#  $L_{p}$-solvability and Hölder regularity for stochastic time fractional
Burgers’ equations driven by multiplicative space-time white noise

Beom-Seok Han  Department of Mathematics, Pohang University of Science and
Technology, 77, Cheongam-ro, Nam-gu, Pohang, Gyeongbuk, 37673, Republic of
Korea<EMAIL_ADDRESS>
###### Abstract.

We present the $L_{p}$-solvability for stochastic time fractional Burgers’
equations driven by multiplicative space-time white noise:

$\partial_{t}^{\alpha}u=a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+\bar{b}^{i}uu_{x^{i}}+\partial_{t}^{\beta}\int_{0}^{t}\sigma(u)dW_{t},\,t>0;\,\,u(0,\cdot)=u_{0},$

where $\alpha\in(0,1)$, $\beta<3\alpha/4+1/2$, and
$d<4-2(2\beta-1)_{+}/\alpha$. The operators $\partial_{t}^{\alpha}$ and
$\partial_{t}^{\beta}$ are the Caputo fractional derivatives of order $\alpha$
and $\beta$, respectively. The process $W_{t}$ is an
$L_{2}(\mathbb{R}^{d})$-valued cylindrical Wiener process, and the
coefficients $a^{ij},b^{i},c$ and $\sigma(u)$ are random.

In addition to the existence and uniqueness of a solution, we also suggest the
Hölder regularity of the solution. For example, for any constant $T<\infty$,
small $\varepsilon>0$, and almost sure $\omega\in\Omega$, we have

$\sup_{x\in\mathbb{R}^{d}}|u(\omega,\cdot,x)|_{C^{\left[\frac{\alpha}{2}\left(\left(2-(2\beta-1)_{+}/\alpha-d/2\right)\wedge
1\right)+\frac{(2\beta-1)_{-}}{2}\right]\wedge 1-\varepsilon}([0,T])}<\infty$

and

$\sup_{t\leq
T}|u(\omega,t,\cdot)|_{C^{\left(2-(2\beta-1)_{+}/\alpha-d/2\right)\wedge
1-\varepsilon}(\mathbb{R}^{d})}<\infty.$

The Hölder regularity of the solution in time changes behavior at $\beta=1/2$.
Furthermore, if $\beta\geq 1/2$, then the Hölder regularity of the solution in
time is $\alpha/2$ times the one in space.

###### Key words and phrases:

Stochastic partial differential equation, Time fractional derivatives,
Stochastic Burgers’ equation, Time fractional Burgers’ equation, Space-time
white noise, Hölder regularity

###### 2020 Mathematics Subject Classification:

35R11, 26A33, 60H15, 35R60

This work was supported by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No. NRF-2021R1C1C2007792) and the
BK21 Fostering Outstanding Universities for Research (FOUR) funded by the
Ministry of Education (MOE, Korea) and the National Research Foundation of
Korea (NRF)

##  1\. Introduction

This article investigates the existence, uniqueness, $L_{p}$-regularity, and
maximal Hölder regularity of a solution to stochastic time fractional Burgers’
equations (STFBEs) driven by space-time white noise. We consider

$\displaystyle\partial_{t}^{\alpha}u=Lu+\bar{b}^{i}uu_{x^{i}}+\partial_{t}^{\beta}\int_{0}^{t}\sigma(u)dW_{t},\quad(\omega,t,x)\in\Omega\times(0,\infty)\times\mathbb{R}^{d};\quad
u(0,\cdot)=u_{0},$ (1.1)

where $\alpha\in(0,1)$, $\beta<\frac{3}{4}\alpha+\frac{1}{2}$, and
$d<4-\frac{2(2\beta-1)_{+}}{\alpha}$. The operators $\partial_{t}^{\alpha}$
and $\partial_{t}^{\beta}$ are the Caputo fractional derivatives of order
$\alpha$ and $\beta$, and the operator $L$ is the second order random
differential operator defined as follows:

$\begin{gathered}(Lu)(\omega,t,x)=a^{ij}(\omega,t,x)u_{x^{i}x^{j}}+b^{i}(\omega,t,x)u_{x^{i}}+c(\omega,t,x)u.\end{gathered}$

The random coefficients $a^{ij}$, $b^{i}$, and $c$ are predictable,
differentiable (or continuous), and bounded functions. The diffusion
coefficient $\sigma(u)=\sigma(\omega,t,x,u)$ is a predictable and measurable
function satisfying growth conditions and Lipschitz continuity in $u$. The
detailed conditions on $a^{ij}$, $b^{i}$, $c$, and $\sigma$ are described in
Assumptions 3.1 and 3.3. The random measure $dW_{t}$ is induced from an
$L_{2}(\mathbb{R}^{d})$-valued cylindrical Wiener process $W_{t}$.

When $\alpha=\beta=1$ in equation (1.1), the equation is said to be a
stochastic Burgers’ equation (SBE) of form

$\partial_{t}u=Lu+\bar{b}uu_{x}+\sigma(u)\dot{W},\quad(\omega,t,x)\in\Omega\times(0,\infty)\times\mathbb{R};\quad
u(0,\cdot)=u_{0},$ (1.2)

where $\dot{W}$ is the space-time white noise. Numerous studies have been
conducted on the equation (1.2), but we only refer to the reader to [13, 14,
29]. In [13], the author proved the uniqueness, existence, and continuity of a
solution to a semilinear equation, including an equation of type (1.2) on the
unit interval $(0,1)$. Additionally, the same properties of a solution on
$\mathbb{R}$ were obtained in [14] when the $L_{2}$ bounded conditions on
$\sigma(u)$ were imposed. In [29], the authors investigated the Hölder
regularity and moment estimates of the random field solution to (1.2) with
$L=\Delta$ and $\bar{b}=-1$.

In contrast, (deterministic) partial differential equations with Caputo
fractional derivatives have been used in many fields, such as electrochemical
processes [5, 19], dielectric polarization [33], viscoelastic materials [32],
biology [31], and physics [11, 18]. Especially, equation (1.1) with
$\alpha\in(0,1)$ and $\sigma(u)=0$ is called a time fractional Burgers’
equation (TFBE), which describes the propagation of waves through viscous
media ([1, 2]). Indeed, various researches have been conducted on numerical
analysis for the TFBE (see [3, 9, 10, 20, 30]). From a mathematical
standpoint, it is reasonable to wonder whether it is possible to demonstrate
the uniqueness and existence of a solution to STFBE (1.1), and also to obtain
the Hölder regularity of the solution. To the best of our knowledge, [36] is
the only study that answers this question. The authors of [36] demonstrate the
existence, uniqueness, and regularity of the mild solution to SBEs with
fractional derivatives in time and space on a bounded domain
$\mathcal{D}\subset\mathbb{R}^{d}$.

