\{}\mathbb{E}\bigg{[}\int_{0}^{t}\int_{0}^{1}g^{p}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{]}\bigg{\\}}^{\frac{1}{p}}$
$\displaystyle\quad\times\bigg{[}\int_{0}^{t}\int_{0}^{\infty}\big{|}(k\ell+k)^{-\frac{1}{2}}e^{-\frac{z^{2}}{4(\ell+1)}}-(k\ell)^{-\frac{1}{2}}e^{-\frac{z^{2}}{4\ell}}\big{|}^{2q}\mathrm{d}z\mathrm{d}\ell\bigg{]}^{\frac{1}{2q}}$
$\displaystyle\leq
C(p,K,T,u^{0})\bigg{[}k^{\frac{3}{2}-q}\int_{0}^{t}\int_{0}^{\infty}\big{|}(\ell+1)^{-\frac{1}{2}}e^{-\frac{z^{2}}{4(\ell+1)}}-\ell^{-\frac{1}{2}}e^{-\frac{z^{2}}{4\ell}}\big{|}^{2q}\mathrm{d}z\mathrm{d}\ell\bigg{]}^{\frac{1}{2q}}$
$\displaystyle\leq C(p,K,T,u^{0})k^{\frac{1}{4}-\frac{3}{2p}},$ (2.34)

provided $q<\frac{3}{2}$ which implies $p>6$. Finally, we consider the term
$J_{2}$ of (2.2) and estimate it using Hölder’s inequality and Lemma 2.2 as

$\displaystyle J_{2}$ $\displaystyle\leq
C(p)\mathbb{E}\bigg{[}\int_{t}^{t+k}g^{p}(s,y,u_{\epsilon,m}(s,y))\mathrm{d}y\mathrm{d}s\bigg{]}^{\frac{1}{p}}\bigg{[}\int_{0}^{k}\int_{0}^{1}\widetilde{p}^{2q}(t-s,x,y)\mathrm{d}y\mathrm{d}s\bigg{]}^{\frac{1}{2q}}$
$\displaystyle\leq
C(p,K,T,u^{0})k^{\frac{1}{p}}\bigg{[}\int_{0}^{k}\int_{0}^{\infty}(t-s)^{-q}e^{-\frac{q(y-x)^{2}}{t-s}}\mathrm{d}y\mathrm{d}s\bigg{]}^{\frac{1}{2q}}$
$\displaystyle\leq C(p,K,T,u^{0})k^{\frac{1}{4}-\frac{1}{2p}},\ \ \text{ for
}\ \ p>6.$ (2.35)

Combining (2.2)-(2.2), we obtain for $p>6$

$\displaystyle\mathopen{}\mathclose{{}\left\\{\mathbb{E}\big{[}\big{|}I_{\epsilon,m}^{4,1}(x+h,t+k)-I_{\epsilon,m}^{4,1}(x,t)\big{|}^{p}\big{]}}\right\\}^{\frac{1}{p}}$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left[h^{\frac{1}{2}-\frac{3}{p}}+k^{\frac{1}{4}-\frac{3}{2p}}}\right],$
(2.36)

For $p>10$, using Kolmogorov’s continuity Theorem (see Corollary 1.2, [38]) we
obtain a modification of $I_{\epsilon,m}^{4,1}(\cdot,\cdot)$ with
$\mathbb{P}$-almost all trajectories being Hölder continuous.

By an application of Markov’s inequality and using uniform bound of the
stochastic term (cf. (2.2)), we have

$\displaystyle\lim_{M\to\infty}\limsup_{m\in\mathbb{N}}\mathbb{P}\bigg{(}|I_{\epsilon,m}^{4}(t,x)|\geq
M\bigg{)}\leq\lim_{M\to\infty}\frac{C(p,K,T,u^{0})}{M^{p}}=0.$

for all $(t,x)\in[0,T]\times[0,1]$. Furthermore, under the parabolic metric
$|(t,x)|=|t|^{\frac{1}{2}}+|x|$, the condition (2.36) implies for any
$\epsilon_{1}>0$, there exists an
$\eta(\epsilon_{1})=\epsilon_{1}^{\frac{3+\tilde{\delta}}{p}-\frac{1}{2}},$
for some $\tilde{\delta}>\frac{p}{2}-3>0$ such that

$\displaystyle\lim_{\eta\to
0}\limsup_{m\in\mathbb{N}}\mathbb{P}\bigg{(}\sup_{|(t-s,x-y)|\leq\eta}|I_{\epsilon,m}^{4}(t,x)-I_{\epsilon,m}^{4}(s,y)|\geq\epsilon_{1}\bigg{)}\leq\lim_{\eta\to
0}\frac{C^{p}\eta^{\frac{p}{2}-3}}{\epsilon_{1}^{p}}=C^{p}\lim_{\eta\to
0}\eta^{\tilde{\delta}}=0,$

for all $(t,x)\in[0,T]\times[0,1]$. Hence, by an application of Theorem 7.3,
[1] yields that $I_{\epsilon,m}^{4}$ is tight in the space
$\mathrm{C}([0,T]\times[0,1])$. ∎

Combining the tightness obtained above, we conclude that the sequence of the
processes

$\displaystyle
u_{\epsilon,m}=I_{m}^{1}+\frac{\alpha}{\delta+1}I_{\epsilon,m}^{2}+\beta
I_{\epsilon,m}^{3}+\sqrt{\epsilon}I_{\epsilon,m}^{4}$

is uniformly tight in $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ for all $m$
and $p>\max\\{6,2\delta+1\\}$. Let us now move to the global well-posedness of
the integral equation (1.2). A similar result can be found in [15] (cf. [18,
20] also).

######  Theorem 2.4.

Assume that Hypothesis 1.1 holds and the initial data
$u^{0}\in\mathrm{L}^{p}([0,1])$, for $p>\max\\{6,2\delta+1\\}$. Then, there
exists a unique $\mathrm{L}^{p}([0,1])$-valued $\mathscr{F}_{t}$-adapted
continuous process $u_{\epsilon}(\cdot)$ satisfying the integral equation
(1.2) such that

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C(T).$ (2.37)

###### Proof.

The proof of uniqueness is quite easy. Let us assume $u_{\epsilon}(\cdot)$ and
$v_{\epsilon}(\cdot)$ be any two solutions of the integral equation (1.2). Let
us define the following stopping times

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

By Theorem 2.1, we obtain $u_{\epsilon}(t)=v_{\epsilon}(t),\
t\in[0,\tau_{1,R}\wedge\tau_{2,R}]$. By the uniform energy estimate (2.37) and
an application of Markov’s inequality yields

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

Hence, we deduce the uniqueness of the solution on the whole interval $[0,T]$.

Let us now establish the existence of the solution $u_{\epsilon}(\cdot)$ in
the interval $[0,T]$. The main ingredient of this proof is the Skorokhod
representation theorem (see Theorem 3, [11]) and Lemma 1.1, [17]. With the
help Lemmas A.3 and 2.3, we obtain the tightness of $u_{\epsilon,m}(\cdot)$,
for each $m$ and $p>\max\\{6,2\delta+1\\}$. For a given pair of subsequences
$\\{u_{\epsilon,l}\\}$ and $\\{u_{\epsilon,k}\\}$, by Prokhorov’s theorem and
Skorokhod’s representation theorem, there exist subsequences $l_{j}$ and
$k_{j}$ (denoted by $u_{\epsilon,l_{j}},\ u_{\epsilon,k_{j}}$) and a
subsequence of random variables
$z_{\epsilon,j}:=\big{(}\widetilde{u}_{\epsilon,l_{j}},\bar{u}_{\epsilon,k_{j}},\bar{\mathrm{W}}_{j}^{\mathrm{Q}}\big{)}_{j\in\mathbb{N}}\in\mathcal{I}$,
where
$\mathcal{I}:=\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))\times\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))\times\mathrm{C}([0,T]\times[0,1])$,
for $p>\max\\{6,2\delta+1\\}$ in some probability space
$(\bar{\Omega},\bar{\mathscr{F}},\bar{\mathbb{P}})$ such that the sequence
$z_{j}$ converges $\bar{\mathbb{P}}$-a.s. in $\mathcal{I}$ to a random
variable
$z_{\epsilon}:=\big{(}\widetilde{u}_{\epsilon},\bar{u}_{\epsilon},\bar{\mathrm{W}}^{\mathrm{Q}}\big{)}$
as $j\to\infty$, with the same distributions of $z_{\epsilon,j}$ and
$\big{(}u_{\epsilon,l_{j}},u_{\epsilon,k_{j}},\mathrm{W}^{\mathrm{Q}}\big{)}$.
Here, the two random fields $\bar{\mathrm{W}}^{\mathrm{Q}}$ and
$\bar{\mathrm{W}}_{j}^{\mathrm{Q}}$ are Brownian fields defined on different
stochastic bases
$\Xi=(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}\\}_{t\geq
0},\bar{\mathbb{P}})$ and
$\bar{\Xi}_{j}=(\bar{\Omega},\bar{\mathscr{F}},\\{\bar{\mathscr{F}}_{t}^{j}\\}_{t\geq
0},\bar{\mathbb{P}})$, respectively, where $\bar{\mathscr{F}}_{t}$ and
$\bar{\mathscr{F}}_{t}^{j}$ are the completion of the $\sigma$-fields
generated by $z_{\epsilon}(t,x)$ and $z_{\epsilon,j}(t,x)$ for all
$s\in[0,t]$, $x\in[0,1]$, respectively. By Lemma 1.1, [17], we only need to
verify that $z_{\epsilon,j}$ converges to $z_{\epsilon}$ weakly. We know that
weak solution and mild solution are equivalent, interested readers can find a
proof in [18] (see Proposition 3.7). Passing the limit $j\to\infty$ in the
weak formulation (1.3.2) with $\widetilde{u}_{\epsilon,j}(\cdot)$ replaced by
$u_{\epsilon}(\cdot)$, for every smooth function
$\phi\in\mathrm{C}^{2}([0,1])$, with $\phi(0)=\phi(1)=0$, we obtain

