# Level of noises and long time behavior of the solution for space-time
fractional SPDE in bounded domains

Jebessa B. Mijena
Department of Mathematics, Georgia College & State University, GA 31061, USA
Email<EMAIL_ADDRESS>Erkan Nane
Department of Mathematics and Statistics, Auburn University, Alabama 36849,
USA
Email<EMAIL_ADDRESS>Alemayehu G. Negash
Department of Mathematics and Statistics, Auburn University, Alabama 36849,
USA
Email<EMAIL_ADDRESS>
###### Abstract

In this paper we study the long time behavior of the solution to a certain
class of space-time fractional stochastic equations with respect to the level
$\lambda$ of a noise and show how the choice of the order $\beta\in(0,\,1)$ of
the fractional time derivative affects the growth and decay behavior of their
solution. We consider both the cases of white noise and colored noise. Our
results extend the main results in Foondun [16] to fractional Laplacian as
well as higher dimensional cases.

Keywords: Space-time fractional SPDE, space-time white noise, space colored
noise, moment bounds in bounded domains.

##  1 Introduction and Statement of the Main results

In 1855, Adolf Fick [11] developed his now-famous principle governing the
transport of mass through diffusive media. In his second law, Fick showed that
$u(t,\,\text{x})$ obeys the classical diffusion equation
$\partial_{t}u=D\partial_{xx}u$ in the spatial dimension, which estimates how
the concentration $u(t,\,\text{x})$ of a diffusive substance varies with space
and time, where $D$ is the diffusion coefficient. The space-time fractional
diffusion equation which is obtained when integer-order derivative operators
in space and time are replaced by fractional counterparts (in Caputo or
Riemann-Liouville sense) has been recently treated by several authors (see,
for example, Mijena and Nane [20], Saichev and Zaslavsky [28], and Mainardi,
Luchko, and Pagnini[19]). The typical form of the space-time fractional
equation is $\partial_{t}^{\beta}u=-(-\Delta)^{\alpha/2}u$, where
$\partial_{t}^{\beta}$ is the Caputo fractional derivative with
$\beta\in(0,\,1)$, $\alpha\in(0,\,2)$ and
$\Delta=\sum_{i=1}^{d}\partial_{x_{i}}^{2}$ is the Laplacian. These equations
can be used to model anomalous diffusion processes or diffusion processes in
non-homogeneous media with random fractal structures (see, for instance,
Meerschaert et al. [22], Meerschaert, Nane, and Vellaisamy [24], Baeumer,
Luks, and Meerschaert [4], Chen, Kim and Kim [7] and the references therein).
Since these space-time fractional equations depend on the fractional
parameters $\beta$ and $\alpha$, in [9] the importance of the continuity of
their solutions with respect to these parameters is discussed. For example, if
the partial derivative in time $\partial_{t}$ in the classical heat equation
$\partial_{t}u=-\Delta u$ is substituted with fractional derivatives
$\partial_{t}^{\beta}$ for $0<\beta<1$, the processes explains the sticking
and trapping behavior of particle, while if the Laplacian $\Delta$ is replaced
with fractional power $-(-\Delta)^{\alpha/2}$ for $0<\alpha<2$, it describes
long particle jumps (see [1]). In [20], Mijena and Nane have recently
introduced time fractional SPDEs, which can be utilized to represent phenomena
with random effects and thermal memory.
Consider the following space-time fractional equation with Dirichlet boundary
conditions (see (2) below for a representation of the solution).

$\left\\{\begin{aligned}
&\partial^{\beta}_{t}u_{t}(\text{x})=-(-\Delta)^{\alpha/2}u(\text{x}),\,\,\text{x}\in
B,\,t>0,\\\ &\,u_{t}(x)=0,\,\,\text{x}\in B^{C},\,\,t>0,\end{aligned}\right.$
(1.1)

where $\alpha\in(0,\,2)$ and $\beta\in(0,\,1)$. The fractional time derivative
is the Caputo derivative which first appeared in [3] and is defined by

$\partial^{\beta}_{t}u_{t}(\text{x})=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}\frac{\partial
u_{r}(\text{x})}{\partial r}\frac{{\rm d}r}{(t-r)^{\beta}}.$ (1.2)

If $u_{0}(\text{x})$ denotes the initial condition to equation (1.1), then the
solution can be written as

$u_{t}(\text{x})=\int_{B}G_{B}^{(\beta)}(t,\,x,\,y)u_{0}(\text{y})\,dy,$

where $G_{B}^{(\beta)}(t,\,x,\,y)$ is the space-time fractional heat kernel
defined in (2.7).
Now consider

$\displaystyle\partial^{\beta}_{t}u_{t}(\text{x})$
$\displaystyle=-(-\Delta)^{\alpha/2}u(\text{x})+f(t,\text{x}),$ (1.3)

with the same initial condition $u_{0}(\text{x})$ and $f(t,\text{x})$ is some
nice function. To get the correct version of (1.3) we will make use of [29,
30, 31]. The fractional Duhamel principle implies that the mild solution to
(1.3) for $t>0$ is given by

$u_{t}(\text{x})=\int_{B}G_{B}^{(\beta)}(t,\,\text{x},\,\text{y})u_{0}(\text{y})\,{\rm
d}\text{y}+\int_{0}^{t}\int_{B}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})\partial_{s}^{1-\beta}f(s,\text{y})\,{\rm
d}\text{y}\,{\rm d}s.$ (1.4)

Using the fractional order integral $I^{\gamma}_{t}$ defined by

$I^{\gamma}_{t}f(t):=\frac{1}{\Gamma(\gamma)}\int_{0}^{t}(t-\tau)^{\gamma-1}f(\tau){\rm
d}\tau,$

and the property

$\partial^{\beta}_{t}I^{\beta}_{t}g(\text{t})=g(\text{t}),$

for every $\beta\in(0,\,1)$, and $g\in L_{\infty}({\mathbb{R}}_{+})$ or $g\in
C({\mathbb{R}}_{+})$, then by the Duhamel’s principle, the mild solution to
(1.3) where the force is $f(t,\,x)=I_{t}^{1-\beta}g(t,\,x)$, will be given by

$\displaystyle u_{t}(\text{x})$
$\displaystyle=\int_{B}G_{B}^{(\beta)}(t,\,x,\,y)u_{0}(\text{y})\,dy+\int_{0}^{t}\int_{B}G_{B}^{(\beta)}(t-s,\,x,\,y)\partial_{s}^{1-\beta}(I^{1-\beta}_{t}g(s,\,\text{y}))\,dy\,ds$
$\displaystyle=\int_{B}G_{B}^{(\beta)}(t,\,x,\,y)u_{0}(\text{y})\,dy+\int_{0}^{t}\int_{B}G_{B}^{(\beta)}(t-s,\,x,\,y)g(s,\,\text{y})\,dy\,ds.$
(1.5)

Recently, Mijena and Nane [20], Foondun, Mijena, and Nane [14], and Foondun
[16] considered the following time fractional stochastic heat equation on the
interval $(0,\,L)$ with Dirichlet boundary condition:

$\left\\{\begin{aligned}
&\partial^{\beta}_{t}u_{t}(x)=\frac{1}{2}\partial_{xx}u_{t}(x)+I_{t}^{1-\beta}[\lambda\sigma(u_{t}(x))\dot{W}(t,\,x)]\;\;\text{for}\;\;0<x<L\;\;\text{and}\;\;t>0\\\
&u_{t}(0)=u_{t}(L)=0\quad\text{for}\quad t>0,\end{aligned}\right.$ (1.6)

where the initial condition $u_{0}:[0,L]\rightarrow{\mathbb{R}}_{+}$ is non-
random and non-negative bounded function which is strictly positive on a set
of positive measures in $[0,\,L]$. $\dot{W}$ denotes a space-time white noise
and $\sigma:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a globally Lipschitz
function satisfying $l_{\sigma}|x|\leqslant|\sigma(x)|\leqslant L_{\sigma}|x|$
where $l_{\sigma}$ and $L_{\sigma}$ are positive constants. $\lambda$ is a
positive parameter known as the level of the noise and will play a significant
part in this paper.
Using Walsh [32], we define the mild solution to (1.6) as the random field
$u=\\{u_{t}(x)\\}_{t>0,x\in B}$ satisfying

$u_{t}(x)=(\mathcal{P}_{B}^{\beta}u_{0})_{t}(x)+\lambda\int_{0}^{L}\int_{0}^{t}p_{B}^{\beta}(t-s,\,x,\,y)\sigma(u_{s}(y))W({\rm
d}s\,,{\rm d}y),$ (1.7)

where $p_{B}^{\beta}(t,x,y)$ denotes the probability density function of the
time-changed killed Brownian motion upon exiting the domain $[0,\,L]$
associated with the fractional time operator, and

$(\mathcal{P}_{B}^{\beta}u_{0})_{t}(x):=\int_{0}^{L}p_{B}^{\beta}(t,\,x,\,y)u_{0}(y){\rm
d}y.$

In Foondun and Nualart [15] and Foondun, Guerngar, and Nane [13], the authors
looked at the behavior of the solution to equation (1.6) for small and large
$\lambda$ when $\boldsymbol{\beta}=1$. They showed that if $\lambda$ is large
enough, the second moment of the solution $u_{t}$ grows exponentially fast;
while if $\lambda$ is small, the second moment of the solution $u_{t}$
eventually decays exponentially. Nualart [27] and Xie [33] have used precise
heat kernel estimates to sharpen the results in [15]. However, in [16] Foondun
has shown that a more complicated situation will occur instead of such phase
transition if fractional time derivative replaces the usual time derivative.
That is, for any fixed $\beta\in(0,\,1)$, the long time behavior of the
supremum of the second moment of the solution to equation (1.6) behaves
differently by considering the cases when $\beta\in(0,\,\frac{1}{2})$ and
$\beta\in(\frac{1}{2},\,1)$ separately, and the reasons for considering these
cases are also explained. For more details about the interpretation of the
results, one can refer [16]. These findings are interesting from an
application standpoint since fractional time derivatives are commonly used in
the modeling of various systems with memory. Therefore, it is very important
to realize that the use of such derivatives can result in considerable change
in the qualitative properties of the solution. The main aim of this work is to
investigate the long time behavior of the solution to (1.8) with respect to
the level of the noise $\lambda$. Our work extend the main results in Foondun
[16] to fractional Laplacian as well as higher dimensional cases.

Consider the following stochastic heat equation on a regular bounded domain
$B$ in ${\mathbb{R}}^{d}$, $d\geqslant 1$ with Dirichlet boundary condition:

$\left\\{\begin{aligned}
&\partial^{\beta}_{t}u_{t}(\text{x})=-\left(-\Delta\right)^{\frac{\alpha}{2}}u_{t}(\text{x})+I_{t}^{1-\beta}[\lambda\sigma(u_{t}(\text{x}))\dot{W}(t,\,\text{x})]\;\;\text{for}\;\;\text{\text{x}}\in
B\;\;\text{and}\;\;t>0\\\ &u_{t}(\text{x})=0\quad\text{for}\quad\text{x}\notin
B\;\;\text{and}\;\;t>0,\end{aligned}\right.$ (1.8)

and the initial condition $u_{0}:B\rightarrow{\mathbb{R}}_{+}$ is a non-random
measurable and bounded function that has support with positive measure inside
B. The operator $-\left(-\Delta\right)^{\frac{\alpha}{2}}$, where
$0<\alpha\leqslant 2$, is the $L^{2}$-generator of a symmetric $\alpha$-stable
process $X_{t}^{B}$ killed when exiting $B$. $\dot{W}$ denotes a space-time
white noise and $\sigma:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a globally
Lipschitz function satisfying
$l_{\sigma}|\text{x}|\leqslant|\sigma(\text{x})|\leqslant
L_{\sigma}|\text{x}|$ where $l_{\sigma}$ and $L_{\sigma}$ are positive
constants. The positive parameter $\lambda$ is called the level of the noise.
$\dot{W}(t,\,\text{x})$ is a space-time white noise with $\text{x}\in B$,
which is assumed to be adapted with respect to a filtered probability space
$(\Omega,\,\mathcal{F},\,\mathcal{F}_{t},\,{\mathbb{P}})$, where $\mathcal{F}$
is complete and the filtration $\\{\mathcal{F}_{t},\,t\geqslant 0\\}$ is right
continuous.
$\dot{W}(t,\,\text{x})$ is a generalized process with covariance given by

${\mathbb{E}}\left[\dot{W}(t,\,\text{x})\dot{W}(s,\,\text{y})\right]=\delta(t-s)\delta(\text{x}-\text{y}).$

That is, $W(f)$ is a random field indexed by function $f\in
L^{2}((0,\,\infty)\times B)$ and for all $f,g\in L^{2}((0,\,\infty)\times B)$,
we have

${\mathbb{E}}\left[W(f)W(g)\right]=\int_{0}^{\infty}\int_{B}f(t,\text{x})g(t,\text{x})\,d\text{x}\,dt.$

Hence $W(f)$ can be represented

$W(f)=\int_{0}^{\infty}\int_{B}f(t,\text{x})W(\,d\text{x}\,dt).$

Note that $W(f)$ is $\mathcal{F}_{t}$-measurable whenever $f$ is supported on
$[0,\,t]\times B$.
The Walsh-Dalang Integrals [8, 32] that is used in equation (1.10) is defined
as follows. We use the Brownian Filtration $\\{\mathcal{F}_{t}\\}$ and the
Walsh-Dalang integrals defined as follows:

  * •

$(t,x)\rightarrow\Phi_{t}(x)$ is an elementary random field when $\exists
0\leqslant a<b$ and an $\mathcal{F}_{a}$-measurable $X\in L^{2}(\Omega)$ and
$\phi\in L^{2}(B)$ such that

$\Phi_{t}(x)=X1_{[a,\,b]}(t)\phi(x)\ \ \,t>0,x\in B.$

  * •

If $h=h_{t}(x)$ is non-random and $\Phi$ is elementary, then

$\int h\Phi dW:=X\int_{(a,\,b)\times B}h_{t}(x)\phi(x)W(dtdx).$

  * •

The stochastic integral is Wiener’s, and it is well defined iff
$h_{t}(x)\phi(x)\in L^{2}([a,\,b]\times B)$.