In this paper, we provide the $L_{p}$ uniqueness, existence, and regularity of
a strong solution to equation (1.1) with random second order differential
operator $L$ on the whole spatial domain $\mathbb{R}^{d}$. Additionally, we
achieve the Hölder regularity of the solution in time and space. In detail, if
$u(\omega,t,x)$ denotes the solution to equation (1.1), then for any bounded
stopping time $\tau\leq T$ and small constant $\varepsilon>0$, almost surely,

$\begin{gathered}\sup_{x\in\mathbb{R}^{d}}|u(\omega,\cdot,x)|_{C^{\left[\frac{\alpha}{2}\left(\left(2-(2\beta-1)_{+}/\alpha-d/2\right)\wedge
1\right)+\frac{(2\beta-1)_{-}}{2}\right]\wedge
1-\varepsilon}([0,\tau])}<\infty,\\\
\sup_{t\leq\tau}|u(\omega,t,\cdot)|_{C^{\left(2-(2\beta-1)_{+}/\alpha-d/2\right)\wedge
1-\varepsilon}(\mathbb{R}^{d})}<\infty.\end{gathered}$ (1.3)

where $a_{+}=(|a|+a)/2$, $a_{-}=(|a|-a)/2$, and $C^{\gamma}(\mathcal{D})$ is
the Hölder spaces. Observe that the behavior of the Hölder regularity of the
solution in time changes from $\beta=1/2$. For example, if $\beta\geq 1/2$,
then the Hölder regularity of the solution in time is $\alpha/2$ times that of
the regularity in space. Additionally, we can recover the the Hölder
regularity results of SBEs by letting $\alpha,\beta\uparrow 1$. These results
are consistent with the well-known results of stochastic heat equations driven
by space-time white noise (e.g. [26, Remark 8.7] or [16, Corollary 3.1]). In
contrast, if $\beta<1/2$, the Hölder regularity in time gains additional
regularity by as much as $1/2-\beta$. (Remark 3.11).

Several remarks about the proof are made. The proof strategy for the main
theorem (Theorem 3.5) is based on [16]. However, some differences exist
because since it is not certain that Itô’s formula and the maximum principle
hold for STFBE (1.1).

Thus, the proof proceeds as follows. As in [16], we focus on proving the
uniqueness and existence of the $L_{p}$ solution in each (large) $p>2$, and
the main difficulty is to demonstrating the existence of the solutions. Hence,
we consider the cut-off form of equation (1.1) to obtain local solutions.
Afterward, we construct a global solution candidate $u$ by pasting the local
solutions (Lemma 4.3 and Remark 4.4). A uniform $L_{p}$ bound of $u$ is
required to show that our the candidate $u$ is a global solution; thus, we
divide the local solution into two parts: the noise-dominating and the
nonlinear-dominating part. To estimate the noise-dominating parts, we employ
the $L_{p}$ bound of the diffusion coefficient $\sigma(u)$ (Lemma 4.5). In
contrast, to control the nonlinear-dominating part (Lemma 4.8), we employ an
inequality similar to the chain rule (Lemma 4.6) and a version of the Grönwall
inequality including the Caputo fractional derivatives (Theorem 4.7).

To obtain the maximal Hölder regularity of the solution to equation (1.1), we
require two components: the Hölder embedding theorem for the solution space
$\mathcal{H}_{p}^{\gamma}(\tau)$ (Theorem 2.16) and the uniqueness of the
solution in $p$ (Theorem 3.9). Indeed, when the $L_{p}$ existence and
uniqueness of a solution are given, we have the Hölder regularity of the
solution in each (large) $p>2$ by employing the Hölder embedding theorem for
the solution space (Theorem 2.16 and Theorem 3.5). The Hölder regularity of
the solution becomes larger as a large $p$ is chosen; thus, we have to select
$p$ that is as large as possible. Therefore, we require the uniqueness of
solutions in $p$ because $p$ varies.

This article is organized as follows. Section 2 introduces the definitions and
properties of space-time white noise, fractional calculus, and stochastic
Banach spaces. Additionally, we present the Hölder embedding theorem for the
solution space $\mathcal{H}_{p}^{\gamma}(\tau)$. Section 3 states the main
results of this article and suggests some remarks. The proof of the main
results is presented in Section 4. Next, Section 5 proves the Hölder embedding
theorem for the solution space $\mathcal{H}_{p}^{\gamma}(\tau)$.

We finish this section with an introduction to the notation used in this
paper. The sets $\mathbb{N}$ and $\mathbb{R}$ are sets of natural and real
numbers, respectively. The set $\mathbb{R}^{d}$ denotes the $d$-dimensional
Euclidean space of points $x=(x^{1},\dots,x^{d})$ for $x^{i}\in\mathbb{R}$.
Throughout this paper, we assume Einstein’s summation convention on
$i,j,k\in\mathbb{N}$. We use $:=$ to denote a definition. For a real-valued
function $f$, we set the following:

$f_{+}:=\frac{|f|+f}{2}\quad\text{and}\quad f_{-}:=\frac{|f|-f}{2}.$

For a normed space $F$, a measure space $(X,\mathcal{M},\mu)$, and
$p\in[1,\infty)$, a space $L_{p}(X,\mathcal{M},\mu;F)$ is a set of $F$-valued
$\mathcal{M}^{\mu}$-measurable functions such that

$\|u\|_{L_{p}(X,\mathcal{M},\mu;F)}:=\left(\int_{X}\|u(x)\|_{F}^{p}\mu(dx)\right)^{1/p}<\infty.$

A set $\mathcal{M}^{\mu}$ is the completion of $\mathcal{M}$ with respect to
the measure $\mu$. For $\gamma\in(0,1]$ and $k=0,1,2,\dots$, a set
$C^{k+\gamma}(\mathbb{R}^{d})$ is the set of $\mathbb{R}$-valued continuous
functions $u=u(x)$ such that

$|u|_{C^{\gamma+k}(\mathbb{R}^{d})}:=\sup_{x\in\mathbb{R}^{d},|\bm{\beta}|=k}\left|D^{\bm{\beta}}u(x)\right|+\sup_{\begin{subarray}{c}x,y\in\mathbb{R}^{d},x\neq
y\\\
|\bm{\beta}|=k\end{subarray}}\frac{\left|D^{\bm{\beta}}u(x)-D^{\bm{\beta}}u(y)\right|}{|x-y|^{\gamma}}<\infty,$

where $\bm{\beta}$ is a multi-index. Similarly, for $\gamma\in(0,1]$ and
$0<T<\infty$, the set $C^{\gamma}([0,T];F)$ is the set of $F$-valued
continuous functions $u$ such that

$|u|_{C^{\gamma}([0,T];F)}:=\sup_{t\in[0,T]}\left|u(t)\right|_{F}+\sup_{\begin{subarray}{c}t,s\in[0,T],\\\
s\neq
t\end{subarray}}\frac{\left|u(t)-u(s)\right|_{F}}{|t-s|^{\gamma}}<\infty.$

For $a,b\in\mathbb{R}$, we set $a\wedge b:=\min\\{a,b\\}$ and $a\vee
b:=\max\\{a,b\\}$. Let $\mathcal{S}=\mathcal{S}(\mathbb{R}^{d})$ denote the
set of Schwartz functions on $\mathbb{R}^{d}$. Let
$N=N(a_{1},a_{2},...,a_{k})$ be a generic constant if $N$ depends only on
$a_{1},a_{2},...,a_{k}$. The constant $N$ can vary line by line. For functions
depending on $\omega$, $t$, and $x$, the argument $\omega\in\Omega$ is
omitted. Finally, for $x\in\mathbb{R}^{d}$,
$\bar{x}^{i}:=(x^{1},\dots,x^{i-1},x^{i+1},\dots,x^{d})$.