$\displaystyle\int_{0}^{1}\widetilde{u}_{\epsilon}(t,y)$
$\displaystyle\phi(y)\mathrm{d}y=\int_{0}^{t}u^{0}(y)\phi(y)\mathrm{d}y+\nu\int_{0}^{t}\int_{0}^{1}\widetilde{u}_{\epsilon}(s,y)\phi^{\prime\prime}(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}\mathfrak{c}(\widetilde{u}_{\epsilon}(t,y))\phi(y)\mathrm{d}y\mathrm{d}s+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\mathfrak{p}(\widetilde{u}_{\epsilon}(s,y))\phi^{\prime}(y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}g(s,y,\widetilde{u}_{\epsilon}(s,y))\phi(y)\bar{\mathrm{W}}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y),\
\ \bar{\mathbb{P}}\text{-a.s.},$ (2.38)

for all $t\in[0,T]$ on the probability space $\bar{\Xi}$. Furthermore, (2.2)
remains valid for $\bar{u}_{\epsilon}$, then by the uniqueness of the solution
we conclude that $\bar{u}_{\epsilon}=\widetilde{u}_{\epsilon}$. Now, applying
Lemma 1.1 [17], we infer that $u_{\epsilon,m}$ converges in
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p>\max\\{6,2\delta+1\\}$ in
probability to some element
$u_{\epsilon}\in\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for
$p>\max\\{6,2\delta+1\\}$. Hence, the proof is over. ∎

######  Remark 2.5.

Let us assume that the initial data $u^{0}\in\mathrm{C}([0,1])$. Using the
above existence and uniqueness result, Lemmas A.1, A.2 and A.3, we deduce that
$u_{\epsilon}$ (solution of the integral equation (1.2)) has a modification
with continuous paths on $[0,T]$ with the values in $\mathrm{C}([0,1])$.

##  3\. Large deviation principle

In this section, first we recall some basic definitions related to the LDP and
then we state a sufficient condition of the LDP, followed by our main Theorem
3.7 of this section.

######  Definition 3.1 (LDP).

Let $\mathrm{I}:\mathcal{E}\to[0,\infty]$ be a rate function on a Polish space
$\mathcal{E}$. The sequence $\\{X_{\epsilon}\\}_{\epsilon\in(0,1]}$ 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 $\mathrm{F}\subset\mathcal{E}$,

$\displaystyle\limsup_{\epsilon\to
0}\epsilon\log\mathbb{P}\big{(}X_{\epsilon}\in\mathrm{F}\big{)}\leq-\inf_{x\in\mathrm{F}}\mathrm{I}(x).$

  2. (2)

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

$\displaystyle\liminf_{\epsilon\to
0}\epsilon\log\mathbb{P}\big{(}X_{\epsilon}\in\mathrm{F}\big{)}\geq-\inf_{x\in\mathrm{O}}\mathrm{I}(x).$

Our proof of LDP for the solution of the system (1.1)-(1.2) is based on a weak
convergence approach. From the work [29], one can obtain the equivalency
between the Laplace principle and LDP.

######  Definition 3.2 (Laplace principle).

The sequence of the random variable $\\{X_{\epsilon}\\}$ on the Polish space
$\mathcal{E}$ is said to satisfy the Laplace principle with the rate function
$\mathrm{I}(\cdot)$ if for each $f\in\mathrm{C}_{b}(\mathcal{E})$ (space of
bounded continuous functions from $\mathcal{E}$ to $\mathbb{R}$),

$\displaystyle\lim_{\epsilon\to
0}\epsilon\log\mathbb{E}\bigg{\\{}\exp\bigg{[}-\frac{f(X_{\epsilon})}{\epsilon}\bigg{]}\bigg{\\}}=-\inf_{x\in\mathcal{E}}\big{\\{}f(x)+\mathrm{I}(x)\big{\\}}.$

On the probability space $(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq
0},\mathbb{P})$, let us define a predictable process
$\phi:\Omega\times[0,T]\to\mathrm{L}^{2}([0,T]\times[0,1])$ and the following
sets:

$\displaystyle\mathfrak{U}^{M}$
$\displaystyle:=\bigg{\\{}\phi\in\mathrm{L}^{2}([0,T]\times[0,1]):\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\leq
M\bigg{\\}},\ M\in\mathbb{N},$ $\displaystyle\mathscr{P}_{2}$
$\displaystyle:=\bigg{\\{}\phi:\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s<\infty,\
\ \mathbb{P}\text{-a.s.}\bigg{\\}},$ $\displaystyle\mathscr{P}_{2}^{M}$
$\displaystyle:=\bigg{\\{}\phi\in\mathscr{P}_{2}:\
\phi(\omega)\in\mathfrak{U}^{M},\ \ \mathbb{P}\text{-a.s.}\bigg{\\}},$

where $\mathfrak{U}^{M}$ is a compact metric space equipped with the weak
topology of $\mathrm{L}^{2}([0,T]\times[0,1])$, $\mathscr{P}_{2}^{M}$ is an
admissible control set. Let $\mathcal{E}$ and $\mathcal{E}_{0}$ be any two
Polish spaces. We assume that the initial data takes values in a compact
subspace of $\mathcal{E}_{0}$ and the solution in $\mathcal{E}$. For every
$\epsilon>0$, define
$\mathscr{G}_{\epsilon}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times[0,1])\to\mathcal{E}$
as a family of measurable maps. Set
$\mathrm{Y}_{\epsilon,u^{0}}=\mathscr{G}_{\epsilon}\big{(}u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}\big{)}$.
Let us provide the sufficient condition for the uniform Laplace principle
(ULP) for the family $\mathrm{Y}_{\epsilon,u^{0}}$.

######  Condition 3.3.

There exists a measurable map
$\mathscr{G}_{0}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times[0,1])\to\mathcal{E}$
such that the following hold:

  1. (1)

For $M\in\mathbb{N}$, let $\phi_{m},\phi\in\mathfrak{U}^{M}$ be such that
$\phi_{m}\to\phi$ and $u^{0}_{m}\to u^{0}$ as $m\to\infty$. Then

$\displaystyle\mathscr{G}_{0}\bigg{(}u^{0}_{m},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi_{m}(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\to\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)},\
\ \text{ as }\ \ m\to\infty.$

  2. (2)

For $\mathrm{M}\in\mathbb{N}$, let
$\phi_{\epsilon},\phi\in\mathscr{P}_{2}^{M}$ be such that $\phi_{\epsilon}$
converges in distribution (as $\mathfrak{U}^{M}$ valued random elements) to
$\phi$, and the initial data $u^{0}_{\epsilon}\to u^{0}$ as $\epsilon\to 0$ in
$\mathcal{E}_{0}$. Then

$\displaystyle\mathscr{G}_{\epsilon}\bigg{(}u^{0}_{\epsilon},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\Rightarrow\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)},\text{
as }\epsilon\to 0,$

where $``\Rightarrow"$ represents the convergence in distribution.

For $\psi\in\mathcal{E}$ and $u^{0}\in\mathcal{E}_{0}$, we define

$\displaystyle\SS_{\psi}:=\bigg{\\{}\phi\in\mathrm{L}^{2}([0,T]\times[0,1]):\
\psi=\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\bigg{\\}},$

and the rate function as

$\mathrm{I}_{u^{0}}(\psi)=\mathopen{}\mathclose{{}\left\\{\begin{aligned}
&\inf_{\psi\in\SS_{\psi}}\bigg{\\{}\frac{1}{2}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}},\
\ &&\text{ if }\ \ \psi\in\mathcal{E},\\\ &+\infty,\ \ &&\text{ if }\ \
\SS_{\psi}=\varnothing,\end{aligned}}\right.$ (3.1)

where

$\displaystyle\inf_{\psi\in\SS_{\psi}}\bigg{\\{}\frac{1}{2}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}}$
$\displaystyle\hskip
28.45274pt=\inf_{\mathopen{}\mathclose{{}\left\\{\phi\in\mathrm{L}^{2}([0,T]\times[0,1]):\psi=\mathscr{G}_{0}(u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s)}\right\\}}\bigg{\\{}\frac{1}{2}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}}$

Let us recall a general criteria for the LDP form [3].

######  Theorem 3.4 (Theorem 7, [3]).

Let
$\mathscr{G}_{0}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times[0,1])\to\mathcal{E}$
be a measurable map, and the Condition 3.3 holds. Then, the family
$\\{\mathrm{Y}_{\epsilon,u^{0}}\\}$ satisfies the Laplace principle on
$\mathcal{E}$ with the rate function $\mathrm{I}_{u^{0}}(\cdot)$ given in
(3.1), uniformly over the initial data $u^{0}$ on the compact subsets of
$\mathcal{E}_{0}$.