  * •

We have Walsh isometry,

${\mathbb{E}}\left(\left|\int h\Phi
dW\right|^{2}\right)=\int_{0}^{\infty}ds\int_{B}dy[h_{s}(y)]^{2}{\mathbb{E}}(|\Phi_{s}(y)|^{2}).$
(1.9)

We can make sense of equation (1.8) using Walsh theory [32] again by using the
integral equation below.

$u_{t}(x)=(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})\sigma(u_{s}(\text{y}))W({\rm
d}s\,,{\rm d}\text{y}),$ (1.10)

where $G_{B}^{(\beta)}(t,\text{x},\text{y})$ denotes the heat kernel of the
space-time fractional diffusion equation with Dirichlet boundary conditions in
(1.1), and

$(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(x):=\int_{B}G_{B}^{(\beta)}(t,\,\text{x},\,\text{y})u_{0}(\text{y})\,{\rm
d}\text{y}.$

If $d<(2\wedge\beta^{-1})\alpha$, the proof for the existence of a unique
random-field solution of (1.8) satisfying

$\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}\leqslant
c_{1}e^{c_{2}t\lambda^{\frac{2\alpha}{\alpha-\beta d}}}$

for all $t>0$ can be found in Foondun et al. [14].
Next we state our main results.
We first note that when $\alpha=2$ and $B=(0,\,1)$, (1.8) becomes (1.6). Thus,
our new results are consistent with that obtained in Foondun [16]

######  Theorem 1.1.

Suppose that $d<(2\wedge\beta^{-1})\alpha$. Let $u_{t}$ denote the unique
solution to (1.10). Then the second moment of $u_{t}$ cannot decay
exponentially fast, regardless of what $\lambda$ is. Indeed, if we further
suppose that $\beta\in(0,\,\frac{1}{2}]$, then as $t$ gets large, $\sup_{x\in
B}{\mathbb{E}}|u_{t}(x)|^{2}$ grows exponentially fast for any $\lambda.$

######  Theorem 1.2.

In Theorem 1.1 if $\beta\in(\frac{1}{2},\,1)$, then there exist a strictly
positive real number $\lambda_{u}$ such that for all $\lambda>\lambda_{u}$,
$\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}$ grows exponentially fast as time
gets large.

Since the Mittage-Leffler function $E_{\beta}(-t^{\beta})$ (Gorenfeo et
al.[18]) behaves as a stretched exponential for $t\rightarrow 0$;

$E_{\beta}(-t^{\beta})\backsimeq 1-\frac{t^{\beta}}{\Gamma(\beta+1)}\backsimeq
e^{-\nicefrac{{t^{\beta}}}{{\Gamma(\beta+1)}}},\,\text{for}\,0<t\leqslant 1,$

and as a polynomial decay for $t\rightarrow\infty$,

$E_{\beta}(-t^{\beta})\backsimeq\frac{\sin({\beta\pi})}{\pi}\frac{\Gamma(\beta)}{t^{\beta}},\text{for}\,t\geqslant
1,$

the polynomial decay behaviour of $E_{\beta}(-t^{\beta})$ illustrates the need
for the sharp condition of $\beta\in(0,\,\frac{1}{2}]$ in Theorem 1.1. So the
representation $G_{B}^{(\beta)}$ in equation (2.7) which is defined interms of
$E_{\beta}(.)$ is crucial to our results.

######  Theorem 1.3.

In Theorem 1.1, suppose that $\beta\in(\frac{1}{2},\,1)$. Suppose also that
either $d<\alpha/2\beta$ or $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly
bounded by a constant $C(B)$, then there exist a strictly positive real number
$\lambda_{l}$ such that for $\lambda<\lambda_{l}$ the quantity
$\sup_{t>0}\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}$ is finite.

######  Remark 1.4.

The assumption of a uniform bound $C(B)$ on the eigenfunctions
$\\{\varphi_{n}\\}_{n\geqslant 1}$ look artificial, however there is no known
bound for these eigenfunctions except the cases given in Lemma 2.1 and the
uniform bounds as in Example 2.9 for the Brownian motion in higher dimensional
rectangular boxes.

From the three results above in line with the interpretation given in Foondun
[16], we note that whenever $\beta\in(0,\frac{1}{2}]$, the killed time changed
$\alpha$-stable process $X_{E_{t}}^{B}$ defined in Section 2 reaches the
boundary of $B$ slower which allows the non-linear term to grow exponentially
for any $\lambda$ when $t$ becomes large. On the other hand, for
$\beta\in(\frac{1}{2},\,1)$, this process does not have time to generate such
growth unless $\lambda$ is large enough, since it proceeds quickly enough to
the boundary.
Fractional parameters play an important role in many problems involving space-
time fractional equations. However, in modeling problems, these fractional
parameters are unknown a priori. As a result, continuity of the solutions with
respect to these parameters is essential for modeling purposes. The following
continuity theorem of the unique solution $u_{t}^{\beta}(x)$ of equation (1.5)
with respect to the parameter $\beta$ is Theorem 4.3(b) of Dang $et\,al.$ [9].

######  Theorem 1.5 (Dang, Nane and Nguyen [9]).

Let $u_{t}^{(\gamma)}$ and $u_{t}^{(\beta)}$ denote the solution to the
following equation for parameters $\gamma,\beta\in\left(0,\,1\right)$ with
$\gamma\rightarrow\beta$. The initial condition $u_{0}$ is the same for both
equations.

$\left\\{\begin{aligned} \frac{\partial^{\beta}}{\partial
t^{\beta}}u_{t}(x)&=\Delta u_{t}(x),\,x\in B,\,t>0,\\\
u_{t}(x)&=0,\,x\in\partial D,\,t>0,\\\ u(0,\,x)&=f(x),\,x\in
B.\end{aligned}\right.$ (1.11)

Then, we have

$\lim_{\gamma\rightarrow\beta}||u_{t}^{(\gamma)}(\text{x})-u_{t}^{(\beta)}(\text{x})||_{H}^{2}=0.$

where $H$ is the Hilbert space of all functions with the bounded norm induced
by

$||f||_{H}=\sqrt{\sum_{k=1}^{\infty}\mu_{k}^{2}|\langle
f,\,\varphi_{k}\rangle|^{2}}.$

######  Remark 1.6.

Meerschaert et al. [24] established the existence of a unique solution for
equation (1.11).

The following continuity theorem of the unique solution $u_{t}^{\beta}(x)$ of
equation (1.6) with respect to the parameter $\beta$ is Theorem 1.3 of Foondun
[16].

######  Theorem 1.7 (Foondun [16]).

Let $u_{t}^{(\beta)}$ and $u_{t}$ denote the solution to equation (1.6) and
the solution to equation (1.6) for $\beta=1$ respectively. The initial
condition $u_{0}$ is the same for both equations. Then, for any $p\geqslant
2$, we have

$\lim_{\beta\rightarrow
1}\sup_{\text{x}\in[0,\,L]}{\mathbb{E}}|u_{t}(x)-u_{t}^{(\beta)}(x)|^{p}=0.$

In the next theorem, we show the continuity of the solution $u_{t}(x)$ of
(1.10) with respect to the fractional parameter $\beta$.

######  Theorem 1.8.

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$. For
$d<\frac{1}{2}\min\\{\frac{1}{\beta},\,\frac{1}{\gamma}\\}\alpha$, let
$u_{t}^{(\gamma)}$ and $u_{t}^{(\beta)}$ denote solutions to equation (1.10)
for parameters $\beta,\gamma\in\left(\frac{1}{2},\,1\right)$ respectively with
$\gamma\rightarrow\beta$. The initial condition $u_{0}$ is the same for both
equations. Then, for any $p\geqslant 2$, we have

$\lim_{\gamma\rightarrow\beta}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}^{(\gamma)}(\mathrm{x})-u_{t}^{(\beta)}(\mathrm{x})|^{p}=0.$

Here we observe that Theorem 1.8 extends Theorem 1.3 in Foondun [16] to
fractional Laplacian case, and Theorem 4.3(b) of Dang et al. [9] to stochastic
case.

The other class of equations that we consider in this paper is equation with
space colored noise stated as

$\left\\{\begin{aligned}
&\partial^{\beta}_{t}u_{t}(\text{x})=-\left(-\Delta\right)^{\frac{\alpha}{2}}u_{t}(\text{x})+I_{t}^{1-\beta}[\lambda\sigma(u_{t}(\text{x}))\dot{F}(t,\,\text{x})]\;\;\text{for}\;\;\text{\text{x}}\in
B\;\;\text{and}\;\;t>0\\\ &u_{t}(\text{x})=0\quad\text{for}\quad\text{x}\notin
B\;\;\text{and}\;\;t>0,\end{aligned}\right.$ (1.12)

and the initial condition $u_{0}:B\rightarrow{\mathbb{R}}_{+}$ is a non-random
measurable and bounded function that has support with positive measure inside
B. The operator $-\left(-\Delta\right)^{\frac{\alpha}{2}}$, where
$0<\alpha\leqslant 2$, is the $L^{2}$-generator of a symmetric $\alpha$-stable
process $X_{t}^{B}$ killed when exiting $B$. The function
$\sigma:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a globally Lipschitz function
satisfying $l_{\sigma}|\text{x}|\leqslant|\sigma(\text{x})|\leqslant
L_{\sigma}|\text{x}|$ where $l_{\sigma}$ and $L_{\sigma}$ are positive
constants. The positive parameter $\lambda$ is called the level of the noise.
The noise $\dot{F}(t,\,x)$ is white in time and colored in space satisfying

$\operatorname{Cov}(\dot{F}(t,\,x),\dot{F}(s,\,y))=\delta_{0}(t-s)f(x,y),$

where $0<f(x,\,y)\leqslant g(x-y)$ and $g$ is a locally integrable function on
${\mathbb{R}}^{d}$ with possible singularity at $0$ satisfying

$\int_{{\mathbb{R}}^{d}}\frac{\hat{g}(\xi)}{1+|\xi|^{\alpha}}{\rm
d}\xi<\infty,$ (1.13)

where $\hat{g}$ denotes the Fourier transform of $g$.

We will need the following non degeneracy condition on the spatial correlation
of the noise.

######  Assumption 1.9.

Assume there exists some positive number $K_{f}$ such that

$\inf_{x,\,y\in B}f(x,\,y)\geqslant K_{f}.$

This assumption is very mild as it is shown by the following examples.

######  Example 1.10.

For the following list of examples Assumption 1.13 is satisfied.

  * •

Riesz Kernel:

$f(x,y)=\frac{1}{|x-y|^{\gamma}}\,\,\text{with}\,\,\gamma<d\wedge\alpha.$

  * •

The Exponential-type kernel: $f(x,y)=exp[-(x\cdot y)]$.

  * •

The Ornstein-Uhlenbeck-type kernels: $f(x,\,y)=exp[-|x-y|^{\delta}]$ with
$\delta\in(0,\,2]$.

  * •

Poisson Kernels:

$f(x,y)=\left(\frac{1}{|x-y|^{2}+1}\right)^{\frac{d+1}{2}}.$

  * •

Cauchy Kernels:

$f(x,y)=\sum_{j=1}^{d}\left(\frac{1}{1+(x_{j}-y_{j})^{2}}\right).$

Following Walsh [32], $u_{t}$ is a mild solution to (1.12) if

$u_{t}(x)=(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})\sigma(u_{s}(\text{y}))F({\rm
d}s\,,{\rm d}\text{y}),$ (1.14)

where $G_{B}^{(\beta)}(t,\text{x},\text{y})$ denotes the heat kernel of the
space-time fractional diffusion equation with Dirichlet boundary conditions in
(1.12), and

$(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(x):=\int_{B}G_{B}^{(\beta)}(t,\,\text{x},\,\text{y})u_{0}(\text{y})\,{\rm
d}\text{y}.$

######  Theorem 1.11.

Suppose that the Dalang condition (1.13) holds. Let $u_{t}$ denote the unique
solution to (1.10). Then no matter what $\lambda$ is, the second moment of
$u_{t}$ cannot decay exponentially fast. In fact, if we further assume that
$\beta\in(0,\,\frac{1}{2}]$ and Assumption 1.9 holds, then as $t$ gets large,
$\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}$ grows exponentially fast for any
$\lambda.$

######  Theorem 1.12.

In Theorem 1.11, if $\beta\in(\frac{1}{2},\,1)$ and Assumption 1.9 hold, then
there exist a strictly positive real number $\lambda_{u}$ such that for all
$\lambda>\lambda_{u}$, $\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}$ grows
exponentially fast as time gets large.