##  2\. Preliminaries

In this section, we introduce the definitions and properties of space-time
white noise, fractional calculus, and stochastic Banach spaces. Throughout
this paper, $(\Omega,\mathcal{F},P)$ is a complete probability space equipped
with a filtration $\\{\mathcal{F}_{t}\\}_{t\geq 0}$. Let
$\\{\mathcal{F}_{t}\\}_{t\geq 0}$ denote a filtration satisfying the usual
conditions. Let $\mathcal{P}$ be the predictable $\sigma$-field related to
$\\{\mathcal{F}_{t}\\}_{t\geq 0}$.

First, we present the space-time white noise $\dot{W}$ to understand the
stochastic part of (1.1).

######  Definition 2.1 (Space-time white noise).

A generalized random field $\dot{W}$ is said to be the space-time white noise
if it is a centered Gaussian random field such that its covariance is given by

$\mathbb{E}\,\dot{W}(h)\,\dot{W}(g)=\int_{0}^{\infty}\int_{\mathbb{R}^{d}}h(t,x)g(t,x)dxdt,\quad\forall
h,g\in L_{2}((0,\infty)\times\mathbb{R}^{d}).$

######  Remark 2.2.

We employ a series of Itô’s stochastic integral to interpret the stochastic
part of equation (1.1). More precisely, let $\\{\eta^{k}:k\in\mathbb{N}\\}$ be
an orthonormal basis on $L_{2}(\mathbb{R}^{d})$. If we define

$w_{t}^{k}:=\int_{0}^{t}\int_{\mathbb{R}^{d}}\eta^{k}(x)\dot{W}(ds,dx)$

using the Walsh integral (see [35]), then $\\{w_{t}^{k}:k\in\mathbb{N}\\}$ is
a set of one dimensional independent Wiener processes. Then, if we set (see
[26, Section 8.3], and [23, Section 7])

$W_{t}:=\sum_{k=1}^{\infty}\eta^{k}w_{t}^{k},$

then $W_{t}$ is an $L_{2}(\mathbb{R}^{d})$-valued cylindrical Wiener process
and $dW_{t}=\sum_{k}\eta^{k}dw_{t}^{k}$. Thus, equation (1.1) can be rewritten
as

$\partial_{t}^{\alpha}u=Lu+\bar{b}^{i}uu_{x^{i}}+\partial_{t}^{\beta}\int_{0}^{t}\sigma(u)\eta^{k}dw^{k}_{t},\quad(\omega,t,x)\in\Omega\times(0,\infty)\times\mathbb{R}^{d};\quad
u(0,\cdot)=u_{0}.$

Next, we review the facts of fractional calculus. For more information, we
refer to the reader to [6, 17, 21, 32].

######  Definition 2.3.

Let $\alpha>0$, and for $\varphi\in L_{1}((0,T))$, the Riemann-Liouville
fractional integral of the order $\alpha$ is defined as follows:

$I_{t}^{\alpha}\varphi(t):=(I^{\alpha}_{t}\varphi)(t):=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\varphi(s)ds\quad\mbox{for
all}\quad t\in(0,T),$

where $\Gamma(\alpha):=\int_{0}^{\infty}t^{\alpha-1}e^{-t}dt$.

######  Remark 2.4.

For any $q\in[1,\infty]$, by Jensen’s inequality

$\|I^{\alpha}\varphi\|_{L_{q}((0,T))}\leq
N(\alpha,p,T)\|\varphi\|_{L_{q}((0,T))}.$ (2.1)

Therefore, $I_{t}^{\alpha}\varphi(t)$ is well-defined and finite for almost
all $t\leq T$. Additionally, Fubini’s theorem implies that, for
$\alpha,\beta\geq 0$, we have

$I^{\alpha+\beta}\varphi(t)=I^{\alpha}I^{\beta}\varphi(t).$

######  Definition 2.5.

For $\alpha>0$, let $n\in\mathbb{N}$ be a nonnegative integer such that
$n-1\leq\alpha<n$. Suppose $\varphi(t)$ is a real-valued function on $[0,T]$
such that $\varphi$ is $(n-1)$-times differentiable and
$(\frac{d}{dt})^{n-1}\varphi$ is absolutely continuous on $[0,T]$.

  1. (i)

The Riemann-Liouville fractional derivative $D_{t}^{\alpha}\varphi$ is defined
as

$D_{t}^{\alpha}\varphi(t):=\frac{1}{\Gamma(n-\alpha)}\frac{d^{n}}{dt^{n}}\int_{0}^{t}(t-s)^{n-\alpha-1}\varphi(s)ds.$

  2. (ii)

The Caputo fractional derivative $\partial_{t}^{\alpha}\varphi$ is defined as

$\displaystyle\partial_{t}^{\alpha}\varphi$
$\displaystyle:=\frac{1}{\Gamma(n-\alpha)}\int_{0}^{t}(t-s)^{n-\alpha-1}\varphi^{(n)}(s)ds$
$\displaystyle:=\frac{1}{\Gamma(n-\alpha)}\frac{d}{dt}\int_{0}^{t}(t-s)^{n-\alpha-1}\left[\varphi^{(n-1)}(s)-\varphi^{(n-1)}(0)\right]ds.$

######  Remark 2.6.

  1. (i)

For any $\alpha,\beta\geq 0$,
$D^{\alpha}_{t}D^{\beta}_{t}\varphi=D^{\alpha+\beta}_{t}\varphi$ and

$D^{\alpha}_{t}I^{\beta}_{t}\varphi=D^{\alpha-\beta}_{t}\varphi
1_{\alpha>\beta}+I_{t}^{\beta-\alpha}\varphi 1_{\alpha\leq\beta}.$

Additionally, if $\alpha\in(0,1)$, $I^{1-\alpha}_{t}\varphi$ is absolutely
continuous, and $I^{1-\alpha}_{t}\varphi(0)=0$, then the following equality
holds:

$I^{\alpha}_{t}D^{\alpha}_{t}\varphi(t)=\varphi(t).$

  2. (ii)

By the definition of fractional derivatives, if
$\varphi(0)=\varphi^{(1)}(0)=\cdots=\varphi^{(n-1)}(0)=0$, then
$D^{\alpha}_{t}\varphi=\partial_{t}^{\alpha}\varphi$.

Below we recall the definitions and properties of stochastic Banach spaces
(for more detail, see [12, 25, 26, 27]). The solution space
$\mathcal{H}_{p}^{\gamma}(T)$ and embedding theorems for
$\mathcal{H}_{p}^{\gamma}(T)$ are suggested.