If we verify Condition 3.3, then by Theorem 3.4, the family
$\\{\mathrm{Y}_{\epsilon,u^{0}}\\}$ satisfies the Laplace principle and by
equivalency we obtain the family $\\{\mathrm{Y}_{\epsilon,u^{0}}\\}$ satisfies
LDP. Now, our focus is to verify the Condition 3.3.

###  3.1. Controlled and skeleton equations

In this subsection, we discuss controlled and skeleton equations. Let us fix
$\mathcal{E}=\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$,
$\mathcal{E}_{0}=\mathrm{L}^{p}([0,1])$ and the solution map of the integral
equation (1.2) represented by
$u_{\epsilon}=\mathscr{G}_{\epsilon}\big{(}u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}\big{)}$.
Let us denote the solution
$u_{\epsilon,\phi}(t,x):=\mathscr{G}_{\epsilon}\big{(}u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\big{)}$
of the following stochastic controlled integral (SCI) equation

$\displaystyle u_{\epsilon,\phi}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\mathfrak{p}(u_{\epsilon,\phi}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{\epsilon,\phi}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s.$
(3.2)

If we choose $\epsilon=0$, then the above SCI equation reduced to skeleton
equation or limiting case. Let us denote the solution
$u_{0,\phi}(t,x):=\mathscr{G}_{0}\big{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\big{)}$
of the following skeleton equation

$\displaystyle u_{0,\phi}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial y}(t-s,x,y)\mathfrak{p}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{0,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s.$
(3.3)

For $\psi\in\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, we define the rate
function

$\displaystyle\mathrm{I}_{u^{0}}(\psi):=\frac{1}{2}\inf_{\mathopen{}\mathclose{{}\left\\{\phi\in\mathrm{L}^{2}([0,T]\times[0,1]):\psi=\mathscr{G}_{0}(u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s)}\right\\}}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s,$
(3.4)

where $\psi$ satisfies the skeleton equation

$\displaystyle\psi(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial y}(t-s,x,y)\mathfrak{p}(\psi(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(\psi(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,\psi(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s.$
(3.5)

The following results discusses the existence and uniqueness of solution to
the above stochastic controlled and skeleton integral equations.

######  Theorem 3.5.

Assume that Hypothesis 1.1 holds and $\phi\in\mathscr{P}_{2}^{M}$. Then, for
any initial data $u^{0}\in\mathrm{L}^{p}([0,1])$, for
$p>\max\\{6,2\delta+1\\}$, there exists a unique
$\mathrm{L}^{p}([0,1])$-valued $\mathscr{F}_{t}$-adapted continuous process
$u_{\epsilon,\phi}(\cdot,\cdot)$ satisfying SCI equation (3.1).

###### Proof.

The proof of this theorem is based on the Girsanov theorem (Theorem 10.14,
[9]).

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

$\displaystyle\frac{\mathrm{d}\widetilde{\mathbb{P}}}{\mathrm{d}\mathbb{P}}:=\exp\bigg{\\{}-\frac{1}{\sqrt{\epsilon}}\int_{0}^{T}\int_{0}^{1}\phi(s,y)\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)-\frac{1}{2\epsilon}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}},$

where $\mathrm{W}^{\mathrm{Q}}(\cdot,\cdot)$ is a white in time and colored in
space noise and the stochastic integral
$\displaystyle\int_{0}^{t}\int_{0}^{1}\phi(s,y)\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
is considered in the sense of Walsh (see [38]). Then the stochastic process

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

is a white in time and colored in space noise under the probability measure
$\widetilde{\mathbb{P}}$. Also,

$\displaystyle\exp\bigg{\\{}-\frac{1}{\sqrt{\epsilon}}\int_{0}^{T}\int_{0}^{1}\phi(s,y)\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)-\frac{1}{2\epsilon}\int_{0}^{T}\int_{0}^{1}|\phi(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{\\}},$

is an exponential martingale and $\widetilde{\mathbb{P}}$ is an 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 probability measure
$\mathbb{P}$. By Girsanov’s theorem, we conclude that
$\widetilde{\mathrm{W}}^{\mathrm{Q}}(\cdot,\cdot)$ is a real valued Wiener
process with respect to the new probability measure $\widetilde{\mathbb{P}}$,
and using the equivalency of the measure, we find that
$\widetilde{\mathrm{W}}^{\mathrm{Q}}(t,x)$ is a white in time and colored in
space noise. On the probability space
$(\Omega,\mathscr{F},\\{\mathscr{F}_{t}\\}_{t\geq 0},\widetilde{\mathbb{P}})$,
using Theorem 2.4 we ensure the existence and uniqueness of the solution of
the SCI equation (3.1) for the newly constructed probability measure
$\widetilde{\mathbb{P}}$. Hence, we obtain the well-posedness for the SCI
equation (3.1) with respect to the probability measure $\mathbb{P}$ also. ∎

######  Theorem 3.6.

Assume that Hypothesis 1.1, and $\phi\in\mathrm{L}^{2}([0,T]\times[0,1])$.
Then, for any initial data $u^{0}\in\mathrm{L}^{p}([0,1])$, for $p\geq
2\delta+1$, there exists a unique solution
$u_{0,\phi}\in\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ to the skeleton
integral equation (3.1).

###### Proof.

Since $\phi\in\mathrm{L}^{2}([0,T]\times[0,1])$, and

$\displaystyle\mathopen{}\mathclose{{}\left\|\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{0,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s}\right\|_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left(\int_{0}^{t}(t-s)^{\frac{1}{p}-1}\|g(s,u_{0,\phi}(s))\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s}\right)^{1/2}\mathopen{}\mathclose{{}\left(\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s}\right)^{1/2}$
$\displaystyle\leq
CK\|\phi\|_{\mathrm{L}^{2}([0,T]\times[0,1])}\mathopen{}\mathclose{{}\left(\int_{0}^{t}(t-s)^{\frac{1}{p}-1}\big{(}1+\|u_{0,\phi}(s)\|_{\mathrm{L}^{2}}\big{)}^{2}\mathrm{d}s}\right)^{1/2}$
$\displaystyle\leq
CKT^{\frac{1}{2p}}\|\phi\|_{\mathrm{L}^{2}([0,T]\times[0,1])}\sup_{t\in[0,T]}\big{\\{}1+\|u_{0,\phi}(t)\|_{\mathrm{L}^{2}}\big{\\}}<\infty,$

the well-posedness of the skeleton integral equation (3.1) can be obtained in
a similar way as in the proof of Proposition 3.4, [20] (see Section 2 above
also) with some minor modifications in the proof. ∎

###  3.2. Proof of LDP for the solution of the system (1.1)-(1.2)

In this subsection, we state and prove the main result of this section. Let us
first state the result.

######  Theorem 3.7.

Assume that Hypothesis 1.1 holds and the initial data
$u^{0}\in\mathrm{L}^{p}([0,1])$, for $p>\max\\{6,2\delta+1\\}$. Then, there
exists a stochastic process $\\{u_{\epsilon}(t,x):\ t\in[0,T],\ x\in[0,1]\\}$
satisfying the Laplace principle on $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$
with the rate function $\mathrm{I}_{u^{0}}(\cdot)$ defined in (3.1), uniformly
over the initial data $u^{0}$ on the compact subsets of $\mathcal{E}_{0}$.

In view of Theorem 3.4, our focus will be on the verification of Condition
3.3. Validation of Condition 3.3 (2) is easy compared to (1). Therefore, we
are providing a proof of verification of Condition 3.3 (2) only.

###### Proof.

First, we establish the convergence of the controlled process. Let $M>0$, and
$\phi_{\epsilon},\phi\in\mathscr{P}_{2}^{M}$ be such that
$\phi_{\epsilon}\Rightarrow\phi$, and the initial data $u^{0}_{\epsilon}\to
u^{0}$ as $\epsilon\to 0$. Our aim is to show that

$u_{\epsilon,\phi_{\epsilon}}=\mathscr{G}_{\epsilon}\bigg{(}u^{0}_{\epsilon},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\Rightarrow
u_{0,\phi}=\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)},$

as $\epsilon\to 0$, where $u_{0,\phi}$ is the solution of the integral
equation (3.1). In order to establish this, we use compactness arguments.

Consider the following integral equation for $(t,x)\in[0,T]\times[0,1]$:

$\displaystyle u_{\epsilon,\phi_{\epsilon}}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}_{\epsilon}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\mathfrak{p}(u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))\phi_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle=:I_{1,\epsilon}+\frac{\alpha}{\delta+1}I_{2,\epsilon}+\beta
I_{3,\epsilon}+\sqrt{\epsilon}I_{4,\epsilon}+I_{5,\epsilon}.$ (3.6)

For the term $I_{1,\epsilon}$, we have
$\|I_{1,\epsilon}\|_{\mathrm{L}^{p}}\leq
C\|u^{0}_{\epsilon}\|_{\mathrm{L}^{p}}$. Then, applying Markov’s inequality,
we deduce

$\displaystyle\mathbb{P}\big{(}\|I_{1,\epsilon}\|_{\mathrm{L}^{p}}>R\big{)}\leq
R^{-1}\mathbb{E}\big{[}\|I_{1,\epsilon}\|_{\mathrm{L}^{p}}\big{]}\leq
CR^{-1}(1+\|u^{0}\|_{\mathrm{L}^{p}}),$

which implies $I_{1,\epsilon}$ is tight in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for any $p\geq 1$.