######  Theorem 1.13.

In Theorem 1.11, suppose that $\beta\in(\frac{1}{2},\,1)$,
$d<\frac{\alpha}{2\beta}$, and $\int_{B\times B}f(x,\,y)\,{\rm d}x\,{\rm
d}y<\infty$. If $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$, then there exist a strictly positive real number
$\lambda_{l}$ such that for $\lambda<\lambda_{l}$ the quantity
$\sup_{t>0}\sup_{x\in B}{\mathbb{E}}|u_{t}(x)|^{2}$ is finite.

######  Corollary 1.14.

In Theorem 1.13, if $d=1$ and $\alpha=2$ then the conclusion of theorem
follows.

######  Corollary 1.15.

In Theorem 1.13, if $d=1$ and $f$ is Riesz Kernel function, then the
conclusion of theorem follows.

######  Theorem 1.16.

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$, and
$\int_{B\times B}f(x,\,y)\,{\rm d}x\,{\rm d}y<\infty$. For
$d<\frac{1}{2}\min\\{\frac{1}{\beta},\,\frac{1}{\gamma}\\}\alpha$, let
$u_{t}^{(\beta)}$ and $u_{t}^{(\gamma)}$ denote solutions to (1.10) for
parameters $\beta,\,\gamma\in\left(\frac{1}{2},\,1\right)$ with
$\gamma\rightarrow\beta$. The initial condition $u_{0}$ is the same for both
equations. Then, for any $p\geqslant 2$, we have

$\lim_{\gamma\rightarrow\beta}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}^{(\gamma)}(\text{x})-u_{t}^{(\beta)}(\text{x})|^{p}=0.$

We now briefly give an outline of the paper. In this paper we employ similar
methods as in Foondun [16] with crucial changes to prove our main results.
This method was also used in [12] and [15] among others. Section 2 contains
estimates and some preliminary results needed for the proof of main results.
Section 3 is devoted to the proof of Theorem 1.1, Theorem 1.2, and Theorem 1.3
while the proofs of Theorem 1.8 is given in Section 4. Further, in Section 5
we give the proof of Theorem 1.11, Theorem 1.12, Theorem 1.13, and Theorem
1.16.

##  2 Preliminaries

In this section we give some preliminary results, which are needed for the
proofs of the main theorems.
Let $X_{t}$ denote a symmetric stable process of index $\alpha\in(0,2)$ in
${\mathbb{R}}^{d}$ and $B$ be a regular bounded open subset of
${\mathbb{R}}^{d}$. Let $X_{t}^{B}$ denote the symmetric stable process killed
upon exiting $B$. The following Hilbert-Schmidt expansion (see Davies [10] )
holds for the probability density function $p_{B}$ of $X_{t}^{B}$,

$p_{B}(t,\,\text{x},\,\text{y}):=\sum_{n=1}^{\infty}e^{-\mu_{n}t}\varphi_{n}(\text{x})\varphi_{n}(\text{y}),$
(2.1)

for all $\text{x},\,\text{y}\in B,\,t>0,$ where $\\{\varphi_{n}\\}_{n\geqslant
0}$ is an orthonormal basis of $L^{2}(B)$, and
$0<\mu_{1}<\mu_{2}\leqslant\mu_{3}\leqslant\cdots$ is a sequence of positive
real numbers such that, for every $n\geqslant 1$,
$P_{t}^{B}\varphi_{n}=e^{-\mu_{n}t}\varphi_{n}$.
From Theorem 2.3 of Blumenthal and Getoor [5] and the proof of Theorem 5.1 of
Chen, Meerschaert, and Nane [6], we infer the following lemma:

######  Lemma 2.1.

[5, 6] For any bounded open subset $B$ of ${\mathbb{R}}^{d}$, the system of
eigenfunctions $\\{\varphi_{n}\\}_{n\geqslant 1}$ and the corresponding
eigenvalues $\\{\mu_{n}\\}_{n\geqslant 1}$ satisfy the following:

  1. a)

$C_{1}n^{\frac{\alpha}{d}}\leqslant\mu_{n}\leqslant C_{2}n^{\frac{\alpha}{d}}$
for every $n\in{\mathbb{N}}$,

  2. b)

$|\varphi_{n}(\mathrm{x})|\leqslant C\mu_{n}^{\frac{d}{2\alpha}}$ for every
$\mathrm{x}\in B$,

where $C$, $C_{1}$, and $C_{2}$ are positive real numbers.

It is also known that

$\displaystyle u_{t}(\mathrm{x})$
$\displaystyle:={\mathbb{E}}^{x}[u_{0}(X_{t}^{B})]$
$\displaystyle=\int_{B}p_{B}(t,\,\text{x},\,\text{y})\,u_{0}(y)\,{\rm d}y,$

solves the heat equation
$\partial_{t}u_{t}(\mathrm{x})=-(-\Delta)^{\alpha/2}u_{t}(\text{x})$ defined
on $B$ with Dirichlet boundary condition and initial condition $u_{0}$.

Let $D=\\{D_{r},r\geqslant 0\\}$ denote a $\beta$-stable subordinator and
$E_{t}$ to be the inverse of a stable subordinator of index $\beta\in(0,\,1)$.
The process $X_{E_{t}}^{B}$ is just a time-changed of the killed
$\alpha$-symmetric stable process $X^{B}_{t}$ and since $\beta\in(0,\,1)$,
$X_{E_{t}}^{B}$ moves more slowly than $X_{t}^{B}.$ It is known that the
density of the time changed process $X_{E_{t}}^{B}$ is given by the
$G^{(\beta)}(t,x)$. By conditioning, we have

$\displaystyle G^{(\beta)}(t,x)=\int_{0}^{\infty}p_{B}(s,x)f_{t}(s)ds,$ (2.2)

where

$\displaystyle f_{t}(s)=t\beta^{-1}s^{-1-1/\beta}g_{\beta}(ts^{-1/\beta}),$
(2.3)

where $g_{\beta}(.)$ (Cf. Meerschaert and Straka [23]) is the density function
of $D_{1}$ and is infinitely differentiable on the entire real line, with
$g_{\beta}(u)=0$ for $u\leqslant 0$; see Meerschaert and Scheffler [25] for
more information about the inverse stable subordinator $E_{t}$.
The function $v_{t}(\text{x}):={\mathbb{E}}^{x}[u_{0}(X_{E_{t}}^{B})]$ solves
the space-time fractional equation

$\left\\{\begin{aligned}
&\partial^{\beta}_{t}v_{t}(\text{x})=-(-\Delta)^{\alpha/2}v_{t}(\text{x})\;\;\text{for}\;\;\text{x}\in
B\;\;\text{and}\;\;t>0\\\ &v_{t}(\text{x})=0\quad\text{for}\;\;\text{x}\notin
B,\end{aligned}\right.$ (2.4)

with initial condition $u_{0}$ (Chen, Meerschaert, and Nane [6]). Thus, we get
the following representation of $v_{t}(x)$

$\displaystyle v_{t}(\text{x})$
$\displaystyle:={\mathbb{E}}^{\text{x}}[u_{0}(X_{E_{t}}^{B})]$
$\displaystyle=\int_{B}\int_{0}^{\infty}\sum_{n=1}^{\infty}e^{-\mu_{n}s}\varphi_{n}(\text{x})\varphi_{n}(\text{y})f_{t}(s)\,u_{0}(\text{y})\,{\rm
d}s{\rm d}\text{y}$
$\displaystyle=\int_{B}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}t^{\beta})\varphi_{n}(\text{x})\varphi_{n}(\text{y})u_{0}(\text{y}){\rm
d}\text{y}$
$\displaystyle=\int_{B}G_{B}^{(\beta)}(t,\,\text{x},\,\text{y})u_{0}(\text{y})\,{\rm
d}\text{y},$ (2.5)

this follows since the Laplace transform of $f_{t}(s)$ is
$\hat{f}_{t}(\lambda)=E_{\beta}(-\lambda t^{\beta})$ where
$E_{\beta}(x)=\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(1+\beta k)}$ is the
Mittag-Leffler function and have the following property,

$\frac{1}{1+\Gamma(1-\beta)x}\leqslant
E_{\beta}(-x)\leqslant\frac{1}{1+\Gamma(1+\beta)^{-1}x}\ \ \ \text{for}\ x>0,$
(2.6)

where $\Gamma(\cdot)$ is the gamma function. Thus, we get the following
expansion,

$G_{B}^{(\beta)}(t,\,\text{x},\,\text{y}):=\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}t^{\beta})\varphi_{n}(\text{x})\varphi_{n}(\text{y}).$
(2.7)

Using (2.3) and a change of variable from (2.7) we obtain

$\displaystyle G_{B}^{(\beta)}(t,\,\text{x},\,\text{y})$
$\displaystyle=\int_{0}^{\infty}\sum_{n=1}^{\infty}e^{-\mu_{n}(t/u)^{\beta}}\varphi_{n}(\text{x})\varphi_{n}(\text{y})g_{\beta}(u){\rm
d}u$ (2.8)
$\displaystyle=\int_{0}^{\infty}p_{B}(\left(\frac{t}{u}\right)^{\beta},\,\text{x},\,\text{y})g_{\beta}(u)\,{\rm
d}u.$ (2.9)

The following Lemma from Mijena and Nane [20] about the $L^{2}$-norm of the
heat kernel which is a crucial ingredient for the existence and uniqueness of
solution of equation (1.8) is also used in proving Theorem 1.8.

######  Lemma 2.2.

(Lemma 1 in [20]) Suppose that $d<2\alpha$, then

$\int_{{\mathbb{R}}^{d}}[G_{t}^{(\beta)}(x)]^{2}dx=C^{*}t^{\frac{-\beta
d}{\alpha}},$ (2.10)

where the constant $C^{*}$ is given by

$C^{*}=\frac{(\nu)^{-\frac{d}{\alpha}}2\pi^{\frac{d}{2}}}{\alpha\Gamma(\frac{d}{2})}\frac{1}{(2\pi)^{d}}\int_{0}^{\infty}z^{\frac{d}{\alpha-1}}({\mathbb{E}}_{\beta}(-z))^{2}dz.$

The following three results are helpful in proving the finiteness result in
Theorem 1.3, and Theorem 1.13.

######  Lemma 2.3.

For $p\geqslant 2$ and $\beta\in\left(\frac{1}{p},\,1\right)$, we have

$\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}{\rm
d}s\lesssim\mu_{n}^{-\frac{1}{\beta}},$ (2.11)

for every $t>0$.

###### Proof.

Using the Mittag-Leffler function bounds given in (2.6), we get

$\displaystyle\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}\,{\rm d}s$
$\displaystyle\leqslant
C\,\int_{0}^{t}\left(1+\Gamma(1+\beta)^{-1}\mu_{n}(t-s)^{\beta}\right)^{-p}\,{\rm
d}s$ $\displaystyle\leqslant
C\,\int_{0}^{t}\left(1\wedge\left(\mu_{n}(t-s)^{\beta}\right)^{-p}\right)\,{\rm
d}s,$ (2.12)

where

$1\wedge\left(\mu_{n}(t-s)^{\beta}\right)^{-p}=\begin{cases}1&\text{if
}\left(\mu_{n}(t-s)^{\beta}\right)^{-p}>1,\\\
\left(\mu_{n}(t-s)^{\beta}\right)^{-p}&\text{if
}\left(\mu_{n}(t-s)^{\beta}\right)^{-p}\leqslant 1.\end{cases}$ (2.13)

Now

$\displaystyle\left(\mu_{n}(t-s)^{\beta}\right)^{-p}\leqslant 1$
$\displaystyle\implies\mu_{n}(t-s)^{\beta}\geqslant 1$
$\displaystyle\implies(t-s)^{\beta}\geqslant\frac{1}{\mu_{n}}$
$\displaystyle\implies
t-s\geqslant\left(\frac{1}{\mu_{n}}\right)^{\frac{1}{\beta}}$
$\displaystyle\implies t-\mu_{n}^{-\frac{1}{\beta}}\geqslant s.$ (2.14)

So, from (2.13) using (2) we obtain

$\displaystyle\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}\,{\rm d}s$
$\displaystyle\leqslant
C\,\int_{0}^{t}\left(1\wedge\left(\mu_{n}(t-s)^{\beta}\right)^{-p}\right)\,{\rm
d}s$ (2.15)
$\displaystyle=C\,\left(\int_{0}^{t-\mu_{n}^{-\frac{1}{\beta}}}\left(\mu_{n}(t-s)^{\beta}\right)^{-p}{\rm
d}s+\int_{t-\mu_{n}^{-\frac{1}{\beta}}}^{t}{\rm d}s\right)$
$\displaystyle=C\,\left(\frac{1}{\mu_{n}^{p}}\int_{0}^{t-\mu_{n}^{-\frac{1}{\beta}}}(t-s)^{-p\beta}{\rm
d}s+\mu_{n}^{-\frac{1}{\beta}}\right)$
$\displaystyle=C\,\left(\frac{1}{\mu_{n}^{p}(1-p\beta)}\left(t^{-p\beta+1}-\left(\frac{1}{\mu_{n}}\right)^{-p+\frac{1}{\beta}}\right)+\mu_{n}^{-\frac{1}{\beta}}\right)$
$\displaystyle\leqslant
C\,\left(\frac{\mu_{n}^{p-\frac{1}{\beta}}}{\mu_{n}^{p}(p\beta-1)}+\mu_{n}^{-\frac{1}{\beta}}\right)$
$\displaystyle\leqslant
C\,\left(\frac{1}{p\beta-1}+1\right)\mu_{n}^{-\frac{1}{\beta}}\,$
$\displaystyle\lesssim\mu_{n}^{-\frac{1}{\beta}},$ (2.16)

since $\frac{t^{-p\beta+1}}{1-p\beta}<0$.