######  Definition 2.7.

Let $p>1$ and $\gamma\in\mathbb{R}$. The space
$H_{p}^{\gamma}=H_{p}^{\gamma}(\mathbb{R}^{d})$ is the set of all tempered
distributions $u$ on $\mathbb{R}$ such that

$\|u\|_{H_{p}^{\gamma}}:=\left\|(1-\Delta)^{\gamma/2}u\right\|_{L_{p}}=\left\|\mathcal{F}^{-1}\left[(1+|\xi|^{2})^{\gamma/2}\mathcal{F}(u)(\xi)\right]\right\|_{L_{p}}<\infty.$

Similarly, $H_{p}^{\gamma}(l_{2})=H_{p}^{\gamma}(\mathbb{R}^{d};l_{2})$ is a
space of $l_{2}$-valued functions $g=(g^{1},g^{2},\cdots)$ such that

$\|g\|_{H_{p}^{\gamma}(l_{2})}:=\left\|\left|(1-\Delta)^{\gamma/2}g\right|_{l_{2}}\right\|_{L_{p}}=\left\|\left|\mathcal{F}^{-1}\left[\left(1+|\xi|^{2}\right)^{\gamma/2}\mathcal{F}(g)(\xi)\right]\right|_{l_{2}}\right\|_{L_{p}}<\infty.$

######  Remark 2.8.

Let $d\in\mathbb{N}$ and $\gamma\in(0,\infty)$. A nonnegative smooth function
$R_{\gamma}(x)$ exists on $\mathbb{R}^{d}$ such that, for $u\in
C_{c}^{\infty}(\mathbb{R}^{d})$,

$\left((1-\Delta)^{-\gamma/2}\,u\right)(x)=\int_{\mathbb{R}^{d}}R_{\gamma}(y)u(x-y)dy$

and

$\left|R_{\gamma}(x)\right|\leq NA_{\gamma,d}(x)1_{|x|\leq
2}+Ne^{-|x|/2}1_{|x|\geq 2},$

where $N=N(\gamma,d)$ is a positive constant and

$A_{\gamma,d}(x)=\begin{cases}|x|^{\gamma-d}+1+O(|x|^{\gamma-d+2})\quad&\mbox{for}\quad
0<\gamma<d,\\\ \log(2/|x|)+1+O(|x|^{2})\quad&\mbox{for}\quad\gamma=d,\\\
1+O(|x|^{\gamma-d})\quad&\mbox{for}\quad\gamma>d.\end{cases}$

For more detail, see [12, Proposition 1.2.5].

We introduce the space of point-wise multipliers in $H_{p}^{\gamma}$.

######  Definition 2.9.

Fix $\gamma\in\mathbb{R}$ and $\alpha\in[0,1)$ such that $\alpha=0$ if
$\gamma\in\mathbb{Z}$ and $\alpha>0$ if $|\gamma|+\alpha$ is not an integer.
Define

$\displaystyle B^{|\gamma|+\alpha}=\begin{cases}B(\mathbb{R})&\quad\text{if
}\gamma=0,\\\ C^{|\gamma|-1,1}(\mathbb{R})&\quad\text{if $\gamma$ is a nonzero
integer},\\\
C^{|\gamma|+\alpha}(\mathbb{R})&\quad\text{otherwise},\end{cases}$
$\displaystyle
B^{|\gamma|+\alpha}(\ell_{2})=\begin{cases}B(\mathbb{R},\ell_{2})&\quad\text{if
}\gamma=0,\\\ C^{|\gamma|-1,1}(\mathbb{R},\ell_{2})&\quad\text{if $\gamma$ is
a nonzero integer},\\\
C^{|\gamma|+\alpha}(\mathbb{R},\ell_{2})&\quad\text{otherwise},\end{cases}$

where $B(\mathbb{R})$ is the space of bounded Borel functions on $\mathbb{R}$,
$C^{|\gamma|-1,1}(\mathbb{R})$ represents the space of $|\gamma|-1$ times
continuous differentiable functions whose derivatives of the $(|\gamma|-1)$th
order derivative are Lipschitz continuous, and $C^{|\gamma|+\alpha}$ is the
real-valued Hölder spaces. The space $B(\ell_{2})$ denotes a function space
with $\ell_{2}$-valued functions instead of real-valued function spaces.

Below we collect the properties of Bessel potential spaces.

######  Lemma 2.10.

Let $\gamma\in\mathbb{R}$ and $p>1$.

  1. (i)

The space $C_{c}^{\infty}(\mathbb{R}^{d})$ is dense in $H_{p}^{\gamma}$.

  2. (ii)

Let $\gamma-d/p=n+\nu$ for some $n=0,1,\cdots$ and $\nu\in(0,1]$. Then, for
any $k\in\\{0,1,\cdots,n\\}$, we have

$|D^{k}u|_{C(\mathbb{R}^{d})}+|D^{n}u|_{\mathcal{C}^{\nu}(\mathbb{R}^{d})}\leq
N\|u\|_{H_{p}^{\gamma}},$ (2.2)

where $\mathcal{C}^{\nu}(\mathbb{R}^{d})$ is the Zygmund space.

  3. (iii)

The operator $D_{i}:H_{p}^{\gamma}\to H_{p}^{\gamma+1}$ is bounded. Moreover,
for any $u\in H_{p}^{\gamma+1}$,

$\left\|D^{i}u\right\|_{H_{p}^{\gamma}}\leq N\|u\|_{H_{p}^{\gamma+1}},$

where $N=N(\gamma,p)$.

  4. (iv)

For $\gamma_{1},\gamma_{2}\in\mathbb{R}$, and $u\in
H_{p}^{\gamma_{1}+\gamma_{2}}$, we have

$\|\Delta^{\gamma_{1}/2}u\|_{H_{p}^{\gamma_{2}}}\leq
N\|u\|_{H_{p}^{\gamma_{1}+\gamma_{2}}},$

where $N=N(\gamma_{1},\gamma_{2})$.

  5. (v)

For $\gamma\in(0,1)$, and $u\in H_{p}^{\gamma}$, we have

$\|(1-\Delta^{\gamma})u\|_{L_{p}}\leq
N\left(\|u\|_{L_{p}}+\|(-\Delta)^{\gamma}u\|_{L_{p}}\right),$

where $N=N(\gamma,p)$.

  6. (vi)

For any $\mu,\gamma\in\mathbb{R}$, the operator
$(1-\Delta)^{\mu/2}:H_{p}^{\gamma}\to H_{p}^{\gamma-\mu}$ is an isometry.

  7. (vii)

Let

$\begin{gathered}\varepsilon\in[0,1],\quad
p_{i}\in(1,\infty),\quad\gamma_{i}\in\mathbb{R},\quad i=0,1,\\\
\gamma=\varepsilon\gamma_{1}+(1-\varepsilon)\gamma_{0},\quad
1/p=\varepsilon/p_{1}+(1-\varepsilon)/p_{0}.\end{gathered}$

Then, we have

$\|u\|_{H^{\gamma}_{p}}\leq\|u\|^{\varepsilon}_{H^{\gamma_{1}}_{p_{1}}}\|u\|^{1-\varepsilon}_{H^{\gamma_{0}}_{p_{0}}}.$

  8. (viii)

Let $u\in H_{p}^{\gamma}$. Then, we have

$\|au\|_{H_{p}^{\gamma}}\leq
N\|a\|_{B^{|\gamma|+\alpha}}\|u\|_{H_{p}^{\gamma}}\quad\text{and}\quad\|bu\|_{H_{p}^{\gamma}(\ell_{2})}\leq
N\|b\|_{B^{|\gamma|+\alpha}(\ell_{2})}\|u\|_{H_{p}^{\gamma}},$

where $N=N(\gamma,p)$ and $B^{|\gamma|+\alpha},B^{|\gamma|+\alpha}(\ell_{2})$
are introduced in Definition 2.9.