Let us consider the term $I_{3,\epsilon}$ and estimate it using (A.1), (A.7),
Minkowski’s and Young’s inequalities as

$\displaystyle\|I_{3,\epsilon}\|_{\mathrm{L}^{p}}=\bigg{\|}\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)\mathfrak{c}(u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\int_{0}^{t}\mathopen{}\mathclose{{}\left\\{(t-s)^{-\frac{\delta}{2p}}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{\delta+1}+\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+(t-s)^{-\frac{\delta}{p}}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{2\delta+1}}\right\\}\mathrm{d}s$
$\displaystyle\leq
C\mathopen{}\mathclose{{}\left\\{T^{1-\frac{\delta}{2p}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{\delta+1}+T\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+T^{1-\frac{\delta}{p}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{2\delta+1}}\right\\},$
(3.7)

for $p\geq 2\delta+1$. In view of Lemma A.3, we only need to show that the
term
$\eta_{\epsilon}=\sup\limits_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{2\delta+1}$
(considering the highest order non-linearity and other terms can be handled in
a same manner) is bounded in probability. Using Markov’s inequality, similar
arguments as in the proofs of Lemma 2.2 (cf. (2.2)) and Theorem 2.4, we obtain

$\displaystyle\lim_{R\to\infty}\sup_{\epsilon\in(0,1]}\mathbb{P}\big{(}\eta_{\epsilon}\geq
R\big{)}=\lim_{R\to\infty}\sup_{\epsilon\in(0,1]}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{2\delta+1}\geq
R\bigg{)}=0.$ (3.8)

Therefore, the term $I_{3,\epsilon}$ is tight in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p\geq 2\delta+1$. Similarly,
using (A.1), (A.7), Minkowski’s and Young’s inequalities, we estimate the term
$I_{2,\epsilon}$ as

$\displaystyle\|I_{2,\epsilon}\|_{\mathrm{L}^{p}}\leq
C\int_{0}^{t}(t-s)^{-\frac{\delta}{2p}-\frac{1}{2}}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{\delta+1}\mathrm{d}s\leq
CT^{\frac{1}{2}-\frac{\delta}{2p}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}^{\delta+1},$
(3.9)

for $p\geq\delta+1$. Again, using similar arguments as in (3.8), we conclude
that $I_{2,\epsilon}$ is tight in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p\geq\delta+1$. Let us move to
the final term $I_{5,\epsilon}$ of the right hand side of (3.2), and estimate
it using (A.1), (A.7), Hypothesis 1.1 (linear growth of $g$), Hölder’s and
Minkowski’s inequalities as

$\displaystyle\|I_{5,\epsilon}\|_{\mathrm{L}^{p}}$
$\displaystyle\leq\bigg{(}\int_{0}^{t}\int_{0}^{1}|\phi_{\epsilon}(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)g^{2}(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\bigg{(}\int_{0}^{t}\int_{0}^{1}|\phi_{\epsilon}(s,y)|^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)g^{2}(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{\frac{p}{2}}}^{\frac{1}{2}}$
$\displaystyle\leq
CT^{\frac{1}{2p}}M^{\frac{1}{2}}K\bigg{(}1+\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}\bigg{)}.$
(3.10)

Thus, the tightness of the term $\ I_{5,\epsilon}$ can be obtained in a
similar way as we have done for $I_{3,\epsilon}$. Also, the tightness of the
penultimate term $I_{4,\epsilon}$ is already established in Lemma 2.3.
Combining the tightness of all terms, we obtain the tightness of
$\\{u_{\epsilon,\phi_{\epsilon}}\\}$ in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p>\max\\{6,2\delta+1\\}$. By
Prokhorov’s theorem, we are able to find a subsequence of controlled process
$\\{u_{\epsilon,\phi_{\epsilon}}\\}$ (still denoted by same index) converges
in distribution to some element say $u_{0,\phi}$ in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p>\max\\{6,2\delta+1\\}$. Now,
we only need to verify that $u_{0,\phi}$ satisfies (3.1), that is, we need to
show that as $\epsilon\to 0$

$\displaystyle I_{1,\epsilon}$
$\displaystyle\Rightarrow\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y,$
$\displaystyle I_{2,\epsilon}$
$\displaystyle\Rightarrow\int_{0}^{t}\int_{0}^{1}\frac{\partial G}{\partial
y}(t-s,x,y)\mathfrak{p}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s,$
$\displaystyle I_{3,\epsilon}$
$\displaystyle\Rightarrow\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s,$
$\displaystyle I_{5,\epsilon}$
$\displaystyle\Rightarrow\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{0,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s,$
$\displaystyle\sqrt{\epsilon}I_{4,\epsilon}$ $\displaystyle\Rightarrow 0,$

in the space $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for
$p>\max\\{6,2\delta+1\\}$. The convergence of the first term $I_{1,\epsilon}$
is trivial, since $u^{0}_{\epsilon}\to u^{0}$ as $\epsilon\to 0$. As
$\\{u_{\epsilon,\phi_{\epsilon}}\\}$ is tight, the family of random elements
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(u_{\epsilon,\phi_{\epsilon}},\mathrm{W}^{\mathrm{Q}},\phi_{\epsilon}}\right)}\right\\}$
is also tight. For
$\mathrm{W}_{\epsilon}^{\mathrm{Q}}:=\mathrm{W}^{\mathrm{Q}}$,
$\epsilon\in(0,1]$, using Skorokhod’s representation theorem (see Theorem 3,
[11]), we can construct a new probability space
$(\widehat{\Omega},\widehat{\mathscr{F}},\\{\widehat{\mathscr{F}}_{t}\\}_{t\geq
0},\widehat{\mathbb{P}})$ and a sequence of random variables
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}},\widehat{\mathrm{W}}^{\mathrm{Q}}_{\epsilon},\widehat{\phi}_{\epsilon}}\right)}\right\\}$
such that

$\displaystyle\mathscr{L}\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}},\widehat{\mathrm{W}}_{\epsilon}^{\mathrm{Q}},\widehat{\phi}_{\epsilon}}\right)}\right\\}=\mathscr{L}\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(u_{\epsilon,\phi_{\epsilon}},\mathrm{W}^{\mathrm{Q}},\phi_{\epsilon}}\right)}\right\\},$

where $\mathscr{L}$ denotes the law, and the sequence of random variables
$\mathopen{}\mathclose{{}\left\\{\mathopen{}\mathclose{{}\left(\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}},\widehat{\mathrm{W}}_{\epsilon}^{\mathrm{Q}},\widehat{\phi}_{\epsilon}}\right)}\right\\}$
converges to a random variable
$\mathopen{}\mathclose{{}\left(\widehat{u}_{0,\widehat{\phi}},\widehat{\mathrm{W}}^{\mathrm{Q}},\widehat{\phi}}\right)$,
$\widehat{\mathbb{P}}$-a.s., in the space

$\displaystyle\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))\times\mathrm{C}([0,T]\times[0,1])\times\mathfrak{U}^{M},\
\ \text{ for }\ \ p>\max\\{6,2\delta+1\\}.$

Moreover, $\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}$ satisfies the
following integral equation:

$\displaystyle\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}_{\epsilon}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\mathfrak{p}(\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}(s,y))\widehat{\mathrm{W}}_{\epsilon}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}(s,y))\widehat{\phi}_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s,\
\widehat{\mathbb{P}}\text{-a.s.},$ (3.11)

and we can use the convergences
$\widehat{u}_{\epsilon,\widehat{\phi}_{\epsilon}}\to\widehat{u}_{0,\widehat{\phi}}$
and $\widehat{\phi}_{\epsilon}\to\widehat{\phi}$, $\widehat{\mathbb{P}}$-a.s.,
to prove that

$\displaystyle\widehat{u}_{0,\widehat{\phi}}(t,x)$
$\displaystyle=\int_{0}^{1}G(t,x,y)u^{0}(y)\mathrm{d}y+\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\mathfrak{p}(\widehat{u}_{0,\widehat{\phi}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(\widehat{u}_{0,\widehat{\phi}}(s,y))\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,\widehat{u}_{0,\widehat{\phi}}(s,y))\widehat{\phi}(s,y)\mathrm{d}y\mathrm{d}s.$
(3.12)

Note that $\mathscr{L}\\{\widehat{\phi}\\}$ and $\mathscr{L}\\{\phi\\}$ are
the same, since $\mathscr{L}\\{\widehat{\phi}_{\epsilon}\\}$ and
$\mathscr{L}\\{\phi_{\epsilon}\\}$ are the same and
$\widehat{\phi}_{\epsilon}\to\widehat{\phi}$, $\widehat{\mathbb{P}}$-a.s.
Therefore, by the uniqueness of the solution of the integral equation (3.2),
we conclude that $\mathscr{L}\\{\widehat{u}_{0,\widehat{\phi}}\\}$ and
$\mathscr{L}\\{u_{0,\phi}\\}$ are the same, which implies
$u_{\epsilon,\phi_{\epsilon}}\Rightarrow
u_{0,\phi}=\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}$.
For simplicity, we assume that $u_{\epsilon,\phi_{\epsilon}}\to u_{0,\phi}$
and $\phi_{\epsilon}\to\phi,\ \widehat{\mathbb{P}}$-a.s., and we need to
verify that
$u_{0,\phi}=\mathscr{G}_{0}\big{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}{\phi}(s,y)\mathrm{d}y\mathrm{d}s\big{)}$
as
$\widehat{u}_{0,\widehat{\phi}}=\mathscr{G}_{0}\big{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\widehat{\phi}(s,y)\mathrm{d}y\mathrm{d}s\big{)}$.