∎

######  Lemma 2.4.

For $p\geqslant 2$, $\beta\in\left(\frac{1}{p},\,1\right)$ and
$d<\frac{\alpha}{\beta}$, we have

$\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}{\rm
d}s<\infty,$ (2.17)

for every $t>0$.

###### Proof.

Using Lemma 2.1(a) and Lemma 2.3, we obtain the following:

$\displaystyle\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}{\rm
d}s$
$\displaystyle=\sum_{n=1}^{\infty}\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}{\rm
d}s$ $\displaystyle\lesssim\sum_{n=1}^{\infty}\mu_{n}^{-\frac{1}{\beta}}\,$
$\displaystyle\lesssim\sum_{n=1}^{\infty}n^{-\frac{\alpha}{\beta d}}<\infty,$
(2.18)

since $\frac{\alpha}{\beta d}>1$.

∎

######  Lemma 2.5.

For $p\geqslant 2$, $\beta\in\left(\frac{1}{p},\,1\right),$ if
$d<\frac{\alpha}{2\beta}$ then we have

$\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}\varphi_{n}^{2}(\text{x})\,{\rm
d}s<\infty,$ (2.19)

for every $t>0$.

###### Proof.

We prove the result using by Lemma 2.1(b) and Lemma 2.3 as follows

$\displaystyle\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}\varphi_{n}^{2}(\text{x})\,{\rm
d}s$
$\displaystyle=\sum_{n=1}^{\infty}\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{p}\varphi_{n}^{2}(\text{x})\,{\rm
d}s$ $\displaystyle\leqslant
C\,\sum_{n=1}^{\infty}\mu_{n}^{\frac{d}{\alpha}}\mu_{n}^{-\frac{1}{\beta}}$
$\displaystyle\lesssim\,\sum_{n=1}^{\infty}n^{\frac{\alpha}{d}(\frac{d}{\alpha}-\frac{1}{\beta})}\,$
$\displaystyle\lesssim\,\sum_{n=1}^{\infty}n^{1-\frac{\alpha}{\beta
d}}<\infty,$ (2.20)

since $\frac{\alpha}{\beta d}>2$. ∎

The following continuity Lemma is crucial to justify the conclusion of Theorem
1.8.

######  Lemma 2.6.

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$, depending on the geometrical characteristics of the domain
$B$, i.e.,

$C(B)=\sup_{n\geqslant 1,\text{x}\in B}\varphi_{n}(\text{x}).$

Fix $t>0$, then for $d<\alpha$ we have

$\lim_{\gamma\rightarrow\beta}\sup_{x\in
B}\left|(\mathcal{G}_{B}^{(\gamma)}u_{0})_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(x)\right|=0,$

where

$(\mathcal{G}_{B}^{(\gamma)}u_{0})_{t}(x):=\int_{B}G_{B}^{(\gamma)}(t,\,\text{x},\,\text{y})u_{0}(\text{y})\,{\rm
d}\text{y}.$

###### Proof.

Since the initial datum $u_{0}$ and $G_{B}^{(\gamma)}(t,\,x,\,y)$ are given to
be bounded from above, it suffices to show that

$\lim_{\gamma\rightarrow\beta}\sup_{x\in
B}\left|G_{B}^{(\gamma)}(t,\,x,\,y)-G_{B}^{(\beta)}(t,\,x,\,y)\right|=0.$

Now using the Laplace transform of the Mittag-Leffler function, for any
$\theta>0$ we have

$\displaystyle\lim_{\gamma\rightarrow\beta}\int_{0}^{\infty}e^{-\theta
t}E_{\gamma}(-\mu_{n}t^{\gamma})\,{\rm d}t$
$\displaystyle=\lim_{\gamma\rightarrow\beta}\frac{\theta^{\gamma-1}}{\theta^{\gamma}+\mu_{n}}=\int_{0}^{\infty}e^{-\theta
t}E_{\beta}(-\mu_{n}t^{\beta})\,{\rm d}t.$

This implies that

$\displaystyle\lim_{\gamma\rightarrow\beta}E_{\gamma}(-\mu_{n}t^{\gamma})=E_{\beta}(-\mu_{n}t^{\beta}).$
(2.21)

Now using the expansions for the heat kernel, we have

$\displaystyle|G_{B}^{(\beta)}(t,\,x,\,y)-G_{B}^{(\gamma)}(t,\,x,\,y)|$
$\displaystyle\leqslant\sum_{n=1}^{\infty}\left[|E_{\beta}(-\mu_{n}t^{\beta})-E_{\gamma}(-\mu_{n}t^{\gamma})||\varphi_{n}(\text{x})||\varphi_{n}(\text{y})|\right].$
(2.22)

Taking limit as $\gamma\rightarrow\beta$ on both sides of (2.22) we obtain the
required result, since each terms in the summation can be bounded by a
quantity independent of $\gamma$ and $\beta$. This is due to the uniform
boundedness of the eigenfunctions $\\{\varphi_{n}\\}_{n\geqslant 1}$ and the
bounds on the Mittag-Leffler function together with Lemma 2.1(a), and summable
as shown below.
For any $t>0$, then we obtain

$\displaystyle\sum_{n=1}^{\infty}\bigg{[}|E_{\beta}(-\mu_{n}t^{\beta})-E_{\gamma}(-\mu_{n}t^{\gamma})|$
$\displaystyle|\varphi_{n}(\text{x})||\varphi_{n}(\text{y})|\bigg{]}\leqslant[C(B)]^{2}\sum_{n=1}^{\infty}\left|E_{\beta}(-\mu_{n}t^{\beta})-E_{\gamma}(-\mu_{n}t^{\gamma})\right|$
$\displaystyle\leqslant[C(B)]^{2}\sum_{n=1}^{\infty}\left[\frac{1}{1+\Gamma(1+\beta)^{-1}\mu_{n}t^{\beta}}+\frac{1}{1+\Gamma(1+\gamma)^{-1}\mu_{n}t^{\gamma}}\right]$
$\displaystyle\leqslant[C(B)]^{2}\sum_{n=1}^{\infty}\frac{1}{\mu_{n}}\left(\frac{1}{t^{\beta}}+\frac{1}{t^{\gamma}}\right)$
$\displaystyle\leqslant
2[C(B)]^{2}\text{max}(1,1/t)\sum_{n=1}^{\infty}n^{-\frac{\alpha}{d}}<\infty,$
(2.23)

where $C$ is a constant positive real number. ∎

The following lemma is useful in the proof of Lemma 2.8(b) and Proposition
6.1.

######  Lemma 2.7.

If $\beta\in(\beta_{0},\beta_{1})$ for some $\beta_{0},\beta_{1}\in(0,1)$ and
$\frac{d}{\alpha}<\lambda<\frac{1}{2\beta}$, then using the inequality
$e^{-z}\leqslant\mathscr{C}_{\mu}z^{-\mu}$, we bound the following integral as
follows:

$\int_{0}^{\infty}e^{-2\theta s/p}\frac{1}{s^{2\lambda\beta}}{\rm d}s\lesssim
p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right],$
(2.24)

where
$0<\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}$
and $\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$.

###### Proof.

Consider the integral

$\displaystyle\int_{0}^{\infty}e^{-2\theta s/p}\frac{1}{s^{2\lambda\beta}}{\rm
d}s$ $\displaystyle=p^{1-2\lambda\beta}\int_{0}^{\infty}\frac{e^{-2\theta
r}}{r^{2\lambda\beta}}{\rm d}r$
$\displaystyle=p^{1-2\lambda\beta}\left[\int_{0}^{1}\frac{e^{-2\theta
r}}{r^{2\lambda\beta}}{\rm d}r+\int_{1}^{\infty}\frac{e^{-2\theta
r}}{r^{2\lambda\beta}}{\rm d}r\right]$ $\displaystyle\lesssim
p^{1-2\lambda\beta}\left[\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}\int_{0}^{1}\frac{1}{r^{2\lambda\beta+2\mu_{0}}}{\rm
d}r+\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}\int_{1}^{\infty}\frac{1}{r^{2\lambda\beta+2\mu_{1}}}{\rm
d}r\right]$ $\displaystyle\lesssim
p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right],$
(2.25)

where $\beta\in(\beta_{0},\beta_{1})$ for some $\beta_{0},\beta_{1}\in(0,1)$,
$\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta\lambda}{2}\Big{)}$
and $\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$, which implies that the proper
integral $\int_{0}^{1}\frac{1}{r^{2\beta\lambda+2\mu_{0}}}dr$ and
$\int_{1}^{\infty}\frac{1}{r^{2\beta\lambda+2\mu_{1}}}dr$ are convergent. ∎

######  Lemma 2.8 (Nane and Tuan, 2022 [26]).

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$, depending on the geometrical characteristics of the domain
$B$, i.e.,

$C(B)=\sup_{n\geqslant 1,\text{x}\in B}\varphi_{n}(\text{x}).$

a) Let $\beta\in(\frac{1}{2},1)$, and
$\frac{d}{\alpha}<\lambda<\frac{1}{2\beta}$. Then there exists a constant
$C=C(\lambda)$ independent of $\beta$ such that

$\displaystyle|G_{B}^{(\beta)}(t,\,x,\,y)|\leqslant C{t^{-\beta\lambda}}.$
(2.26)

b) Let $\beta\in(\beta_{0},\beta_{1})$ for some $\beta_{0},\beta_{1}\in(0,1)$.
Let also
$0<\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}$
and $\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$. Then for
$d<\frac{\alpha}{2\beta}$, there exists positive constant $C$ which depends on
$\mu_{0},\mu_{1},p$ and independent of $\beta$ such that

$\displaystyle\bigg{[}\int_{0}^{\infty}\int_{B}$ $\displaystyle
e^{-\frac{2\theta s}{p}}|G_{B}^{(\beta)}(s,\,x,\,y)|^{2}{\rm d}y\,{\rm
d}s\,\bigg{]}^{\frac{p}{2}}\lesssim
Cp^{\frac{p(1-2\lambda\beta_{0})}{2}}\left(\theta^{-2\mu_{0}}+\theta^{-2\mu_{1}}\right)^{p/2}.$
(2.27)

###### Proof.

(a) Using the fact that
$E_{\beta}(-\mu_{n}t^{\beta})\leqslant\frac{C}{(1+\mu_{n}t^{\beta})^{\lambda}}$
for any $\frac{d}{\alpha}<\lambda<\frac{1}{2\beta}$ and $C$ is independent of
$\beta$, we find that

$\displaystyle|G_{B}^{(\beta)}(t,\,x,\,y)|$
$\displaystyle=\bigg{|}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}s^{\beta})\varphi_{n}(x)\varphi_{n}(y)\bigg{|}\lesssim
C\sum_{n=1}^{\infty}|E_{\beta}(-\mu_{n}s^{\beta})|$ $\displaystyle\lesssim
C\sum_{n=1}^{\infty}\frac{1}{\mu_{n}^{\lambda}}\frac{1}{t^{\beta\lambda}}\lesssim
C{t^{-\beta\lambda}},$ (2.28)

where we used the eigenfunction expansion of $G_{B}^{(\beta)}(t,\,x,\,y)$ and
that the series
$\sum_{n=1}^{\infty}\frac{1}{\mu_{n}^{\lambda}}\leqslant\sum_{n=1}^{\infty}n^{-\frac{\alpha\lambda}{d}}$
is convergent since $\lambda>\frac{d}{\alpha}$.
(b) Using part (a) of this lemma together with Lemma 2.7, we obtain

$\displaystyle\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}\big{|}G_{B}^{(\beta)}(s,\,x,\,y)\big{|}^{2}{\rm d}y\,{\rm d}s$
$\displaystyle\lesssim\int_{0}^{\infty}e^{-2\theta
s/p}\frac{1}{s^{2\lambda\beta}}{\rm d}s$ $\displaystyle\lesssim
p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right],$
(2.29)

where $\beta\in(\beta_{0},\beta_{1})$ for some $\beta_{0},\beta_{1}\in(0,1)$,
$\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta\lambda}{2}\Big{)}$
and $\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$, which implies that the proper
integral $\int_{0}^{1}\frac{1}{r^{2\beta\lambda+2\mu_{0}}}dr$ and
$\int_{1}^{\infty}\frac{1}{r^{2\beta\lambda+2\mu_{1}}}dr$ are convergent. It
follows from (2) that for any $p\geqslant 0$

$\displaystyle\bigg{[}\int_{0}^{\infty}\int_{B}e^{-2\theta s/p}|$
$\displaystyle G_{B}^{(\beta)}(s,\,x,\,y)|^{2}\,{\rm d}s\,{\rm
d}y\bigg{]}^{p/2}\lesssim
p^{\frac{p(1-2\lambda\beta_{0})}{2}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right]^{p/2},$

which allows us to deduce (2.27). That is, each terms in the summation can be
bounded by a quantity independent of $\beta$.