###### Proof.

The above results are well known. For (i), (iii), (vi), and (vii), see
Theorems 13.3.7 (i), 13.8.1, 13.3.7 (ii), and Exercise 13.3.20 of [27],
respectively. In the case of (ii) and (iv), see [34]. For (v), see Theorems
1.3.6 and 1.3.8 of [12]. For (viii), we refer the reader to [26, Lemma 5.2].

∎

######  Definition 2.11 (Stochastic Banach spaces).

Let $\tau\leq T$ be a bounded stopping time, $p\geq 2$, and
$\gamma\in\mathbb{R}$. Set $\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$}:=\\{(\omega,t):0<t\leq\tau(\omega)\\}$
and define

$\displaystyle\mathbb{H}_{p}^{\gamma}(\tau):=L_{p}\left(\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$},\mathcal{P},dP\times
dt;H_{p}^{\gamma}\right),$
$\displaystyle\mathbb{H}_{p}^{\gamma}(\tau,l_{2}):=L_{p}\left(\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$},\mathcal{P},dP\times
dt;H_{p}^{\gamma}(l_{2})\right),$ $\displaystyle
U_{p}^{\gamma}:=L_{p}\left(\Omega,\mathcal{F}_{0},H_{p}^{\gamma}\right).$

We write $u\in\mathbb{H}_{p}^{\gamma}$ if $u\in\mathbb{H}_{p}^{\gamma}(\tau)$
exists for any bounded stopping time $\tau$. Additionally, if $\gamma=0$, then
we use $\mathbb{L}$ instead of $\mathbb{H}$,
$\|f\|_{\mathbb{L}_{p}(\tau)}:=\|f\|_{\mathbb{H}_{p}^{0}(\tau)}$. The norm of
each space is defined naturally, for example,

$\|f\|_{\mathbb{H}_{p}^{\gamma}(\tau)}:=\left(\mathbb{E}\int_{0}^{\tau}\|f(t,\cdot)\|_{H_{p}^{\gamma}}^{p}\,dt\right)^{1/p}.$

Lemma 2.12 exhibits the relation between the stochastic and fractional
integrals, which is employed when $I_{t}^{\alpha}$ or $D_{t}^{\alpha}$ is
applied to the stochastic part of the SPDEs.

######  Lemma 2.12.

Let $T<\infty$ be a constant.

  1. (i)

Let $\alpha\geq 0$ and $h\in L_{2}(\Omega\times[0,T],\mathcal{P};l_{2})$.
Then, the equality

$I^{\alpha}\left(\sum_{k=1}^{\infty}\int_{0}^{\cdot}h^{k}(s)dw_{s}^{k}\right)(t)=\sum_{k=1}^{\infty}\left(I^{\alpha}\int_{0}^{\cdot}h^{k}(s)dw_{s}^{k}\right)(t)$

holds for all $t\leq T$ almost surely and in $L_{2}(\Omega\times[0,T])$, where
the series on both sides converge in probability.

  2. (ii)

If $\alpha\geq 0$ and $h_{n}\to h$ in
$L_{2}(\Omega\times[0,T],\mathcal{P};l_{2})$ as $n\to\infty$, then

$\sum_{k=1}^{\infty}\left(I^{\alpha}\int_{0}^{\cdot}h_{n}^{k}dw_{s}^{k}\right)(t)\to\sum_{k=1}^{\infty}\left(I^{\alpha}\int_{0}^{\cdot}h^{k}dw_{s}^{k}\right)(t)$

in probability uniformly on $[0,T]$.

  3. (iii)

If $\alpha>1/2$ and $h\in L_{2}(\Omega\times[0,T],\mathcal{P};l_{2})$, then
$\left(I^{\alpha}\sum_{k=1}^{\infty}\int_{0}^{\cdot}h^{k}(s)dw_{s}^{k}\right)(t)$
is differentiable in $t$ and

$\frac{\partial}{\partial
t}\left(I^{\alpha}\sum_{k=1}^{\infty}\int_{0}^{\cdot}h^{k}(s)dw_{s}^{k}\right)(t)=\frac{1}{\Gamma(\alpha)}\sum_{k=1}^{\infty}\int_{0}^{t}(t-s)^{\alpha-1}h^{k}(s)dw_{s}^{k}$

(a.e.) on $\Omega\times[0,T]$.

###### Proof.

See Lemmas 3.1 and 3.3 of [7]. ∎

Fix a small $\kappa_{0}>0$. For $\alpha\in(0,1)$ and $\beta<\alpha+1/2$, set

$c_{0}:=\frac{(2\beta-1)_{+}}{\alpha}+\kappa_{0}1_{\beta=1/2}.$ (2.3)

Next, we introduce the solution spaces (for more detail, see Definitions 2.9
and 2.12 in [25]).

######  Definition 2.13 (Solution spaces).

Let $\tau\leq T$ be a bounded stopping time, $\alpha\in(0,1)$,
$\beta<\alpha+1/2$, $\gamma\in\mathbb{R}$, and $p\geq 2$.

  1. (i)

For $u\in\mathbb{H}_{p}^{\gamma}(\tau)$, we write
$u\in\mathcal{H}^{\gamma}_{p}(\tau)$ if $u_{0}\in U_{p}^{\gamma}$,
$f\in\mathbb{H}_{p}^{\gamma-2}(\tau)$, and
$g\in\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})$ such that

$\displaystyle\partial_{t}^{\alpha}u(t,x)=f(t,x)+\partial_{t}^{\beta}\int_{0}^{t}g^{k}(s,x)dw_{s}^{k},\quad
0<t\leq\tau;\quad u(0,\cdot)=u_{0}$

in the sense of distribution. In other words, for any $\phi\in\mathcal{S}$,
the equality

$\displaystyle(u(t,\cdot),\phi)$
$\displaystyle=(u_{0},\phi)+I_{t}^{\alpha}(f,\phi)+I_{t}^{\alpha-\beta}\sum_{k=1}^{\infty}\int_{0}^{t}(g^{k}(s,\cdot),\phi)dw_{s}^{k}$
(2.4)

holds for a.e. $(\omega,t)\in\Omega\times[0,\tau]$. If
$\alpha-\beta\in(-1/2,0)$, we regard $I_{t}^{\alpha-\beta}$ as
$\frac{\partial}{\partial t}I_{t}^{\alpha-\beta+1}$. The norm in
$\mathcal{H}_{p}^{\gamma}(\tau)$ is defined as follows:

$\|u\|_{\mathcal{H}_{p}^{\gamma}(\tau)}:=\|u\|_{\mathbb{H}_{p}^{\gamma}(\tau)}+\|u_{0}\|_{U_{p}^{\alpha,\gamma}}+\inf_{f,g}\left\\{\|f\|_{\mathbb{H}_{p}^{\gamma-2}(\tau)}+\|g\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}\right\\}.$
(2.5)

  2. (ii)