We consider the term $I_{2,\epsilon}$, and estimate it using similar
calculations as in the proof of Proposition 3.3, [20] (see Step 2), as

$\displaystyle\bigg{\|}I_{2,\epsilon}-\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,\cdot,y)\mathfrak{p}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{\delta}{2p}}\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{\delta}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s$
(3.13) $\displaystyle\leq
CT^{\frac{1}{2}-\frac{\delta}{2p}}\bigg{(}\sup_{s\in[0,T]}\big{\\{}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{\\}}^{\delta}\bigg{)}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}},$
(3.14)

for $p\geq\delta+1$. Since $u_{\epsilon,\phi_{\epsilon}}\to u_{0,\phi},\
\widehat{\mathbb{P}}$-a.s., therefore passing $\epsilon\to 0$ in the above
inequality, we obtain

$\displaystyle I_{2,\epsilon}\to\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial y}(t-s,x,y)\mathfrak{p}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s.$
(3.15)

Now, we consider the term $I_{3,\epsilon}$, and estimate it using similar
arguments as in the proof of Proposition 3.3, [20], to find

$\displaystyle\bigg{\|}I_{3,\epsilon}-\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)\mathfrak{c}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C\int_{0}^{t}\bigg{\\{}(t-s)^{-\frac{\delta}{2p}}(1+\delta)(1+\gamma)\big{(}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{\delta}$
$\displaystyle\qquad+\gamma+(t-s)^{-\frac{\delta}{p}}(1+2\delta)\big{(}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{2\delta}\bigg{\\}}\|u_{\epsilon,\phi_{\epsilon}}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s$
(3.16) $\displaystyle\leq
C\bigg{(}T^{1-\frac{\delta}{2p}}(1+\delta)(1+\gamma)\sup_{s\in[0,T]}\big{\\{}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{\\}}^{\delta}+T\gamma$
$\displaystyle\qquad+T^{1-\frac{\delta}{p}}(1+2\delta)\sup_{s\in[0,T]}\big{\\{}\|u_{\epsilon,\phi_{\epsilon}}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{\\}}^{2\delta}\bigg{)}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}},$
(3.17)

for $p\geq 2\delta+1$. Using the convergence $u_{\epsilon,\phi_{\epsilon}}\to
u_{0,\phi},\ \widehat{\mathbb{P}}$-a.s., as $\epsilon\to 0$ in the above
inequality, we conclude that

$\displaystyle
I_{3,\epsilon}\to\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\mathfrak{c}(u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s.$
(3.18)

Now, we consider the term $I_{5,\epsilon}$, and estimate it using the triangle
inequality as

$\displaystyle\bigg{\|}I_{5,\epsilon}-\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)g(s,y,u_{0,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(p)\bigg{\|}\int_{0}^{t}\int_{0}^{1}\big{[}G(t-s,\cdot,y)\big{(}g(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))-g(s,y,u_{0,\phi}(s,y))\big{)}\big{]}\phi_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\quad+C(p)\bigg{\|}\int_{0}^{t}\int_{0}^{1}G(t-s,\cdot,y)g(s,y,u_{0,\phi}(s,y))\big{(}\phi_{\epsilon}(s,y)-\phi(s,y)\big{)}\mathrm{d}y\mathrm{d}s\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle=:I_{5,\epsilon}^{1}+I_{5,\epsilon}^{2}.$ (3.19)

Using Hypothesis 1.1, (A.1), (A.7), Cauchy-Schwarz, Minkowski’s and Young’s
inequalities, we estimate the first term term $I_{5,\epsilon}^{1}$ of the
right hand side of the above inequality as

$\displaystyle\|I_{5,\epsilon}^{1}\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C(p)M^{\frac{1}{2}}\bigg{\|}\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)\big{(}g(s,y,u_{\epsilon,\phi_{\epsilon}}(s,y))-g(s,y,u_{0,\phi}(s,y))\big{)}^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(p,L)T^{\frac{1}{2p}}M^{\frac{1}{2}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}$
$\displaystyle\to 0\ \ \widehat{\mathbb{P}}\text{-a.s.,}\ \ \text{ as }\ \
\epsilon\to 0.$ (3.20)

Using the convergence $\phi_{\epsilon}\to\phi$, $\widehat{\mathbb{P}}$-a.s,
Hypothesis 1.1, Hölder’s, Minkowski’s and Young’s inequalities, we estimate
the final term $I_{5,\epsilon}^{1}$ of the right hand side of (3.2) as

$\displaystyle\|I_{5,\epsilon}^{2}\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C(p)\bigg{(}\int_{0}^{t}\int_{0}^{1}\big{|}\phi_{\epsilon}(s,y)-\phi(s,y)\big{|}^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\qquad\times\bigg{\|}\bigg{(}\int_{0}^{t}\int_{0}^{1}G^{2}(t-s,\cdot,y)g^{2}(s,y,u_{0,\phi}(s,y))\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\bigg{\|}_{\mathrm{L}^{p}}$
$\displaystyle\leq
C(p,K)T^{\frac{1}{2p}}\bigg{(}\int_{0}^{t}\int_{0}^{1}\big{|}\phi_{\epsilon}(s,y)-\phi(s,y)\big{|}^{2}\mathrm{d}y\mathrm{d}s\bigg{)}^{\frac{1}{2}}\sup_{s\in[0,T]}\big{\\{}1+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{\\}}$
$\displaystyle\to 0\ \ \widehat{\mathbb{P}}\text{-a.s.,}\ \ \text{ as }\ \
\epsilon\to 0.$ (3.21)

Combining (3.2)-(3.2), we obtain

$\displaystyle
I_{5,\epsilon}\to\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{0,\phi}(s,y))\phi(s,y)\mathrm{d}y\mathrm{d}s.$
(3.22)

Finally, the process $I_{4,\epsilon}$ has a modification in the space
$\mathrm{C}([0,T]\times[0,1])$ from Lemma 2.3, therefore
$\sqrt{\epsilon}I_{4,\epsilon}\to 0,\ \widehat{\mathbb{P}}$-a.s. Therefore, we
have established that the stochastic controlled process
$u_{\epsilon,\phi}(\cdot)$ (solution of (3.1)) converges to the solution
$u_{0,\phi}(\cdot)$ to the skeleton equation (3.1) along a subsequence. By the
uniqueness, the result holds for every subsequence $\epsilon_{m}$. The
arbitrariness of $\epsilon_{m}$ implies

$\displaystyle
u_{\epsilon,\phi_{\epsilon}}=\mathscr{G}_{\epsilon}\bigg{(}u^{0}_{\epsilon},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi_{\epsilon}(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\Rightarrow
u_{0,\phi}=\mathscr{G}_{0}\mathopen{}\mathclose{{}\left(u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s}\right),$

and Condition 3.3 (2) is verified.

For the verification of Condition 3.3 (1), it is sufficient to combine the
well-posedness of the skeleton equation (4.8) and the convergence of
$u_{0,\phi_{m}}$. Hence, one can conclude the proof of the theorem.. ∎

##  4\. Uniform large deviation principle

In this section, we recall the basic definitions of ULDP (in particular,
Freidlin-Wentzell uniform large deviation principle (FWULDP), see [12]) and
equicontinuous uniform Laplace principle (EULP). Later, we provide a
sufficient condition under which EULP holds. Let us start with the following
additional information.

Let $(\mathcal{E},d)$ be a Polish space and $\mathcal{E}_{0}$ is a set. For
any $u^{0}\in\mathcal{E}_{0}$, define a rate function,
$\mathrm{I}_{u^{0}}:\mathcal{E}\to[0,\infty]$ (cf. [2, 29, 34]). Let
$\Lambda_{u^{0}}(s_{0})$ denote the level set

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

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

If $h:=h(t,x)$ is a random field and $\mathcal{E}$ is a function space (will
fix later), then $h$-a.s. in $\mathcal{E}$ means that $h$ has a stochastic
version in $\mathcal{E}$-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

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

######  Definition 4.1 (FWULDP).

A family of $\mathcal{E}$-valued random variables
$\\{X_{\epsilon,u^{0}}\\}_{u^{0}\in\mathcal{E}_{0},\epsilon\in(0,1]}$ is said
to satisfy a _Freidlin-Wentzell uniform large deviation principle_ with the
speed $a(\epsilon)$ and the rate function $\mathrm{I}_{u^{0}}(\cdot)$,
uniformly over $\mathscr{T}$ if the following hold:

  1. (1)

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

$\displaystyle\liminf_{\epsilon\to
0}\inf_{u^{0}\in\mathrm{A}}\inf_{\psi\in\Lambda_{u^{0}}(t_{0})}\bigg{\\{}a(\epsilon)\log\mathbb{P}\big{(}d(X_{\epsilon,u^{0}},\psi)<\eta\big{)}+\mathrm{I}_{u^{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

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

$\displaystyle\limsup_{\epsilon\to
0}\sup_{u^{0}\in\mathrm{A}}\sup_{s\in[0,t_{0}]}\bigg{\\{}a(\epsilon)\log\mathbb{P}\big{(}\mathrm{dist}_{\mathcal{E}}(X_{\epsilon,u^{0}},\Lambda_{u^{0}}(s))\geq\eta\big{)}+s\bigg{\\}}\leq
0.$

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

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

######  Definition 4.2 (EULP).

A family of $\mathcal{E}$-valued random variables
$\\{X_{\epsilon,u^{0}}\\}_{u^{0}\in\mathcal{E}_{0},\epsilon\in(0,1]}$ is said
to satisfy an _equicontinuous uniform Laplace principle_ with the speed
$a(\epsilon)$ and the rate function $\mathrm{I}_{u^{0}}(\cdot)$, uniformly
over $\mathscr{T}$ if for any $\mathrm{A}\in\mathscr{T}$ and any equibounded
and equicontinuous family $\mathrm{B}\subset\mathrm{C}_{b}(\mathcal{E})$,

$\displaystyle\lim_{\epsilon\to
0}\sup_{h\in\mathrm{B}}\sup_{u^{0}\in\mathrm{A}}\bigg{|}a(\epsilon)\log\mathbb{E}\bigg{[}\exp\bigg{(}-\frac{h(X_{\epsilon,u^{0}})}{a(\epsilon)}\bigg{)}\bigg{]}+\inf_{f\in\mathcal{E}}\big{\\{}h(f)+\mathrm{I}_{u^{0}}(\psi)\big{\\}}\bigg{|}=0.$
(4.1)

An equivalence between ULDP and EULP is established in [29].