∎

Next we give an example where the sufficient condition of Lemma 2.6 is
satisfied.

######  Example 2.9.

Let $X_{t}$ denote a Brownian motion in ${\mathbb{R}}^{2}$ and $X_{t}^{B}$
denote the Brownian motion killed upon exiting the rectangular domain
$B:=[0,\,L_{1}]\times[0,\,L_{2}]$. The eigenvalues of the Dirichlet Laplacian
are $\mu_{m,n}=(\frac{m\pi}{L_{1}})^{2}+(\frac{n\pi}{L_{2}})^{2}$ and
$\varphi_{m,n}(x,y):=\varphi_{m}(x)\varphi_{n}(y)=\frac{2}{\sqrt{L_{1}L_{2}}}\sin(\frac{m\pi
x}{L_{1}})\sin(\frac{n\pi y}{L_{2}})$ are the corresponding eigenfunction so
that $|\varphi_{m,n}(x.y)|\leqslant\frac{2}{\sqrt{L_{1}L_{2}}}$ for all
$(x,y)\in B$. A similar result is valid for the Brownian motion in higher
dimensional rectangular boxes.

##  3 Proof of Theorem 1.1,Theorem 1.2, and Theorem 1.3

In this section we give the proof of first three main theorems of the paper.
We denote

$\Lambda(\theta):=\int_{0}^{\infty}e^{-\theta
t}E_{\beta}(-\mu_{1}t^{\beta})^{2}\,{\rm d}t,$ (3.1)

for $\theta>0$. Here it is crucial to note that we can use the bounds in (2.6)
to observe that $\Lambda(\theta)$ tends to infinity as $\theta$ approaches to
zero if and only if $2\beta\leqslant 1$.

###### Proof of Theorem 1.1.

Let us first establish the second statement of the theorem. From the mild
formulation given in (1.10) and using Stochastic Fubini theorem we set

$\displaystyle\langle u_{t},\varphi_{1}\rangle$
$\displaystyle=\int_{B}u_{t}(x)\varphi_{1}(\text{x})\,{\rm d}\text{x}$
$\displaystyle=E_{\beta}(-\mu_{1}t^{\beta})\langle u_{0},\varphi_{1}\rangle\,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\lambda\int_{B}\int_{0}^{t}E_{\beta}(-\mu_{1}(t-s)^{\beta})\varphi_{1}(\text{y})\sigma(v_{s}(\text{y}))W({\rm
d}s\,,{\rm d}\text{y}).$ (3.2)

Taking the second moment of (3) and using the Ito-Walsh isometry, we get

$\displaystyle{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}$
$\displaystyle=E_{\beta}(-\lambda_{1}t^{\beta})^{2}\langle
u_{0},\varphi_{1}\rangle^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\lambda^{2}\int_{B}\int_{0}^{t}E_{\beta}(-\lambda_{1}(t-s)^{\beta})^{2}\varphi_{1}^{2}(\text{y}){\mathbb{E}}|\sigma(u_{s}(\text{y}))|^{2}{\rm
d}s{\rm d}\text{y}.$ (3.3)

Now using (3) and the assumption on $\sigma$ gives

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle=\Lambda(\theta)\,\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}\Lambda(\theta)\,\int_{0}^{\infty}e^{-\theta
t}\int_{B}{\mathbb{E}}|\sigma(u_{s}(\text{y}))|^{2}\,\varphi_{1}^{2}(\text{y}){\rm
d}\text{y}\,{\rm d}t.$ $\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}l_{\sigma}^{2}\Lambda(\theta)\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (3.4)

If $\beta\in(0,\,\frac{1}{2}]$, since $\Lambda(\theta)$ tends to infinity as
$\theta$ goes to zero, we can choose $\theta$ small enough and from (3) we
obtain

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+2\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (3.5)

This implies that that for sufficiently small $\theta$, we have

$\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t=\infty,$

which implies that for sufficiently large $t$, ${\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}$ grows exponentially. Now using Cauchy-Schwarz
inequality we get the following,

$\displaystyle{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}$
$\displaystyle\leqslant{\mathbb{E}}\left(\left(\int_{B}|u_{t}(x)|^{2}\,{\rm
d}x\right)\left(\int_{B}|\varphi_{1}(x)|^{2}\,{\rm d}x\right)\right)$
$\displaystyle\lesssim\sup_{\text{x}\in B}{\mathbb{E}}|u_{t}(\text{x})|^{2}.$

We can thus conclude that $\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}$ too grows exponentially fast for all
values of $\lambda$.
Note that the first part of the conclusion of the theorem merely follows from
the fact that the second term of (3.5) is positive and that the first term
cannot have exponential decay. ∎

###### Proof of Theorem 1.2.

Following the lines for the proof of Theorem 1.1, we get the inequality (3);

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}l_{\sigma}^{2}\Lambda(\theta)\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\,\varphi_{1}\rangle^{2}\,{\rm d}t.$ (3.6)

Since $\beta\in(\frac{1}{2},\,1)$, the function $\Lambda(\theta)$ is bounded.
Thus, for any fixed $\theta>0$ we can find sufficiently large $\lambda_{u}$ so
that for all $\lambda\geqslant\lambda_{u}$, the above inequality (3.6) yields

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\frac{3}{2}\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (3.7)

The result of the theorem is obtained by following the arguments of the
remaining part of the proof of Theorem 1.1. ∎

###### Proof of Theorem 1.3.

We establish the required result using Ito-Walsh isometry and the global
Lipschitz assumption on $\sigma$. The second moment of the mild formulation
given by (1.10) becomes:

$\displaystyle{\mathbb{E}}|u_{t}(x)|^{2}$
$\displaystyle=|(\mathcal{G}_{B}^{(\beta)}u)_{t}(\text{x})|^{2}+\lambda^{2}\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}{\mathbb{E}}|\sigma(u_{s}(\text{y}))|^{2}{\rm
d}s\,{\rm d}\text{y}$
$\displaystyle\leqslant|(\mathcal{G}_{B}^{(\beta)}u)_{t}(\text{x})|^{2}+\lambda^{2}L_{\sigma}^{2}\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}{\mathbb{E}}|u_{s}(\text{y})|^{2}{\rm
d}s\,{\rm d}\text{y}$ $\displaystyle:=J_{1}+J_{2}.$ (3.8)

We observe $J_{1}$ is bounded by the square of the same constant as initial
condition, since
$\int_{B}G_{B}^{(\beta)}(t,\,x)\,{\rm
d}x\leqslant\int_{{\mathbb{R}}^{d}}G^{(\beta)}(t,\,x)\,{\rm d}x=1$ and the
initial datum is assumed to be bounded above by a constant. It remains to
bound $J_{2}$. Since $\\{\varphi_{n}\\}_{{n}\geqslant 1}$ is an orthonormal
sequence, for each fixed $t>0$ we obtain

$\displaystyle J_{2}$
$\displaystyle:=\lambda^{2}L_{\sigma}^{2}\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}{\mathbb{E}}|u_{s}(\text{y})|^{2}{\rm
d}s\,{\rm d}\text{y}$ $\displaystyle\leqslant
a_{t}\lambda^{2}L_{\sigma}^{2}\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{2}\varphi_{n}^{2}(\text{x})\,{\rm
d}s<\infty,$ (3.9)

by Lemma 2.5 in the case of $d<\alpha/2\beta$, where
$a_{t}=\sup_{0<s<t}\sup_{\text{x}\in B}{\mathbb{E}}|u_{s}(\text{x})|^{2}$. In
the case the eigenfunctions $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly
bounded by a constant $C(B)$, it follows from the above that

$\displaystyle J_{2}:$ $\displaystyle\leqslant
a_{t}\lambda^{2}L_{\sigma}^{2}\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{2}\varphi_{n}^{2}(\text{x})\,{\rm
d}s$ $\displaystyle\leqslant
a_{t}\lambda^{2}L_{\sigma}^{2}\left(C(B)\right)^{2}\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{2}{\rm
d}s\leqslant a_{t}\lambda^{2}L_{\sigma}^{2}\left(C(B)\right)^{2}C(ML),$ (3.10)

where $a_{t}=\sup_{0<s<t}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}$ and
$C(ML)=\displaystyle\int_{0}^{t}\sum_{n=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{2}{\rm
d}s$. By Lemma 2.4 $C(ML)<\infty$ is a finite constant independent of $t$.
That is, since $\beta\in(\frac{1}{2},1)$, we can choose $\lambda_{l}$
sufficiently small that for all $\lambda\leqslant\lambda_{l}$, the above
estimates shows

$\displaystyle\sup_{0<s<t}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}\lesssim
1+\frac{1}{2}\sup_{0<s<t}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}.$

This shows that $\sup_{0<t<\infty}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}$ is finite and hence the conclusion of the
theorem follows. ∎

##  4 Proof of Theorem 1.8

In this section we give the proof of Theorem 1.8 and the following proposition
is used in its proof. We can use the bounds of $G_{B}^{(\beta)}(t,\,x,\,y)$ to
show the following proposition holds:

######  Proposition 4.1.

For $d<\frac{\alpha}{\beta}$, let $u_{t}^{(\beta)}$ be a solution to equation
(1.10) for parameters $\beta\in\left(\frac{1}{2},\,1\right)$. Then for some
$\theta$, the supremum on the $p^{th}$ moment of the solution
$u_{s}^{(\beta)}(x)$ is given by

$\sup_{t>0,\,x\in B}e^{-\theta t}{\mathbb{E}}|u_{s}^{(\beta)}(x)|^{p},$ (4.1)

is bounded above by a constant independent of $\beta$.

###### Proof.

We use the mild formulation in equation (1.10) and follow similar computations
to those used in the above proof. Now consider

$u_{t}^{(\beta)}(x)=(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})\sigma(u_{s}^{(\beta)}(\text{y}))W({\rm
d}s\,,{\rm d}\text{y}),$ (4.2)

Now applying the inequality $(a+b)^{p}\leqslant 2^{p}(a^{p}\,+b^{p})$ for any
$a,b\geqslant 0$ and using Burkholder-Davis-Gundy inequality, the $p^{th}$
moment of (4.2) becomes

${\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}\leqslant
2^{p}|(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})|^{p}+(2\lambda)^{p}\left[\int_{B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})|^{2}\left[{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))|^{p}\right]^{\frac{2}{p}}{\rm
d}s\,{\rm d}\text{y}\right]^{\frac{p}{2}},$ (4.3) $\displaystyle e^{-\theta
t}{\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}$ $\displaystyle\lesssim e^{-\theta
t}|(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})|^{p}+e^{-\theta
t}\left[\int_{B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})|^{2}\left[{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))|^{p}\right]^{\frac{2}{p}}{\rm
d}s\,{\rm d}\text{y}\right]^{\frac{p}{2}}$ $\displaystyle:=J_{1}+J_{2}.$ (4.4)

Using Lemma 2.2 and since $\sigma$ is globally Lipschitz inequality, the
second term becomes

$\displaystyle J_{2}$ $\displaystyle:=e^{-\theta
t}\left[\int_{B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})|^{2}\left[{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))|^{p}\right]^{\frac{2}{p}}{\rm
d}s\,{\rm d}\text{y}\right]^{\frac{p}{2}}$ $\displaystyle\leqslant
L_{\sigma}^{p}\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}e^{\frac{-2\theta
t}{p}}[{\mathbb{E}}|u_{s}^{(\beta)}(\text{y})|^{p}]^{2/p}{\rm d}s\,{\rm
d}y\right]^{p/2}$ $\displaystyle\leqslant
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}e^{\frac{-2\theta(t-s)}{p}}{\rm
d}s\,{\rm d}y\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\Gamma\left(1-\frac{\beta
d}{\alpha}\right)\theta^{\frac{\beta d}{\alpha}-1}\right]^{p/2},$ (4.5)

where $b_{t}(\theta)=\sup_{0<s<t}\sup_{\text{z}\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\beta)}(\text{z})|^{p}$. Now we fix $\theta>0$
sufficiently large so that

$\displaystyle J_{2}\lesssim\frac{1}{2}b_{t}(\theta).$ (4.6)

Thus, we have

$\displaystyle e^{-\theta t}{\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}$
$\displaystyle\lesssim 1+\frac{1}{2}b_{t}(\theta).$ (4.7)

Taking the supremum of the left side of (4.7) the result follows. ∎

We are now ready to prove Theorem 1.8.