We say $u\in\mathcal{H}_{p,loc}^{\gamma}(\tau)$ if there exists a sequence
$\tau_{n}\uparrow\tau$ such that $u\in\mathcal{H}_{p}^{\gamma}(\tau_{n})$ for
each $n$. We write $u=v$ in $\mathcal{H}_{p,loc}^{\gamma}(\tau)$ if a sequence
of bounded stopping times $\tau_{n}\uparrow\tau$ exists such that $u=v$ in
$\mathcal{H}_{p}^{\gamma}(\tau_{n})$ for each $n$. We omit $\tau$ if
$\tau=\infty$. In other words,
$\mathcal{H}_{p,loc}^{\gamma}=\mathcal{H}_{p,loc}^{\gamma}(\infty).$

######  Remark 2.14.

If $\alpha-\beta\geq 0$, the stochastic part of (2.4) is considered

$\displaystyle
I_{t}^{\alpha-\beta}\sum_{k=1}^{\infty}\int_{0}^{t}(g^{k}(s,\cdot),\phi)dw_{s}^{k}$
$\displaystyle=\sum_{k=1}^{\infty}I_{t}^{\alpha-\beta}\int_{0}^{t}(g^{k}(s,\cdot),\phi)dw_{s}^{k}.$

Otherwise, if $\alpha-\beta\in(-1/2,0)$, we regard $I_{t}^{\alpha-\beta}$ as
$\frac{\partial}{\partial t}I_{t}^{\alpha-\beta+1}$. Then, by Lemma 2.12
(iii), the stochastic part of (2.4) is

$\displaystyle
I_{t}^{\alpha-\beta}\left(\sum_{k=1}^{\infty}\int_{0}^{t}(g^{k}(s,\cdot),\phi)dw_{s}^{k}\right)$
$\displaystyle=\frac{\partial}{\partial
t}\left(I^{\alpha-\beta+1}\sum_{k=1}^{\infty}\int_{0}^{t}(g^{k}(s,\cdot),\phi)dw_{s}^{k}\right)$
$\displaystyle=\frac{1}{\Gamma(\alpha-\beta+1)}\sum_{k=1}^{\infty}\int_{0}^{t}(t-s)^{\alpha-\beta+1}(g^{k}(s,\cdot),\phi)dw_{s}^{k}.$

Below, we provide the properties of the solution space
$\mathcal{H}_{p}^{\gamma}(\tau)$.

######  Theorem 2.15.

Let $\tau\leq T$ be a bounded stopping time.

  1. (i)

For $\nu\in\mathbb{R}$, the map
$(1-\Delta)^{\nu/2}:\mathcal{H}_{p}^{\gamma+2}(\tau)\to\mathcal{H}_{p}^{\gamma-\nu+2}(\tau)$
is an isometry.

  2. (ii)

If $\gamma\in\mathbb{R}$, $\alpha\in(0,1)$, $\beta<\alpha+1/2$, and $p\geq 2$,
then $\mathcal{H}_{p}^{\gamma}(\tau)$ is a Banach space with the norm
$\|\cdot\|_{\mathcal{H}_{p}^{\gamma}(\tau)}$.

###### Proof.

Despite the fact that the definition of $U_{p}^{\gamma}$ is distinct, the
proof is a repeat of [25, Theorem 2.14] with $\tau$ instead of $T$.

∎

Next, we suggest the Hölder embedding theorems for
$u\in\mathcal{H}_{p}^{\gamma}(\tau)$. The proof of Theorem 2.16 is contained
in Section 5.

######  Theorem 2.16.

Let $\tau\leq T$ be the bounded stopping time, $\gamma\in\mathbb{R}$,
$\alpha\in(0,1)$, $\beta<\alpha+1/2$, and

$p>2\vee\frac{1}{\alpha}\vee\frac{1}{\alpha-\beta+1/2}.$ (2.6)

Suppose $u\in\mathcal{H}_{p}^{\gamma}(\tau)$. Assume $\alpha$, $\beta$, $\mu$,
and $\nu$ satisfy

$\frac{1}{\alpha p}<\mu<\frac{(\alpha(\nu+c_{0}/2)-\beta)\wedge
1/2+1/2}{\alpha}\quad\text{and}\quad\frac{1}{\alpha p}<\nu<1-\frac{c_{0}}{2},$
(2.7)

where $c_{0}$ is the constant introduced in (2.3). Then, for $u\in
C^{\alpha\mu-1/p}([0,\tau];H_{p}^{\gamma-2\nu})$ almost surely and

$\mathbb{E}\|u\|^{p}_{C^{\alpha\mu-1/p}\left([0,\tau];H_{p}^{\gamma-2\nu}\right)}\leq
N\|u\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau)},$ (2.8)

where $N=N(\alpha,\beta,\gamma,d,p,T)$.

######  Remark 2.17.

Theorem 2.16 is consistent with the previous results ([26, Theorem 7.2]). In
other words, if we let $\alpha,\beta\uparrow 1$ in Theorem 2.16, conditions
(2.6) and (2.7), and the result (2.8) approach those of the case of
$\alpha=\beta=1$.

By combining Lemma 2.10 (ii) and Theorem 2.16, we have the Hölder embedding
results of solution space $\mathcal{H}_{p}^{(2-c_{0}-d/2)\wedge 1}(\tau)$
which is a preparation to obtain the maximum Hölder regularity of solutions.

######  Corollary 2.18.

Let $\tau\leq T$ be a bounded stopping time, $\alpha\in(0,1)$,
$\beta<\alpha+1/2$, and $0<\gamma<(2-c_{0}-d/2)\wedge 1$, where $c_{0}$ is
introduced in (2.3). Suppose $p$ satisfies (2.6) and
$u\in\mathcal{H}_{p}^{\gamma}(\tau)$. If $\alpha,\beta,\gamma,\nu,d,$ and $p$
satisfy

$\frac{1}{\alpha p}<\mu<\frac{(\alpha(\nu+c_{0}/2)-\beta)\wedge
1/2+1/2}{\alpha}\quad\text{and}\quad\frac{1}{\alpha
p}<\nu<\left(\frac{\gamma}{2}-\frac{d}{2p}\right)\wedge\left(1-\frac{c_{0}}{2}\right),$
(2.9)

then $u\in C^{\alpha\mu-1/p}([0,\tau];C^{\gamma-2\nu-d/p})$ almost surely and

$\mathbb{E}\|u\|^{p}_{C^{\alpha\mu-1/p}\left([0,\tau];C^{\gamma-2\nu-d/p}(\mathbb{R}^{d})\right)}\leq
N\|u\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau)},$

where $N=N(\alpha,\beta,\gamma,d,p,T)$.