######  Theorem 4.3 (Theorem 2.10, [29]).

EULP and FWULDP are equivalent.

Let us provide a sufficient condition for a family of random variables to
satisfy EULP. Then by Theorem 4.3 we obtain FWULDP. For any $M>0$ and $T>0$,
let $\mathscr{P}_{2}^{M}$ be the collection of $\mathscr{F}_{t}$-adapted
processes such that
$\mathbb{P}(\|\phi\|_{\mathrm{L}^{2}([0,T]\times[0,1])}\leq M)=1$ (see Section
3 also).

######  Condition 4.4.

Assume that there exists a family of measurable maps
$\mathscr{G}_{\epsilon}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times[0,1])\to\mathcal{E}$,
indexed by $\epsilon\in(0,1]$, $u^{0}\in\mathcal{E}_{0}$. Let
$\mathrm{W}^{\mathrm{Q}}(\cdot,\cdot)$ be the noise which is white in time and
colored in space and
$X_{\epsilon,\phi}:=\mathscr{G}_{\epsilon}\big{(}u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\big{)}$.
Define
$\mathscr{G}_{0}:\mathcal{E}_{0}\times\mathrm{C}([0,T]\times[0,1])\to\mathcal{E}$,
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 $\mathrm{B}\in\mathscr{T},\ M>0$ and $\eta>0$,

$\displaystyle\lim_{\epsilon\to
0}\sup_{u^{0}\in\mathrm{B}}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\mathopen{}\mathclose{{}\left\\{d\bigg{(}\mathscr{G}_{\epsilon}\bigg{(}u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)},\mathscr{G}_{0}\bigg{(}u^{0},\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s\bigg{)}\bigg{)}>\eta}\right\\}$
$\displaystyle\hskip 284.52756pt=0.$ (4.2)

######  Theorem 4.5 (Theorem 2.13, [29]).

Assume that Condition 4.4 holds. Then, the family
$X_{\epsilon,u^{0}}:=\mathscr{G}_{\epsilon}(u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}})$
satisfies the EULP uniformly over $\mathscr{T}$ with the speed
$a(\epsilon)=\epsilon$, and the rate function
$\mathrm{I}_{u^{0}}:\mathcal{E}\to\mathbb{R}$ given in (3.4).

###  4.1. A priori bound on the controlled process

In this subsection, we introduce the controlled and skeleton systems. For any
$\epsilon\in(0,1]$ and the initial data $u^{0}\in\mathrm{L}^{p}([0,1])$, for
$p>\max\\{6,2\delta+1\\}$, there exists a measurable mapping
$\mathscr{G}_{\epsilon}:\mathrm{L}^{p}([0,1])\times\mathrm{C}([0,T]\times[0,1])\to\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$
such that
$u_{\epsilon}:=\mathscr{G}_{\epsilon}(u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}})$.

Let us consider the following controlled SGBH equation:

$\displaystyle\frac{\partial u_{\epsilon,\phi}(t,x)}{\partial t}$
$\displaystyle=\nu\frac{\partial^{2}u_{\epsilon,\phi}(t,x)}{\partial
x^{2}}+\frac{\alpha}{\delta+1}\frac{\partial}{\partial
x}\mathfrak{p}(u_{\epsilon,\phi}(t,x))+\beta\mathfrak{c}(u_{\epsilon,\phi}(t,x))+g(t,x,u_{\epsilon,\phi}(t,x))\phi(t,x)$
$\displaystyle\quad+\sqrt{\epsilon}g(t,x,u_{\epsilon,\phi}(t,x))\frac{\partial^{2}\mathrm{W}^{\mathrm{Q}}(t,x)}{\partial
x\partial t},$ (4.3)

with initial and boundary conditions given in (1.2) and $g$ satisfying
Hypothesis 1.1.

The global solvability of above controlled SGBH equation can be obtained from
Theorem 2.4, with the help of Girsonov’s theorem (see Theorem 3.5). The
process $u_{\epsilon,\phi}(\cdot)$ satisfies the following SCI equation (3.1).
If the noise is absent in the equation (3.1), then the equation reduces to the
skeleton equation (3.1) and the solution is denoted by $u_{0,\phi}(\cdot)$.
Finally, for any $\psi\in\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ given in
(3.1), the rate function $\mathrm{I}_{u^{0}}(\psi)$ is defined by (3.4).

######  Theorem 4.6 (Existence and uniqueness of the controlled process).

Let $\phi\in\mathscr{P}_{2}^{M},$ for some $M\in\mathbb{N}$. For $\epsilon\geq
0$ and $u^{0}\in\mathrm{L}^{p}([0,1])$, for $p>\max\\{6,2\delta+1\\}$ define a
solution mapping $\mathscr{G}_{\epsilon}$ such that

$\displaystyle\displaystyle
u_{\epsilon,\phi}:=\mathscr{G}_{\epsilon}\mathopen{}\mathclose{{}\left(u^{0},\sqrt{\epsilon}\mathrm{W}^{\mathrm{Q}}+\int_{0}^{\cdot}\int_{0}^{\cdot}\phi(s,y)\mathrm{d}y\mathrm{d}s}\right),$

then $u_{\epsilon,\phi}(\cdot)$ is the unique solution of the SCI equation
(3.1).

###  4.2. Main Theorems

In this subsection, we prove that the family of solutions
$u_{\epsilon}(\cdot)$ of the integral equation (1.2) satisfies ULDP in two
different topologies, $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for
$p>\max\\{6,2\delta+1\\}$ and $\mathrm{C}([0,T]\times[0,1])$.

######  Theorem 4.7 (ULDP in the $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$
topology).

Let $p>\max\\{6,2\delta+1\\}$. The family $\\{u_{\epsilon,\phi}\\}$ satisfies
a ULDP on $\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$ with rate function
$\mathrm{I}_{u^{0}}(\cdot)$ given by (3.4), where the uniformity is over
$\mathrm{L}^{p}([0,1])$-bounded sets of initial conditions, if the following
hold:

  1. (1)

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

$\displaystyle\liminf_{\epsilon\to 0}\inf_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\inf_{\psi\in\Lambda_{u^{0}}(t_{0})}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\|\displaystyle
u_{\epsilon,\phi}-\psi\|_{\mathrm{C}([0,T];\mathrm{L}^{p}([0,1])}<\eta\big{)}+\mathrm{I}_{u^{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

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

$\displaystyle\limsup_{\epsilon\to 0}\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{s\in[0,t_{0}]}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\mathrm{dist}_{\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))}(u_{\epsilon,u^{0}},\Lambda_{u^{0}}(s))\geq\eta\big{)}+s\bigg{\\}}\leq
0.$

######  Theorem 4.8 (ULDP in the $\mathrm{C}([0,T]\times[0,1])$ topology).

The family $\\{\displaystyle u_{\epsilon,\phi}\\}$ satisfies a ULDP on
$\mathrm{C}([0,T]\times[0,1])$ with rate function $\mathrm{I}_{u^{0}}(\cdot)$
given by (3.4), where the uniformity is over $\mathrm{C}([0,1])$-bounded sets
of initial conditions, if the following hold:

  1. (1)

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

$\displaystyle\liminf_{\epsilon\to 0}\inf_{\|u^{0}\|_{\mathrm{C}([0,1])}\leq
N}\inf_{\psi\in\Lambda_{u^{0}}(t_{0})}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\|\displaystyle
u_{\epsilon,\phi}-\psi\|_{\mathrm{C}([0,T]\times[0,1])}<\eta\big{)}+\mathrm{I}_{u^{0}}(\psi)\bigg{\\}}\geq
0.$

  2. (2)

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

$\displaystyle\limsup_{\epsilon\to 0}\sup_{\|u^{0}\|_{\mathrm{C}([0,1])}\leq
N}\sup_{s\in[0,t_{0}]}\bigg{\\{}\epsilon\log\mathbb{P}\big{(}\mathrm{dist}_{\mathrm{C}([0,T]\times[0,1])}(u_{\epsilon,u^{0}},\Lambda_{u^{0}}(s))\geq\eta\big{)}+s\bigg{\\}}\leq
0.$