###### Proof of Theorem 1.8.

From the mild formulation of the solutions, we have

$\displaystyle u_{t}^{(\gamma)}(x)-u_{t}^{(\beta)}(x)$
$\displaystyle=(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\gamma)}(t-s,\,x,\,y)\sigma(u_{s}^{(\gamma)}(y))W({\rm
d}s\,{\rm d}y)$
$\displaystyle-\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)\sigma(u_{s}^{(\beta)}(y))W({\rm
d}s\,{\rm d}y)$
$\displaystyle=(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)$
$\displaystyle+\lambda\int_{B}\int_{0}^{t}[G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)]\sigma(u_{s}^{(\gamma)}(y))W({\rm
d}s\,{\rm d}y)$ (4.8)
$\displaystyle+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]W({\rm
d}s\,{\rm d}y).$

Now applying the inequality $(a+b+c)^{p}\leqslant
3^{p}(a^{p}\,+b^{p}\,+c^{p})$ for any $a,b,c\geqslant 0$ and using Burkholder-
Davis-Gundy inequality, the $p^{th}$ moment of (4) becomes

$\displaystyle{\mathbb{E}}|u_{t}^{(\gamma)}(x)-$ $\displaystyle
u_{t}^{(\beta)}(x)|^{p}\leqslant
3^{p}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}$
$\displaystyle+(3\lambda)^{p}\left[\int_{0}^{t}\int_{B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)|^{2}[{\mathbb{E}}|\sigma(u_{s}^{(\gamma)}(y))|^{p}]^{2/p}\,{\rm
d}y\,{\rm d}s\right]^{p/2}$ (4.9)
$\displaystyle+(3\lambda)^{p}\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)^{2}[{\mathbb{E}}[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]^{p}]^{2/p}{\rm
d}s\,{\rm d}y\right]^{p/2}.$

For a fixed $\theta>0$ the inequality in (4) becomes

$\displaystyle e^{-\theta t}{\mathbb{E}}|u_{t}^{(\gamma)}(x)-$ $\displaystyle
u_{t}^{(\beta)}(x)|^{p}\lesssim e^{-\theta
t}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}$
$\displaystyle+e^{-\theta
t}\left[\int_{0}^{t}\int_{B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)|^{2}[{\mathbb{E}}|\sigma(u_{s}^{(\gamma)}(y))|^{p}]^{2/p}\,{\rm
d}y\,{\rm d}s\right]^{p/2}$ $\displaystyle+e^{-\theta
t}\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)^{2}[{\mathbb{E}}[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]^{p}]^{2/p}{\rm
d}s\,{\rm d}y\right]^{p/2}.$ $\displaystyle:=J_{1}+J_{2}+J_{3}.$

Let us first work on the third term. Using Lemma 2.2 and since $\sigma$ is
globally Lipschitz, we have

$\displaystyle J_{3}$ $\displaystyle:=e^{-\theta
t}\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)^{2}[{\mathbb{E}}[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]^{p}]^{2/p}{\rm
d}s\,{\rm d}y\right]^{p/2}$ $\displaystyle\leqslant
L_{\sigma}^{p}\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}e^{\frac{-2\theta
t}{p}}[{\mathbb{E}}|u_{s}^{(\gamma)}(\text{y})-u_{s}^{(\beta)}(\text{y})|^{p}]^{2/p}{\rm
d}s\,{\rm d}y\right]^{p/2}$ $\displaystyle\leqslant
L_{\sigma}^{p}b(\theta)\left[\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})^{2}e^{\frac{-2\theta(t-s)}{p}}{\rm
d}s\,{\rm d}y\right]^{p/2}$ $\displaystyle\leqslant
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{\infty}\int_{{\mathbb{R}}^{n}}G^{(\beta)}(s,\,\text{x},\,\text{y})^{2}e^{\frac{-2\theta
s}{p}}{\rm d}y\,{\rm d}s\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{\infty}s^{-\frac{\beta
d}{\alpha}}e^{\frac{-2\theta s}{p}}\,{\rm d}s\right]^{p/2}$
$\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{\infty}s^{-\frac{\beta
d}{\alpha}}e^{-\theta s}\,{\rm d}s\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\Gamma\left(1-\frac{\beta
d}{\alpha}\right)\theta^{\frac{\beta d}{\alpha}-1}\right]^{p/2},$ (4.10)

where $b_{t}(\theta)=\sup_{0<s<t}\sup_{\text{z}\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\gamma)}(\text{z})-u_{s}^{(\beta)}(\text{z})|^{p}$. Now
we fix $\theta>0$ sufficiently large so that

$\displaystyle J_{3}\lesssim\frac{1}{2}b_{t}(\theta).$ (4.11)

Next we work on $J_{2}$. Since $(a+b)^{2}\leqslant 4(a^{2}+b^{2})$ for
$a,b\geqslant 0$, using Proposition 4.1, Lemma 2.8(b) and the assumption on
$\sigma$, we obtain

$\displaystyle J_{2}$ $\displaystyle:=e^{-\theta
t}\left[\int_{0}^{t}\int_{B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)|^{2}[{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(y))|^{p}]^{2/p}\,{\rm
d}y\,{\rm d}s\right]^{p/2}$ $\displaystyle\lesssim\sup_{t>0,\,x\in
B}e^{-\theta
t}{\mathbb{E}}|u_{s}^{(\beta)}(x)|^{p}\left[\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}|G_{B}^{(\gamma)}(s,\,x,\,y)-G_{B}^{(\beta)}(s,\,x,\,y)|^{2}\,{\rm
d}s\,{\rm d}y\right]^{p/2}$
$\displaystyle\lesssim\left[\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}|G_{B}^{(\gamma)}(s,\,x,\,y)-G_{B}^{(\beta)}(s,\,x,\,y)|^{2}\,{\rm
d}s\,{\rm d}y\right]^{p/2}$
$\displaystyle\lesssim\left[\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}|G_{B}^{(\gamma)}(s,\,x,\,y)|^{2}{\rm d}s\,{\rm
d}y+\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}|G_{B}^{(\beta)}(s,\,x,\,y)|^{2}\,{\rm d}s\,{\rm d}y\right]^{p/2}$
$\displaystyle\lesssim\left[p^{1-2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}}\left[\frac{\mathscr{C}_{\mu_{0}^{{}^{\prime}}}^{2}\theta^{-2\mu_{0}^{{}^{\prime}}}}{1-2\lambda^{{}^{\prime}}-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}^{{}^{\prime}}}^{2}\theta^{-2\mu_{1}^{{}^{\prime}}}}{2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}+2\mu_{1}^{{}^{\prime}}-1}\right]+p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right]\right]^{p/2},$
(4.12)

where $\beta\in(\beta_{0},\beta_{1})$ and
$\gamma\in(\beta_{0}^{{}^{\prime}},\beta_{1}^{{}^{\prime}})$ for some
$\beta_{0},\beta_{1},\beta_{0}^{{}^{\prime}},\beta_{1}^{{}^{\prime}}\in(0,1)$,
$\frac{d}{\alpha}<\lambda<\frac{1}{2\beta}$,
$\frac{d}{\alpha}<\lambda^{{}^{\prime}}<\frac{1}{2\gamma}$,
$\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}$,
$\mu_{0}^{{}^{\prime}}<\min\Big{(}\frac{1}{2}-\lambda^{{}^{\prime}},~{}~{}\frac{1-2\gamma_{1}\lambda^{{}^{\prime}}}{2}\Big{)}$,
$\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$, and
$\mu_{1}^{{}^{\prime}}>\frac{1-2\beta_{0}^{{}^{\prime}}\lambda^{{}^{\prime}}}{2}$,
where the bound on right hand side of (4) doesn’t depend on $\beta$ and
$\gamma$.
We combine the above estimates and obtain the following:

$\displaystyle\sup_{x\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\gamma)}(x)-u_{s}^{(\beta)}(x)|^{p}$
$\displaystyle\lesssim e^{-\theta
t}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}+\frac{1}{2}b_{t}(\theta)$
$\displaystyle+\left[\int_{0}^{\infty}\int_{B}e^{-2\theta
s/p}|G_{B}^{(\gamma)}(s,\,x,\,y)-G_{B}^{(\beta)}(s,\,x,\,y)|^{2}\,{\rm
d}s\,{\rm d}y\right]^{p/2}.$

Since Lemma 2.6, and Lemma 2.8(b) allows us to use the dominated convergence
theorem, taking $\gamma\rightarrow\beta$ the result of the theorem follows. ∎

##  5 Proof of Theorem 1.11, Theorem 1.12, and Theorem 1.13

###### Proof of Theorem 1.11.

First, we establish the second statement of the theorem. From the mild
formulation given in (1.10) and using Stochastic Fubini theorem, we set

$\displaystyle\langle u_{t},\varphi_{1}\rangle$
$\displaystyle=E_{\beta}(-\mu_{1}t^{\beta})\langle
u_{0},\varphi_{1}\rangle\,+\lambda\int_{B}\int_{0}^{t}E_{\beta}(-\mu_{1}(t-s)^{\beta})\varphi_{1}(\text{y})\sigma(v_{s}(\text{y}))F({\rm
d}s\,,{\rm d}\text{y}).$ (5.1)

Taking the second moment of (5.1), we get

$\displaystyle{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}$
$\displaystyle=E_{\beta}(-\lambda_{1}t^{\beta})^{2}\langle
u_{0},\varphi_{1}\rangle^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\lambda^{2}\int_{B\times
B}\int_{0}^{t}E_{\beta}(-\lambda_{1}(t-s)^{\beta})^{2}\varphi_{1}(\text{y})\varphi_{1}(\text{z}){\mathbb{E}}|\sigma(u_{s}(\text{y}))\sigma(u_{s}(\text{z}))|\,f(y,\,z)\,{\rm
d}s{\rm d}\text{y}{\rm d}\text{z}.$ (5.2)

Now using (5) and the assumption on $\sigma$ gives

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle=\Lambda(\theta)\,\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}\Lambda(\theta)\,\int_{0}^{\infty}e^{-\theta
t}\int_{B\times B}\varphi_{1}(\text{y})\varphi_{1}(\text{z})\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathbb{E}}|\sigma(u_{t}(\text{y}))\sigma(u_{t}(\text{z}))|f(y,\,z){\rm
d}\text{y}\,\text{z}\,{\rm d}t.$
$\displaystyle\geqslant\Lambda(\theta)\,\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}l_{\sigma}^{2}K_{f}\Lambda(\theta)\,\int_{0}^{\infty}e^{-\theta
t}\int_{B\times B}\varphi_{1}(\text{y})\varphi_{1}(\text{z})\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathbb{E}}|u_{t}(\text{y})u_{t}(\text{z})|{\rm
d}\text{y}\,{\rm d}\text{z}\,{\rm d}t$ $\displaystyle=\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}l_{\sigma}^{2}K_{f}\Lambda(\theta)\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (5.3)

If $\beta\in(0,\,\frac{1}{2}]$, since $\Lambda(\theta)$ tends to infinity as
$\theta$ goes to zero, we can choose $\theta$ small enough and from (5) we
obtain

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\frac{1}{2}\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (5.4)

This implies that for sufficiently small $\theta$, we have

$\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t=\infty,$

which also implies that for sufficiently large $t$, ${\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}$ grows exponentially. Now using Cauchy-Schwarz
inequality we get the following,

$\displaystyle{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}$
$\displaystyle={\mathbb{E}}\left|\int_{B}u_{t}(x)\,\varphi_{1}(x)\,{\rm
d}x\right|^{2}$
$\displaystyle\leqslant{\mathbb{E}}\left(\left(\int_{B}|u_{t}(x)|^{2}\,{\rm
d}x\right)\left(\int_{B}\varphi_{1}^{2}(x)\,{\rm d}x\right)\right)$
$\displaystyle\lesssim\sup_{\text{x}\in B}{\mathbb{E}}|u_{t}(\text{x})|^{2}.$

We can thus conclude that $\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}$ too grows exponentially fast for all
values of $\lambda$.
Note that the first part of the conclusion of theorem merely follows from the
fact that the second term of (5) is positive and that the first term cannot
have exponential decay. ∎

###### Proof of Theorem 1.12.

Using the same notations as in the proof of Theorem 1.11, we observe that the
inequality (5) holds;

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+\lambda^{2}l_{\sigma}^{2}K_{f}\Lambda(\theta)\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\,\varphi_{1}\rangle^{2}\,{\rm d}t.$ (5.5)

Since $2\beta>1$, the function $\Lambda(\theta)$ is bounded. Thus, for any
fixed $\theta>0$ there exists sufficiently large $\lambda_{u}$ so that for all
$\lambda\geqslant\lambda_{u}$, from the inequality (5.5) we obtain

$\displaystyle\int_{0}^{\infty}e^{-\theta t}{\mathbb{E}}\langle
u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t$
$\displaystyle\gtrsim\Lambda(\theta)\langle
u_{0},\varphi_{1}\rangle^{2}+2\int_{0}^{\infty}e^{-\theta
t}{\mathbb{E}}\langle u_{t},\varphi_{1}\rangle^{2}\,{\rm d}t.$ (5.6)

Again, following similar lines of arguments as in the proof of Theorem 1.11,
we get result of the current theorem. ∎

###### Proof of Theorem 1.13.

Using $(a+b)^{2}\leqslant 4(a^{2}+b^{2})$ for $a,b\geqslant 0$, and applying
Burkholder’s inequality, the second moment of the mild formulation given by
(1.14) becomes:

$\displaystyle{\mathbb{E}}|u_{t}(x)|^{2}$ $\displaystyle\leqslant
4|(\mathcal{G}_{B}^{(\beta)}u)_{t}(\text{x})|^{2}+4\lambda^{2}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})\times~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathbb{E}}|\sigma(u_{s}(\text{y}))\sigma(u_{s}(\text{z}))|\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{y}\,{\rm d}\text{z}$ $\displaystyle\leqslant
4|(\mathcal{G}_{B}^{(\beta)}u)_{t}(\text{x})|^{2}+4\lambda^{2}L_{\sigma}^{2}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})\times~{}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\mathbb{E}}|u_{s}(\text{y})u_{s}(\text{z})|\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{y}\,{\rm d}\text{z}$ $\displaystyle:=J_{1}+J_{2}.$ (5.7)