###### Proof.

For the proof, we employ Lemma 2.10 (ii) and Theorem 2.16. Then, we have

$\mathbb{E}\|u\|^{p}_{C^{\alpha\mu-1/p}\left([0,\tau];C^{\gamma-2\nu-d/p}(\mathbb{R}^{d})\right)}\leq\mathbb{E}\|u\|^{p}_{C^{\alpha\mu-1/p}\left([0,\tau];H_{p}^{\gamma-2\nu}(\mathbb{R}^{d})\right)}\leq
N\|u\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau)}.$

Thus, the corollary is proved. ∎

##  3\. Main Results

This section presents the uniqueness, existence, $L_{p}$-regularity, and
Hölder regularity of the solution to the following equation:

$\displaystyle\partial_{t}^{\alpha}u$
$\displaystyle=Lu+\bar{b}^{i}uu_{x^{i}}+\partial^{\beta}_{t}\sum_{k}\int_{0}^{t}\sigma(u)\eta^{k}dw_{s}^{k},\quad
t>0;\quad u(0,\cdot)=u_{0},$ (3.1)

where $Lu=a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu$. The coefficients $a^{ij}$,
$b^{i}$, and $c$ are
$\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d})$-measurable, $\bar{b}^{i}$ is
$\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d-1})$-measurable, and $a^{ij}$,
$b^{i}$, $c$, and $\bar{b}^{i}$ (and their derivatives) are uniformly bounded
(see Assumption 3.1). Additionally, we assume the coefficient $\bar{b}^{i}$ is
independent of $x^{i}$. Indeed, because $\bar{b}^{i}$ is independent of
$x^{i}$, we can employ the fundamental theorem of calculus to control the
nonlinear term $\bar{b}^{i}uu_{x^{i}}$ (see Remark 3.2). Moreover, the
diffusion coefficient $\sigma(u)$ is dominated by an $\mathbb{L}_{p}$ function
$h$ (see Assumption 3.3) and it is used to obtain a uniform $L_{p}$ bound of
the local solutions (see Remark 3.4).

In Theorem 3.5, we obtain the existence and uniqueness of a solution in
$\mathcal{H}_{p}^{\gamma}$, where $\gamma\in(0,2-c_{0}-d/2)\wedge 1$. The
components of equation (3.1) affect the properties of the solution $u$. For
example, if $\alpha$, $\beta$, $d$, and $p$ are given, the regularity $\gamma$
is determined. Remarks 3.6, 3.7, and 3.8 provide explanations for these
relations.

Additionally, we have the maximal Hölder regularity of the solution in
Corollary 3.10. We employ the Hölder embedding theorem for solution spaces
(Corollary 2.18). Furthermore, depending on the range of $\beta$, the behavior
of the Hölder regularity of the solution in time varies. In detail, when
$\beta\geq 1/2$, then the Hölder regularity of the solution in space is
$\alpha/2$ times of the Hölder regularity of the solution in time. Moreover,
if we consider the case $\alpha,\beta\uparrow 1$, then the Hölder regularity
in time and space approaches $1/4$ and $1/2$, which are the results of the
SPDEs driven by space-time white noise (e.g. [26, Remark 8.7] or [16,
Corollary 3.1]). In the case of $\beta<1/2$, $1/2-\beta$ of the Hölder
regularity in time is obtained due to the regularity of the stochastic
integral (Remark 3.11).

The following are assumptions on coefficients.

######  Assumption 3.1.

  1. (i)

The coefficients $a^{ij}=a^{ij}(t,x)$, $b^{i}=b^{i}(t,x)$, and $c=c(t,x)$ are
$\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d})$-measurable.

  2. (ii)

The coefficient
$\bar{b}^{i}(t,\bar{x}^{i})=\bar{b}^{i}(t,x^{1},\dots,x^{i-1},x^{i+1},\dots,x^{d})$
is $\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d-1})$-measurable.

  3. (iii)

There exists $K>0$ such that

$K^{-1}|{\xi}|^{2}\leq a^{ij}(t,x)\xi^{i}\xi^{j}\leq
K|{\xi}|^{2}\quad\text{for
all}\quad(\omega,t,x)\in\Omega\times[0,\infty)\times\mathbb{R}^{d},\quad\xi\in\mathbb{R}^{d},$
(3.2)

and

$\sum_{i,j}\left|a^{ij}(t,\cdot)\right|_{C^{2}(\mathbb{R}^{d})}+\sum_{i}\left|b^{i}(t,\cdot)\right|_{C^{2}(\mathbb{R}^{d})}+|c(t,\cdot)|_{C^{2}(\mathbb{R}^{d})}+\sum_{i}\left|\bar{b}^{i}(t,\cdot)\right|_{C^{2}(\mathbb{R}^{d-1})}\leq
K$ (3.3)

for all $(\omega,t)\in\Omega\times[0,\infty)$.

######  Remark 3.2.

To prove the existence of a global solution, we need to acquire a uniform
$L_{p}$ bound of the local solutions. Thus, we separate the local solutions
into two parts: noise-dominating and nonlinear-dominating parts. In this
remark, we consider the nonlinear-dominating parts related to
$\bar{b}^{i}uu_{x^{i}}$.

If coefficient $\bar{b}^{i}$ is independent of $x^{i}$, coefficient
$\bar{b}^{i}$ can be taken out of the integral for $x^{i}$. Then, by the
fundamental theorem of calculus to $x^{i}$, the nonlinear term
$\bar{b}^{i}uu_{x^{i}}$ is eliminated in the $L_{p}$ estimate of the
nonlinear-dominating part of the local solutions. Thus, the nonlinear-
dominating parts are controlled by the initial data and diffusion coefficient
$\sigma(u)$ (for more information, see Lemma 4.8).

To introduce the assumptions on the diffusion coefficient, we may assume
$p\geq 2$.

######  Assumption 3.3 ($p$).

  1. (i)

The coefficient $\sigma(t,x,u)$ is
$\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d})\times\mathcal{B}(\mathbb{R})$-measurable.

  2. (ii)

There exists a constant $K$ such that

$|\sigma(t,x,u)-\sigma(t,x,v)|\leq K|u-v|\quad\text{for
all}\quad(\omega,t,x)\in\Omega\times[0,\infty)\times\mathbb{R}^{d},\quad
u,v\in\mathbb{R}.$

  3. (iii)

There exists a $\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d})$-measurable
function $h\in\mathbb{L}_{p}$ such that

$|\sigma(t,x,u)|\leq|h(t,x)|\quad\text{for
all}\quad(\omega,t,x)\in\Omega\times[0,\infty)\times\mathbb{R}^{d},\quad
u\in\mathbb{R}.$ (3.4)

######  Remark 3.4.

As mentioned in Remark 3.2, we divide the local solutions into two parts, and
the nonlinear-dominating parts are controlled by the initial data $u_{0}$ and
diffusion coefficients $\sigma(u)$. Then, to deal with the noise-dominating
term and the terms including $\sigma(u)$, we employ the function $h(t,x)$
introduced in Assumption 3.3 $(p)$ (iii). Indeed, the terms related to the
diffusion coefficient $\sigma(u)$ are controlled by the initial data and $h$
so that a uniform $L_{p}$ bound of $u$ is obtained (see Lemmas 4.5 and 4.8).

Next, we introduce the main results.

######  Theorem 3.5.

Let

$\alpha\in(0,1),\quad\beta<\frac{3}{4}\alpha+\frac{1}{2},\quad
d<4-2c_{0},\quad 0<\gamma<(2-c_{0}-d/2)\wedge 1$ (3.5)

and

$p=2^{k}\quad\text{for some}\quad k\in\mathbb{N}\quad\text{and}\quad
p>2\vee\frac{1}{\alpha}\vee\frac{1}{\alpha-\beta+1/2}\vee\frac{2+\alpha
d}{\alpha\gamma}\vee\frac{d}{1-\gamma},$ (3.6)

where $c_{0}$ are the constants introduced in (2.3). Suppose Assumptions 3.1
and 3.3 ($p$) hold. If $u_{0}\in U_{p}^{\gamma}$, then there exists a unique
solution $u\in\mathcal{H}_{p,loc}^{\gamma}$ satisfying (3.1). Furthermore, for
$\mu$ and $\nu$ satisfying (2.9), and bounded stopping time $\tau\leq T$, we
have

$u\in
C^{\alpha\mu-1/p}([0,\tau];C^{\gamma-2\nu-d/p})\quad\text{and}\quad\|u\|_{C^{\alpha\mu-\frac{1}{p}}\left([0,\tau];C^{\gamma-2\nu-d/p}\right)}<\infty$
(3.7)

almost surely.