######  Proposition 4.9 (Global estimate).

For any $p>\max\\{6,2\delta+1\\}$ there exists a constant $C>0$ depending on
$p,M,$ and $T$, such that whenever
$\epsilon\in(0,1],u^{0}\in\mathrm{L}^{p}([0,1])$ and
$\phi\in\mathscr{P}_{2}^{M},$

$\displaystyle\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}^{p}+$
$\displaystyle\int_{0}^{T}\||u_{\epsilon,\phi}(s)|^{\frac{p-2}{p}}\partial_{x}u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{2}}^{2}ds+\int_{0}^{T}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p+2\delta}}^{p+2\delta}ds\bigg{]}$
$\displaystyle\quad\leq C(1+\|u^{0}\|_{\mathrm{L}^{p}}^{p}).$ (4.4)

###### Proof.

Proof can be obtained in a similar way as we did for (2.2) in Lemma 2.2. ∎

######  Corollary 4.10.

The random field
$\sup\limits_{t\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(t)\|_{\mathrm{L}^{p}}^{p}$,
for $p>\max\\{6,2\delta+1\\}$ is bounded in probability, that is, for any
given $T>0$, $M>0$, and $N>0$,

$\displaystyle\lim_{R\to\infty}\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\sup_{\epsilon\in(0,1]}\mathbb{P}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,\phi_{\epsilon}}(t)\|_{\mathrm{L}^{p}}^{p}>R\bigg{]}=0.$
(4.5)

###### Proof.

An application of Markov’s inequality and the use of global estimate (4.9)
leads to the proof of (4.5). ∎

######  Theorem 4.11 (Uniform convergence in probability).

For any $T>0,\ \eta>0,\ N>0$, $M>0$ and $p>\max\\{6,2\delta+1\\}$

$\displaystyle\lim_{\epsilon\to 0}\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\|u_{\epsilon,\phi}-u_{0,\phi}\|_{\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))}>\eta\bigg{)}=0.$
(4.6)

###### Proof.

For any $\phi\in\mathscr{P}_{2}^{M},\ \|u^{0}\|_{\mathrm{L}^{p}}\leq N$, for
$p>\max\\{6,2\delta+1\\}$ and $\epsilon\in(0,1]$, we have

$\displaystyle u_{\epsilon,\phi}(t,x)-$ $\displaystyle u_{0,\phi}(t,x)$
$\displaystyle=\frac{\alpha}{\delta+1}\int_{0}^{t}\int_{0}^{1}\frac{\partial
G}{\partial
y}(t-s,x,y)\big{(}\mathfrak{p}(u_{\epsilon,\phi}(s,y))-\mathfrak{p}(u_{0,\phi}(s,y))\big{)}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\beta\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big{(}\mathfrak{c}(u_{\epsilon,\phi}(s,y))-\mathfrak{c}(u_{0,\phi}(s,y))\big{)}\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)\big{(}g(s,y,u_{\epsilon,\phi}(s,y))-g(s,y,u_{0,\phi}(s,y))\big{)}\phi(s,y)\mathrm{d}y\mathrm{d}s$
$\displaystyle\quad+\sqrt{\epsilon}\int_{0}^{t}\int_{0}^{1}G(t-s,x,y)g(s,y,u_{\epsilon,\phi}(s,y))\mathrm{W}^{\mathrm{Q}}(\mathrm{d}s,\mathrm{d}y)$
$\displaystyle=:\frac{\alpha}{\delta+1}J_{\epsilon}^{1}(t,x)+\beta
J_{\epsilon}^{2}(t,x)+J_{\epsilon}^{3}(t,x)+\sqrt{\epsilon}J_{\epsilon}^{4}(t,x).$
(4.7)

Now, we consider $J_{\epsilon}^{1}(\cdot,\cdot)$ and estimate it using similar
arguments as in (3.13) to find

$\displaystyle\|J_{\epsilon}^{1}(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{\delta}{2p}}\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{\delta}\|u_{\epsilon,\phi}(s)-u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s,$
(4.8)

for $p\geq\delta+1$. For any $P>0,\ \epsilon\in(0,1],\
\phi\in\mathscr{P}_{2}^{M},\ \|u^{0}\|_{\mathrm{L}^{p}}\leq N$ and $T>0$, we
define

$\displaystyle\Upsilon_{P}^{\epsilon}:=\bigg{\\{}\omega\in\Omega:\
\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}\leq P,\
\sup_{t\in[0,T]}\|u_{0,\phi}(t)\|_{\mathrm{L}^{p}}\leq P\bigg{\\}}.$ (4.9)

Then, applying Hölder’s inequality in (4.8) and using (4.9), we arrive at

$\displaystyle\|J_{\epsilon}^{1}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}\leq
C(p,\delta,T,P)\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}(t-s)^{-\frac{1}{2}-\frac{\delta}{2p}}\|u_{\epsilon,\phi}(s)-u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s,$
(4.10)

where $\chi_{A}$ is the characteristic function of the set $A$. Let us
consider the term $J_{\epsilon}^{2}(\cdot,\cdot)$, and estimate it using
similar arguments as in (3.16) to obtain

$\displaystyle\|J_{\epsilon}^{2}(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}\bigg{\\{}(t-s)^{-\frac{\delta}{2p}}(1+\delta)(1+\gamma)\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{\delta}+\gamma$
$\displaystyle\qquad+(t-s)^{-\frac{\delta}{p}}(1+2\delta)\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{p}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\big{)}^{2\delta}\bigg{\\}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\mathrm{d}s,$
(4.11)

for $p\geq 2\delta+1$. By (4.9) and using Hölder’s inequality, we have

$\displaystyle\|J_{\epsilon}^{2}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\leq
C(p,\delta,\gamma,T,P)\chi_{\Upsilon_{P}^{\epsilon}}\int_{0}^{t}\bigg{\\{}(t-s)^{-\frac{\delta}{2p}}(1+\delta)(1+\gamma)+\gamma+(t-s)^{-\frac{\delta}{p}}(1+2\delta)\bigg{\\}}$
$\displaystyle\qquad\times\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s.$
(4.12)

Using Lemma A.1, the fact that $\phi\in\mathscr{P}_{2}^{M}$ and Hölder’s
inequality, we find

$\displaystyle\|J_{\epsilon}^{3}(t)\|_{{\mathrm{L}^{p}}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\leq
C\chi_{\Upsilon_{P}^{\epsilon}}\bigg{\\{}\int_{0}^{t}(t-s)^{-\frac{1}{2}+\frac{1}{2p}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}\|\phi(s)\|_{\mathrm{L}^{2}}\mathrm{d}s\bigg{\\}}^{p}$
$\displaystyle\leq
C\chi_{\Upsilon_{P}^{\epsilon}}\bigg{(}\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}$
$\displaystyle\leq
CM^{\frac{p}{2}}\chi_{\Upsilon_{P}^{\epsilon}}\bigg{(}\int_{0}^{t}(t-s)^{\big{(}-1+\frac{1}{p}\big{)}\big{(}\frac{2}{p}+\frac{p-2}{p}\big{)}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}^{2}\mathrm{d}s\bigg{)}^{\frac{p}{2}}$
$\displaystyle\leq
C(p)M^{\frac{p}{2}}T^{\frac{p-2}{2p}}\chi_{\Upsilon_{P}^{\epsilon}}\bigg{(}\int_{0}^{t}(t-s)^{-1+\frac{1}{p}}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{p}}^{p}\mathrm{d}s\bigg{)}.$
(4.13)

Combining (4.2)-(4.2) and applying Gronwall’s inequality, we find

$\displaystyle\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\leq
C\epsilon^{\frac{p}{2}}\sup_{t\in[0,T]}\|J_{\epsilon}^{4}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}$
$\displaystyle\quad\times\exp\bigg{\\{}C\int_{0}^{T}\mathopen{}\mathclose{{}\left[|t-s|^{-\frac{1}{2}-\frac{\delta}{2p}}+|t-s|^{-\frac{\delta}{2p}}+\gamma+|t-s|^{-\frac{\delta}{p}}+|t-s|^{-1+\frac{1}{p}}}\right]\mathrm{d}s\bigg{\\}}$
$\displaystyle\leq
C(\alpha,\beta,\delta,\gamma,p,T,M,P)\epsilon^{\frac{p}{2}}\sup_{t\in[0,T]}\|J_{\epsilon}^{4}(t)\|_{\mathrm{L}^{p}}^{p}\chi_{\Upsilon_{P}^{\epsilon}}.$
(4.14)

Let $\varpi>0$. Using Proposition 4.9, there exists $P>0$ large enough such
that

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

Using (4.2) and (4.2), for any
$\epsilon\in(0,1],u^{0}\in\mathrm{L}^{p}([0,1])$ and
$\phi\in\mathscr{P}_{2}^{M}$, we obtain

$\displaystyle\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}>\eta\bigg{)}$
$\displaystyle\leq\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\big{[}1-\mathbb{P}(\Upsilon_{P}^{\epsilon})\big{]}+\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\eta\bigg{)}$
$\displaystyle\leq\varpi+\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sqrt{\epsilon}\sup_{t\in[0,T]}\|J_{\epsilon}^{4}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\frac{\eta}{C(\alpha,\beta,\delta,\gamma,p,T,M,P)}\bigg{)}.$
(4.16)