We observe $J_{1}$ is bounded by the square of the same constant as initial
condition, since $\int_{B}G_{B}^{(\beta)}(t,\,x)\,{\rm
d}x\leqslant\int_{{\mathbb{R}}^{d}}G^{(\beta)}(t,\,x)\,{\rm d}x=1$ and the
initial datum is assumed to be bounded above by a constant. It remains to
bound $J_{2}$. Now if $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded
by a constant $C(B)$,for each fixed $t>0$, by Cauchy-Schwarz inequality, Lemma
2.1 and Lemma 2.3 we obtain

$\displaystyle J_{2}$ $\displaystyle:=4\lambda^{2}L_{\sigma}^{2}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z}){\mathbb{E}}|u_{s}(\text{y})u_{s}(\text{z})|\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{z}\,{\rm d}\text{y}$ $\displaystyle\leqslant
4\lambda^{2}L_{\sigma}^{2}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})({\mathbb{E}}|u_{s}(\text{y})|^{2})^{\frac{1}{2}}({\mathbb{E}}|u_{s}(\text{z})|^{2})^{\frac{1}{2}}\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{z}\,{\rm d}\text{y}$ $\displaystyle\leqslant
4\lambda^{2}L_{\sigma}^{2}\int_{0}^{t}\sup_{x\in
B}{\mathbb{E}}[|u_{s}(\text{x})|^{2}]\int_{B\times
B}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{z}\,{\rm d}\text{y}$ $\displaystyle\leqslant
4\lambda^{2}L_{\sigma}^{2}a_{t}\int_{0}^{t}\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}E_{\beta}(-\mu_{n}(t-s)^{\beta})E_{\beta}(-\mu_{k}(t-s)^{\beta})\varphi_{n}(\text{x})\varphi_{k}(\text{x})\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int_{B\times
B}\varphi_{n}(\text{y})\varphi_{k}(\text{z})\,f(y,\,z)\,{\rm d}s\,{\rm
d}\text{z}\,{\rm d}\text{y}$ $\displaystyle\leqslant
4\lambda^{2}L_{\sigma}^{2}a_{t}\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left(\int_{0}^{t}E_{\beta}(-\mu_{n}(t-s)^{\beta})^{2}\,{\rm
d}s\right)^{\frac{1}{2}}\left(\int_{0}^{t}E_{\beta}(-\mu_{k}(t-s)^{\beta})^{2}\,{\rm
d}s\right)^{\frac{1}{2}}\varphi_{n}(\text{x})\varphi_{k}(\text{x})\,\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\int_{B}\varphi_{n}(\text{y})\left[\int_{B}\varphi_{k}(\text{z})\,f(y,\,z)\,{\rm
d}\text{z}\right]\,{\rm d}\text{y}$ $\displaystyle\lesssim
C_{3}C(B)^{4}\lambda^{2}L_{\sigma}^{2}a_{t}\sum_{n=1}^{\infty}\sum_{k=1}^{\infty}\left(\mu_{n}^{-\frac{1}{2\beta}}\right)\left(\mu_{k}^{-\frac{1}{2\beta}}\right)\,\int_{B}\left[\int_{B}f(y,\,z)\,{\rm
d}\text{z}\right]\,{\rm d}\text{y}$ $\displaystyle\lesssim
C_{3}\lambda^{2}L_{\sigma}^{2}a_{t}\left(\sum_{n=1}^{\infty}n^{-\frac{\alpha}{2d\beta}}\right)\left(\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2d\beta}}\right)\,\int_{B\times
B}f(y,\,z)\,{\rm d}\text{z}\,{\rm d}\text{y}$ $\displaystyle\lesssim
C\lambda^{2}L_{\sigma}^{2}a_{t},$ (5.8)

where
$C=C_{3}\left(\sum_{n=1}^{\infty}n^{-\frac{\alpha}{2d\beta}}\right)\left(\sum_{k=1}^{\infty}k^{-\frac{\alpha}{2d\beta}}\right)\,\int_{B\times
B}f(y,\,z)\,{\rm d}\text{z}\,{\rm d}\text{y}<\infty$, and
$a_{t}=\sup_{0<s<t}\sup_{x\in B}{\mathbb{E}}[|u_{s}(\text{x})|^{2}$. Since
$\beta\in(\frac{1}{2},1)$, we can choose $\lambda_{l}$ sufficiently small that
for all $\lambda\leqslant\lambda_{l}$, the above estimates shows

$\displaystyle\sup_{0<s<t}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}\lesssim
1+\frac{1}{2}\sup_{0<s<t}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}.$

This shows that $\sup_{0<t<\infty}\sup_{\text{x}\in
B}{\mathbb{E}}|u_{t}(\text{x})|^{2}$ is finite and hence the conclusion of the
theorem follows. ∎

###### Proof of Corollary 1.14.

Since $\frac{1}{\beta}>1$, if $\alpha=2$ and $d=1$, (for the case of the
eigenvalues $\mu_{n}=(\frac{n\pi}{L})^{2}$ and eigenvectors
$\varphi_{n}(x)=(\frac{2}{L})^{\frac{1}{2}}\sin{(\frac{n\pi x}{L})}$ of
Dirichlet Laplacian corresponding to a Brownian Motion killed upon exiting the
domain $[0,\,L]$ so that
$|\varphi_{n}(x)|\leqslant(\frac{2}{L})^{\frac{1}{2}}$ for each $n$) from (3)
using Lemma 2.1 we obtain the result of the corollary. ∎

###### Proof of Corollary 1.15.

In this case, we note that $C_{4}\,\int_{B}\left[\int_{B}f(y,\,z)\,{\rm
d}\text{z}\right]\,{\rm d}\text{y}<\infty$, since

$\displaystyle\int_{B\times B}f(y,\,z)\,{\rm d}\text{z}\,{\rm d}\text{y}$
$\displaystyle=2\frac{L^{2-\gamma}}{(1-\gamma)\,(2-\gamma)}<\infty.$ (5.9)

∎

##  6 Proof of Theorem 1.16

In this section we give the proof of Theorem 1.16 and the following result is
useful to guarantee the use of dominated convergence theorem in the proof of
Theorem 1.16.

######  Proposition 6.1.

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$, depending on the geometrical characteristics of the domain
$B$, i.e.,

$C(B)=\sup_{n\geqslant 1,\text{x}\in B}\varphi_{n}(\text{x}).$

Suppose that $\int_{B\times B}f(x,\,y)\,{\rm d}x\,{\rm d}y<\infty$. For
$\beta\in(\frac{1}{2},1)$ and $\frac{d}{\alpha}<\lambda<\frac{1}{2\beta}$, let
$0<\mu_{0}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta_{1}\lambda}{2}\Big{)}<\min\Big{(}\frac{1}{2}-\lambda,~{}~{}\frac{1-2\beta\lambda}{2}\Big{)}$
and $\mu_{1}>\frac{1-2\beta_{0}\lambda}{2}$ where
$\beta\in(\beta_{0},\,\beta_{1})$ for some $\beta_{0},\beta_{1}\in(0,\,1)$.
Also for $\gamma\in(\frac{1}{2},1)$ and
$\frac{d}{\alpha}<\lambda^{{}^{\prime}}<\frac{1}{2\gamma}$, let
$0<\mu_{0}^{{}^{\prime}}<\min\Big{(}\frac{1}{2}-\lambda^{{}^{\prime}},~{}~{}\frac{1-2\beta_{1}^{{}^{\prime}}\lambda^{{}^{\prime}}}{2}\Big{)}<\min\Big{(}\frac{1}{2}-\lambda^{{}^{\prime}},~{}~{}\frac{1-2\gamma\lambda^{{}^{\prime}}}{2}\Big{)}$,
and
$\mu_{1}^{{}^{\prime}}>\frac{1-2\beta_{0}^{{}^{\prime}}\lambda^{{}^{\prime}}}{2}$
where $\gamma\in(\beta_{0}^{{}^{\prime}},\,\beta_{1}^{{}^{\prime}})$ for some
$\beta_{0}^{{}^{\prime}},\beta_{1}^{{}^{\prime}}\in(0,\,1)$. Then for
$d<\frac{1}{2}\min\\{\frac{1}{\beta},\,\frac{1}{\gamma}\\}\alpha$, there exist
positive a constant $C$ that depends on
$\mu_{0},\,\mu_{0}^{{}^{\prime}},\,\mu_{1},\,\mu_{1}^{{}^{\prime}},$ and $p$,
and independent of $\beta$ and $\gamma$ such that

$\displaystyle\bigg{[}\int_{0}^{t}$ $\displaystyle\int_{B\times
B}e^{-2\theta(t-s)/p}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle f(y,\,z)\,{\rm d}y\,{\rm d}z\,{\rm
d}s\,\bigg{]}^{\frac{p}{2}}\lesssim
C\left[p^{1-2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}}\left(\theta^{-2\mu_{0}^{{}^{\prime}}}+\theta^{-2\mu_{1}^{{}^{\prime}}}\right)+p^{1-2\lambda\beta_{0}}\left(\theta^{-2\mu_{0}}+\theta^{-2\mu_{1}}\right)\right]^{p/2}.$
(6.1)

###### Proof.

Since $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a constant
$C(B)$ and $\int_{B\times B}f(x,\,y)\,{\rm d}x\,{\rm d}y<\infty$, for each
$t>0$, using Lemma 2.7, Lemma 2.8(a) and the fact that $(a+b)^{2}\leqslant
2(a^{2}+b^{2})$ for $a,b\geqslant 0$, we bound the integral term as follows:

$\displaystyle\int_{0}^{t}$ $\displaystyle\int_{B\times
B}e^{-2\theta(t-s)/p}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}f(y,\,z)\,{\rm
d}y\,{\rm d}z\,{\rm d}s$ $\displaystyle\lesssim\int_{0}^{t}\int_{B\times
B}e^{-2\theta(t-s)/p}[(t-s)^{-2\gamma\lambda^{{}^{\prime}}}+2(t-s)^{-(\beta\lambda+\gamma\lambda^{{}^{\prime}})}+(t-s)^{-2\beta\lambda}]f(y,\,z)\,{\rm
d}y\,{\rm d}z\,{\rm d}s$ $\displaystyle\lesssim\int_{0}^{\infty}\int_{B\times
B}e^{-2\theta
s/p}[s^{-\gamma\lambda^{{}^{\prime}}}+s^{-\beta\lambda}]^{2}f(y,\,z)\,{\rm
d}y\,{\rm d}z\,{\rm d}s$ $\displaystyle\lesssim\int_{0}^{\infty}e^{-2\theta
s/p}[s^{-2\gamma\lambda^{{}^{\prime}}}+s^{-2\beta\lambda}]\,{\rm d}s$
$\displaystyle\lesssim
p^{1-2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}}\left[\frac{\mathscr{C}_{\mu_{0}^{{}^{\prime}}}^{2}\theta^{-2\mu_{0}^{{}^{\prime}}}}{1-2\lambda^{{}^{\prime}}-2\mu_{0}^{{}^{\prime}}}+\frac{\mathscr{C}_{\mu_{1}^{{}^{\prime}}}^{2}\theta^{-2\mu_{1}^{{}^{\prime}}}}{2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}+2\mu_{1}^{{}^{\prime}}-1}\right]+p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right],$
(6.2)

where $\mu_{0}$ and $\mu_{0}^{{}^{\prime}}$ are as given in the hypothesis of
the proposition. It follows from (6) that for any $p\geqslant 0$ we obtain

$\displaystyle\bigg{[}\int_{0}^{t}$ $\displaystyle\int_{B\times
B}e^{-2\theta(t-s)/p}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle f(y,\,z)\,{\rm d}y\,{\rm d}z\,{\rm d}s\bigg{]}^{p/2}$
$\displaystyle\lesssim\left[p^{1-2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}}\left[\frac{\mathscr{C}_{\mu_{0}^{{}^{\prime}}}^{2}\theta^{-2\mu_{0}^{{}^{\prime}}}}{1-2\lambda^{{}^{\prime}}-2\mu_{0}^{{}^{\prime}}}+\frac{\mathscr{C}_{\mu_{1}^{{}^{\prime}}}^{2}\theta^{-2\mu_{1}^{{}^{\prime}}}}{2\lambda^{{}^{\prime}}\beta_{0}^{{}^{\prime}}+2\mu_{1}^{{}^{\prime}}-1}\right]+p^{1-2\lambda\beta_{0}}\left[\frac{\mathscr{C}_{\mu_{0}}^{2}\theta^{-2\mu_{0}}}{1-2\lambda-2\mu_{0}}+\frac{\mathscr{C}_{\mu_{1}}^{2}\theta^{-2\mu_{1}}}{2\lambda\beta_{0}+2\mu_{1}-1}\right]\right]^{p/2},$

which allows us to deduce (6.1). That is, each terms in the summation can be
bounded by a quantity independent of $\gamma$ and $\beta$.
∎

###### Proof of Theorem 1.16.

From the mild formulation of the solutions, we have

$\displaystyle u_{t}^{(\gamma)}(x)-u_{t}^{(\beta)}(x)$
$\displaystyle=(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\gamma)}(t-s,\,x,\,y)\sigma(u_{s}^{(\gamma)}(y))F({\rm
d}s\,{\rm d}y)$
$\displaystyle-\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)\sigma(u_{s}^{(\beta)}(y))F({\rm
d}s\,{\rm d}y)$
$\displaystyle=(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)$
$\displaystyle+\lambda\int_{B}\int_{0}^{t}[G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)]\sigma(u_{s}^{(\gamma)}(y))F({\rm
d}s\,{\rm d}y)$ (6.3)
$\displaystyle+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]F({\rm
d}s\,{\rm d}y).$

Now applying the inequality $(a+b+c)^{p}\leqslant
3^{p}(a^{p}\,+b^{p}\,+c^{p})$ for any $a,b,c\geqslant 0$ and using Burkholder-
Davis-Gundy inequality, the $p^{th}$ moment of (6) becomes

$\displaystyle{\mathbb{E}}|u_{t}^{(\gamma)}(x)-$ $\displaystyle
u_{t}^{(\beta)}(x)|^{p}\leqslant
3^{p}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}$
$\displaystyle+(3\lambda)^{p}{\mathbb{E}}\left|\int_{B}\int_{0}^{t}[G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)]\sigma(u_{s}^{(\gamma)}(y))F({\rm
d}s\,{\rm d}y)\right|^{p}$ (6.4)
$\displaystyle+(3\lambda)^{p}{\mathbb{E}}\left|\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)[\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y))]F({\rm
d}s\,{\rm d}y)\right|^{p}.$ $\displaystyle{\mathbb{E}}|u_{t}^{(\gamma)}(x)-$
$\displaystyle u_{t}^{(\beta)}(x)|^{p}\leqslant
3^{p}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}$
$\displaystyle+(3\lambda)^{p}\bigg{[}\int_{0}^{t}\int_{B\times
B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\mathbb{E}}|\sigma(u_{s}^{(\gamma)}(y))\sigma(u_{s}^{(\gamma)}(z))|^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}$ (6.5)
$\displaystyle+(3\lambda)^{p}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\mathbb{E}}[(\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y)))(\sigma(u_{s}^{(\gamma)}(z))-\sigma(u_{s}^{(\beta)}(z)))]^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$

For a fixed $\theta>0$ the inequality in (6) becomes

$\displaystyle e^{-\theta t}$
$\displaystyle{\mathbb{E}}|u_{t}^{(\gamma)}(x)-u_{t}^{(\beta)}(x)|^{p}\lesssim
e^{-\theta
t}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}$
$\displaystyle+e^{-\theta t}\bigg{[}\int_{0}^{t}\int_{B\times
B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\mathbb{E}}|\sigma(u_{s}^{(\gamma)}(y))\sigma(u_{s}^{(\gamma)}(z))|^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}$ $\displaystyle+e^{-\theta
t}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[{\mathbb{E}}[(\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y)))(\sigma(u_{s}^{(\gamma)}(z))-\sigma(u_{s}^{(\gamma)}(z)))]^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$ $\displaystyle:=J_{1}+J_{2}+J_{3}.$
(6.7)

Let us first work on the third term. Since $\sigma$ is globally Lipschitz,
using Lemma 2.4, and Lemma 2.8 (a), we have

$\displaystyle J_{3}$ $\displaystyle:=e^{-\theta t}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bigg{[}{\mathbb{E}}[(\sigma(u_{s}^{(\gamma)}(y))-\sigma(u_{s}^{(\beta)}(y)))(\sigma(u_{s}^{(\gamma)}(z))-\sigma(u_{s}^{(\gamma)}(z)))]^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$ $\displaystyle\lesssim
L_{\sigma}^{p}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{-2\theta
t}{p}}[{\mathbb{E}}[(u_{s}^{(\gamma)}(y)-u_{s}^{(\beta)}(y))(u_{s}^{(\gamma)}(z)-u_{s}^{(\beta)}(z))]^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$ $\displaystyle\lesssim
L_{\sigma}^{p}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{-2\theta
t}{p}}({\mathbb{E}}(u_{s}^{(\gamma)}(y)-u_{s}^{(\beta)}(y))^{p})^{\frac{1}{p}}({\mathbb{E}}(u_{s}^{(\gamma)}(z)-u_{s}^{(\beta)}(z))^{p})^{\frac{1}{p}}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)e^{\frac{-2\theta(t-s)}{p}}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{t}(t-s)^{-2\beta\lambda}e^{\frac{-2\theta(t-s)}{p}}\,{\rm
d}s\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2}$
$\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{\infty}s^{-2\beta\lambda}e^{\frac{-2\theta
s}{p}}\,{\rm d}s\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2}$
$\displaystyle\lesssim
L_{\sigma}^{p}p^{1-2\beta\lambda}b_{t}(\theta)\left[\int_{0}^{\infty}\frac{e^{-2\theta
r}}{r^{2\beta\lambda}}\,{\rm d}r\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm
d}z\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}p^{\frac{p(1-2\beta\lambda)}{2}}b_{t}(\theta)(2\theta)^{\frac{p(2\beta\lambda-1)}{2}}\Gamma(1-2\beta\lambda)^{\frac{p}{2}}\left[\int_{B\times
B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2},$ (6.8)

where $b_{t}(\theta)=\sup_{0<s<t}\sup_{\text{z}\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\gamma)}(\text{z})-u_{s}^{(\beta)}(\text{z})|^{p}$. For
$t>0$, we fix $\theta>0$ sufficiently large so that

$\displaystyle J_{3}\lesssim\frac{1}{2}b_{t}(\theta).$ (6.9)

Next we work on $J_{2}$. Using Proposition 6.2 below and the assumption on
$\sigma$, we obtain

$\displaystyle J_{2}$ $\displaystyle:=e^{-\theta
t}\bigg{[}\int_{0}^{t}\int_{B\times
B}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)||G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bigg{[}[{\mathbb{E}}|\sigma(u_{s}^{(\gamma)}(y))\sigma(u_{s}^{(\gamma)}(z))|^{p/2}]^{2/p}\,f(y,\,z)\,{\rm
d}y\,{\rm d}z\,{\rm d}s\bigg{]}^{p/2}$ $\displaystyle\lesssim\sup_{t>0,\,x\in
B}e^{-\theta
t}{\mathbb{E}}|u_{s}^{(\gamma)}(x)|^{p}\bigg{[}\int_{0}^{t}\int_{B\times
B}e^{-2\theta(t-s)/p}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\,f(y,\,z)\,{\rm
d}y\,{\rm d}z\,{\rm d}s\bigg{]}^{p/2}.$ (6.10)

We combine the above estimates and obtain the following.

$\displaystyle\sup_{x\in B}e^{-\theta s}$
$\displaystyle{\mathbb{E}}|u_{s}^{(\gamma)}(x)-u_{s}^{(\beta)}(x)|^{p}\lesssim
e^{-\theta
t}|(\mathcal{G}_{B}^{(\gamma)}u)_{t}(x)-(\mathcal{G}_{B}^{(\beta)}u)_{t}(x)|^{p}+\frac{1}{2}b_{t}(\theta)+$
$\displaystyle\sup_{t>0,\,x\in B}e^{-\theta
t}{\mathbb{E}}|u_{s}^{(\gamma)}(x)|^{p}\bigg{[}\int_{0}^{t}\int_{B\times
B}e^{-2\theta(t-s)/p}|G_{B}^{(\gamma)}(t-s,\,x,\,y)-G_{B}^{(\beta)}(t-s,\,x,\,y)|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}|G_{B}^{(\gamma)}(t-s,\,x,\,z)-G_{B}^{(\beta)}(t-s,\,x,\,z)|\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$

Since Lemma 2.6 and Proposition 6.1 allow us to use dominated convergence
theorem, taking $\gamma\rightarrow\beta$ the result of the theorem follows. ∎

The following proposition is used in the proof of Theorem 1.16. We use the
bounds of $G_{B}^{(\beta)}(t,\,x,\,y)$ to show the proposition holds:

######  Proposition 6.2.

Assume that $\\{\varphi_{n}\\}_{n\geqslant 1}$ are uniformly bounded by a
constant $C(B)$. Suppose that $d<\frac{\alpha}{2\beta}$, and $\int_{B\times
B}f(x,\,y)\,{\rm d}x\,{\rm d}y<\infty$. For some $\theta$, the supremum on the
$p^{th}$ moment of the solution $u_{s}^{(\beta)}(x)$ to equation (1.12) given
by

$\sup_{0\leqslant s\leqslant t,\,x\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\beta)}(x)|^{p}$ (6.11)

is bounded above by a constant independent of $\beta$.

###### Proof.

We use the mild formulation in equation (1.14) and follow similar computations
to those used in the above proof. Now consider

$u_{t}^{(\beta)}(x)=(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})+\lambda\int_{B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})\sigma(u_{s}^{(\beta)}(\text{y}))F({\rm
d}s\,,{\rm d}\text{y}),$ (6.12)

Now applying the inequality $(a+b)^{p}\leqslant 2^{p}(a^{p}\,+b^{p})$ for any
$a,b\geqslant 0$ and using Burkholder-Davis-Gundy inequality, the $p^{th}$
moment of (6.12) becomes

$\displaystyle{\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}$ $\displaystyle\leqslant
2^{p}|(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})|^{p}+(2\lambda)^{p}\bigg{[}\int_{B\times
B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bigg{[}{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))\sigma(u_{s}^{(\beta)}(\text{z}))|^{p/2}\bigg{]}^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{y}\,{\rm d}\text{z}\bigg{]}^{\frac{p}{2}},$ (6.13)
$\displaystyle e^{-\theta t}{\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}$
$\displaystyle\lesssim e^{-\theta
t}|(\mathcal{G}_{B}^{(\beta)}u_{0})_{t}(\text{x})|^{p}+e^{-\theta
t}\bigg{[}\int_{B\times
B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})|\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bigg{[}{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))\sigma(u_{s}^{(\beta)}(\text{z}))|^{p/2}\bigg{]}^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{y}\,{\rm d}\text{z}\bigg{]}^{\frac{p}{2}}$
$\displaystyle:=J_{1}+J_{2}.$ (6.14)

Since $\sigma$ is globally Lipschitz, using Lemma 2.4, and Lemma 2.8(a), the
second term becomes

$\displaystyle J_{2}$ $\displaystyle:=e^{-\theta t}\bigg{[}\int_{B\times
B}\int_{0}^{t}|G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{y})G_{B}^{(\beta)}(t-s,\,\text{x},\,\text{z})|\bigg{[}{\mathbb{E}}|\sigma(u_{s}^{(\beta)}(\text{y}))\sigma(u_{s}^{(\beta)}(\text{z}))|^{p/2}\bigg{]}^{2/p}\,f(y,\,z)\,{\rm
d}s\,{\rm d}\text{y}\,{\rm d}\text{z}\bigg{]}^{\frac{p}{2}},$
$\displaystyle\lesssim L_{\sigma}^{p}\bigg{[}\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\times$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}e^{\frac{-2\theta
t}{p}}[({\mathbb{E}}[(u_{s}^{(\beta)}(y))^{p})^{\frac{1}{p}}({\mathbb{E}}[(u_{s}^{(\beta)}(z))]^{p})^{\frac{1}{p}}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\bigg{]}^{p/2}.$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{B\times
B}\int_{0}^{t}G_{B}^{(\beta)}(t-s,\,x,\,y)G_{B}^{(\beta)}(t-s,\,x,\,z)\,e^{\frac{-2\theta(t-s)}{p}}\,f(y,\,z)\,{\rm
d}s\,{\rm d}y\,{\rm d}z\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{t}(t-s)^{-2\beta\lambda}e^{\frac{-2\theta(t-s)}{p}}\,{\rm
d}s\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2}$
$\displaystyle\lesssim
L_{\sigma}^{p}b_{t}(\theta)\left[\int_{0}^{\infty}s^{-2\beta\lambda}e^{\frac{-2\theta
s}{p}}\,{\rm d}s\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2}$
$\displaystyle\lesssim
L_{\sigma}^{p}p^{1-2\beta\lambda}b_{t}(\theta)\left[\int_{0}^{\infty}\frac{e^{-2\theta
r}}{r^{2\beta\lambda}}\,{\rm d}s\,\int_{B\times B}f(y,\,z){\rm d}y\,{\rm
d}z\right]^{p/2}$ $\displaystyle\lesssim
L_{\sigma}^{p}p^{\frac{p(1-2\beta\lambda)}{2}}b_{t}(\theta)(2\theta)^{\frac{p(2\beta\lambda-1)}{2}}\Gamma(1-2\beta\lambda)^{\frac{p}{2}}\left[\int_{B\times
B}f(y,\,z){\rm d}y\,{\rm d}z\right]^{p/2},$ (6.15)

where $b_{t}(\theta)=\sup_{0<s<t}\sup_{\text{z}\in B}e^{-\theta
s}{\mathbb{E}}|u_{s}^{(\beta)}(\text{z}|^{p}$. For $t>0$, we fix $\theta>0$
sufficiently large so that

$\displaystyle J_{2}\lesssim\frac{1}{2}b_{t}(\theta).$ (6.16)

Thus, we have

$\displaystyle e^{-\theta t}{\mathbb{E}}|u_{t}^{(\beta)}(x)|^{p}$
$\displaystyle\lesssim 1+\frac{1}{2}b_{t}(\theta).$ (6.17)

Taking the supremum of the left side of (6.17) the result follows. ∎

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