Due to the linear growth of $g$, we have for any $t\in[0,T]$

$\displaystyle\sup_{\epsilon\in(0,1]}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|g(t,\cdot,u_{\epsilon,\phi}(t,\cdot))\|_{\mathrm{L}^{p}}^{p}\bigg{]}\leq
C\bigg{(}1+\sup_{\epsilon\in(0,1]}\mathbb{E}\bigg{[}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)\|_{\mathrm{L}^{p}}^{p}\bigg{]}\bigg{)}<\infty.$
(4.17)

Therefore, Lemma 2.3 ensures that for any $\|u^{0}\|_{\mathrm{L}^{p}}\leq N$
and $\phi\in\mathscr{P}_{2}^{M}$ the family

$\\{J_{\epsilon}^{4}(t):\epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T]\times[0,1])$, for $p>6$. Since we have the
embedding
$\mathrm{C}([0,T]\times[0,1])\hookrightarrow\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$,
the family $\\{J_{\epsilon}^{4}(\cdot)\\}$ is also tight in the space
$\mathrm{C}([0,T];\mathrm{L}^{p}([0,1]))$, for $p>6$. Now, applying
Prokhorov’s Theorem, we get that for any sequences $\epsilon_{m}\rightarrow
0,\phi_{m}\in\mathscr{P}_{2}^{M}$ and $\|u^{0}_{m}\|_{\mathrm{L}^{p}}\leq N$,
there exists a subsequence (relabeled $(\epsilon_{m},\phi_{m},u^{0}_{m})$)
such that $J_{\epsilon_{m}}^{4}(t)$ converges in distribution. Now, if we
multiply by the sequence $\sqrt{\epsilon_{m}}\downarrow 0$, then it is obvious
that

$\sqrt{\epsilon_{m}}\sup_{0\leq t\leq
T}\|J_{\epsilon_{m}}^{4}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon_{m}}}\Rightarrow
0\ \ \text{ as }\ \ m\rightarrow\infty.$

Therefore, the original sequence $\sqrt{\epsilon_{m}}\sup\limits_{0\leq t\leq
T}\|J_{\epsilon_{m}}^{4}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon_{m}}}\Rightarrow
0$ as $m\to\infty$. Since the limit is zero which is a constant, therefore it
converges in probability also, that is,

$\lim_{\epsilon\to
0}\mathbb{P}\bigg{(}\sqrt{\epsilon}\sup_{t\in[0,T]}\|J_{\epsilon}^{4}(t)\|_{\mathrm{L}^{p}}\chi_{\Upsilon_{P}^{\epsilon}}>\frac{\eta}{C(\alpha,\beta,\delta,\gamma,p,T,M,P)}\bigg{)}=0.$

Now, due to arbitrary choice of $\epsilon_{m}\downarrow
0,\phi_{m}\in\mathscr{P}_{2}^{M},$ and $\|u^{0}_{m}\|_{\mathrm{L}^{p}}\leq N$,
we conclude that

$\displaystyle\lim_{\epsilon\to 0}\sup_{\|u^{0}\|_{\mathrm{L}^{p}}\leq
N}\sup_{\phi\in\mathscr{P}_{2}^{M}}\mathbb{P}\bigg{(}\sup_{t\in[0,T]}\|u_{\epsilon,\phi}(t)-u_{0,\phi}(t)\|_{\mathrm{L}^{p}}>\eta\bigg{)}\leq\varpi.$
(4.18)

Since the choice of $\varpi$ is arbitrary, therefore (4.6) holds. ∎

###### Proof of Theorem 4.7.

Combining Theorems 4.6 and 4.11, we obtain the required result. ∎

Let us now move to the proof of Theorem 4.8, that is, the ULDP in the
$\mathrm{C}([0,T]\times[0,1])$ topology.

######  Lemma 4.12.

For any $\|u^{0}\|_{\mathrm{C}([0,1])}\leq N,$ the family

$\mathopen{}\mathclose{{}\left\\{u_{\epsilon,\phi}-u_{0,\phi}:\phi\in\mathscr{P}_{2}^{M},\epsilon\in(0,1]}\right\\}$

is tight in $\mathrm{C}([0,T]\times[0,1])$.

###### Proof.

Let us recall (4.2), that is,

$\displaystyle
u_{\epsilon,\phi}(t,x)-u_{0,\phi}(t,x)=\frac{\alpha}{\delta+1}J_{\epsilon}^{1}(t,x)+\beta
J_{\epsilon}^{2}(t,x)+J_{\epsilon}^{3}(t,x)+\sqrt{\epsilon}J_{\epsilon}^{4}(t,x).$

Using similar calculations as in Lemma A.1 (1), Taylor’s formula and the
embedding
$\mathrm{L}^{q}([0,1])\hookrightarrow\mathrm{L}^{p}([0,1])$, for $q\geq p$, we
obtain

$\displaystyle\|J_{\epsilon}^{1}(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{2}+\frac{1}{2p}}\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{q}}\big{)}^{\delta}\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{q}}\big{)}\mathrm{d}s,$
(4.19)

for $q\in[\delta+1,p]$. Using similar arguments to the above inequality, we
also have

$\displaystyle\|J_{\epsilon}^{2}(t)\|_{\mathrm{L}^{p}}$ $\displaystyle\leq
C\int_{0}^{t}(t-s)^{-\frac{1}{p}+\frac{1}{2p}}\bigg{\\{}(1+\delta)(1+\gamma)\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{q}}\big{)}^{\delta}$
$\displaystyle\quad+\gamma+(1+2\delta)\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{q}}\big{)}^{2\delta}\bigg{\\}}\big{(}\|u_{\epsilon,\phi}(s)\|_{\mathrm{L}^{q}}+\|u_{0,\phi}(s)\|_{\mathrm{L}^{q}}\big{)}\mathrm{d}s,$
(4.20)

for $q\in[2\delta+1,p]$. By Corollary 4.10, we obtain that the terms appearing
in the left hand side of the estimates (4.19) and (4.2) are uniformly bounded
in probability over bounded subsets of initial conditions in
$\mathrm{C}([0,1])$ and for all $\epsilon\in(0,1]$. Therefore, by Lemma A.3,
for all $\phi\in\mathscr{P}_{2}^{M}$ and $\epsilon\in(0,1]$, the family

$\\{J_{\epsilon}^{i}(\cdot):\|u^{0}\|_{\mathrm{C}([0,1])}<R\\}$

are tight in $\mathrm{C}([0,T]\times[0,1])$ for $i=1,2.$ Using (4.17), Lemma
2.3 guarantees that for any $\|u^{0}\|_{\mathrm{C}([0,1])}\leq N$ and
$\phi\in\mathscr{P}_{2}^{M}$ the family

$\\{J_{\epsilon}^{4}(t):\epsilon\in(0,1]\\}$

is tight in $\mathrm{C}([0,T]\times[0,1])$.

The penultimate term $J_{\epsilon}^{3}(\cdot)$ in the right hand side of (4.2)
is not covered by Lemma A.3. Using Lemma 3.1 (iv) and Proposition 3.5, [18],
for $\rho=\infty,\ \gamma=2,\ q=2,\ p=2,\ \kappa=3/4,\ |\delta|=0$ and
$\ell_{1}\in(0,1/2)$, we find

$\displaystyle|J_{\epsilon}^{3}(t,x)-J_{\epsilon}^{3}(t,x+z)|$
$\displaystyle\leq
C|z|^{\ell_{1}}\bigg{(}\int_{0}^{t}\big{\|}\big{(}u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\big{)}\phi(s)\big{\|}_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C|z|^{\ell_{1}}\sup_{s\in[0,t]}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{\infty}}\bigg{(}\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C(M)|z|^{\ell_{1}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{\infty}}.$

Similarly, Hölder’s continuity in time $t$ follows from Lemma 3.1 (iv) and
Proposition 3.5, [18], for $\rho=\infty,\ \gamma=2,\ q=2,\ p=2,\ \kappa=3/4,\
|\delta|=0$ and $\ell_{2}\in(0,1/4)$, since

$\displaystyle\|J_{\epsilon}^{3}(t)-J_{\epsilon}^{3}(s)\|_{\mathrm{L}^{\infty}}$
$\displaystyle\leq
C|t-s|^{\ell_{2}}\bigg{(}\int_{0}^{t}\big{\|}\big{(}u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\big{)}\phi(s)\big{\|}_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C|t-s|^{\ell_{2}}\sup_{s\in[0,t]}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{\infty}}\bigg{(}\int_{0}^{t}\|\phi(s)\|_{\mathrm{L}^{2}}^{2}\mathrm{d}s\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C(M)|t-s|^{\ell_{2}}\sup_{s\in[0,T]}\|u_{\epsilon,\phi}(s)-u_{0,\phi}(s)\|_{\mathrm{L}^{\infty}}.$

By the above estimates, we obtain that $J_{\epsilon}^{3}(\cdot)$ is Hölder
continuous in both time and space and the Hölder norm is also uniformly
bounded over $\epsilon\in(0,1],\ \phi\in\mathscr{P}_{2}^{M}$ and
$\|u^{0}\|_{\mathrm{C}([0,1])}\leq N$. Thus, the family
$\\{J_{\epsilon}^{3}(\cdot):\|u^{0}\|_{\mathrm{C}([0,1])}\leq N,\
\epsilon\in(0,1],\ \phi\in\mathscr{P}_{2}^{M}\\}$ is also tight in the space
$\mathrm{C}([0,T]\times[0,1])$. ∎