###### Proof.

See Proof of Theorem 3.5 in Section 4. ∎

######  Remark 3.6.

  1. (i)

We assume

$\alpha\in(0,1)$

because an inequality acting like the chain rule is employed to deal with the
nonlinear-dominating part of the local solution (see Lemma 4.6).

  2. (ii)

The conditions

$\beta<3\alpha/4+1/2\quad\text{and}\quad d<4-2c_{0}$

are expected to obtain the uniqueness and existence of solutions to SPDEs with
Caputo time fractional derivatives and space-time white noise even for the
semilinear case. For example, see [23, Section 7]. Additionally, observe that
the choice of $\alpha$ and $\beta$ allows $d=1,2,3$, where $c_{0}$ is the
constant introduced in (2.3).

######  Remark 3.7.

  1. (i)

For the existence and uniqueness of local solutions, we impose

$\gamma\in(0,2-c_{0}-d/2).$ (3.8)

Heuristically, if $u$ is a measurable, continuous, and bounded solution to
equation (3.1), then for given $T<\infty$, we can define a bounded stopping
time as follows:

$\tau_{m}:=\inf\left\\{t\geq 0:\sup_{x\in\mathbb{R}^{d}}|u(t,x)|\geq
m\right\\}\wedge T.$

Then, the solution $u$ satisfies the localized version of equation (3.1) on
$(0,\tau_{m})$. In other words,

$\partial_{t}^{\alpha}u=Lu+\frac{1}{2}\bar{b}^{i}\left((|u|\wedge
m)^{2}\right)_{x^{i}}+\partial^{\beta}_{t}\sum_{k}\int_{0}^{t}\sigma(u)\eta^{k}dw_{s}^{k}$
(3.9)

holds on $0<t<\tau_{m}$ with $u(0,\cdot)=u_{0}$. Then, as (3.9) is a
semilinear equation, (3.8) has to be satisfied by [23, Theorem 7.1] (for more
detail, see [23, Section 7] and [25, Section 5].

  2. (ii)

The following condition

$\gamma\in(0,1)$ (3.10)

is assumed due to the nonlinear term $\bar{b}^{i}uu_{x^{i}}$ lowering the
regularity of the solution. Even for SBEs ($\alpha=\beta=1$), the condition in
(3.10) is required (for more information, see [13, 14, 15, 16, 29]).

######  Remark 3.8.

  1. (i)

To obtain the local solution, we employ the $L_{p}$ theory for the semilinear
equation (see [26, Theorem 5.1]). When we control the nonlinear term
$\bar{b}^{i}uu_{x^{i}}$ in the $L_{p}$ estimate, the kernel of
$(1-\Delta)^{-\frac{\gamma-1}{2}}$ has to be controlled. Hence,

$p>\frac{d}{1-\gamma}$

is imposed (see Lemma 4.3).

  2. (ii)

We require $\mathbb{R}$-valued continuous solutions to consider the cut-off
version of equation (3.1). Therefore, we assume

$p>2\wedge\frac{1}{\alpha}\wedge\frac{1}{\alpha-\beta+1/2}\wedge\frac{2+\alpha
d}{\alpha\gamma}$

which is required to apply the Hölder embedding theorem for
$\mathcal{H}_{p}^{\gamma}$ (see Theorem 2.16 and Corollary 2.18).

  3. (iii)

As mentioned in Remark 3.6 (i), we employ an inequality similar to the chain
rule. To apply (4.13) instead of chain rule for the Caputo fractional
derivative, we assume

$p=2^{k}$

for some $k\in\mathbb{N}$.

To achieve the maximal Hölder regularity, we require the uniqueness of the
solution in $p$.

######  Theorem 3.9.

Suppose all the conditions of Theorem 3.5 hold. Let
$u\in\mathcal{H}_{p,loc}^{\gamma}$ be the solution of equation (3.1)
introduced in Theorem 3.5. If $q>p$, $u_{0}\in U_{q}^{\alpha,\gamma}$, and
Assumption 3.3 ($q$) hold, then $u\in\mathcal{H}_{q,loc}^{\gamma}$.

###### Proof.

See Proof of Theorem 3.9 in Section 4. ∎

Finally, we obtain the maximal Hölder regularity of the solution by combining
Theorems 3.5 and 3.9. Recall that $c_{0}$ is introduced in (2.3).

######  Corollary 3.10.

Suppose $\alpha$, $\beta$, $d$, and $\gamma$ satisfy (3.5), and
$u_{0}\in\cap_{p>2}U_{p}^{(2-c_{0}-d/2)\wedge 1}$, and
$h\in\cap_{p>2}\mathbb{L}_{p}$ satisfies (3.4). Then, for any bounded stopping
time $\tau\leq T$, (1.3) holds almost surely.

###### Proof.

When $\alpha$, $\beta$, $d$, and $\gamma$ are given in (3.5), we choose $p$ as
in (3.6). For each $p$, there exists a unique solution
$u_{p}\in\mathcal{H}^{\gamma}_{p,loc}$ to equation (3.1). Due to Theorem 3.9,
$u_{p}\in\mathcal{H}^{\gamma}_{q,loc}$ for any $q\geq p$ so that we write $u$
instead of $u_{p}$ and $u$ is independent of $p$. Thus, by letting $p$ large
in (3.7), we have (1.3). Thus, the corollary is proved. ∎

######  Remark 3.11.

  1. (i)

If $1/2\leq\beta<\alpha+1/2$, the Hölder regularity in space is $\alpha/2$
times that in time. Furthermore, we can recover the Hölder regularity results
of SBEs ($\alpha=\beta=1$) by considering the case $\alpha,\beta\uparrow 1$.
We cite [29, Proposition 5.1] or [16, Corollary 3.1] for reader’s convenience.

  2. (ii)

If $\beta<1/2$, then the Hölder regularity in time obtains additional
regularity by as much as $1/2-\beta$. This phenomenon is caused by the
stochastic integral of equation (3.1) adding the Hölder regularity of noise in
time almost $1/2$, and $\partial_{t}^{\beta}$ reducing the regularity of the
noise by $\beta$.

##  4\. Proof of Theorems 3.5 and 3.9

We assume that all conditions in Theorem 3.5 hold for the remainder of this
section.

To establish the existence of a global solution, we need to obtain the
uniqueness and existence of local solutions (Lemma 4.3). With these local
solutions, we build a candidate for a global solution. More precisely, we
paste the local solutions and demonstrate that the local existence time
explodes almost surely (Lemma 4.9). To prove that the local existence time
explodes almost surely, we demonstrate that a uniform $L_{p}$ bound of local
solutions exists. In detail, we separate the local solution into noise- and
nonlinear-dominating parts. The noise-dominating part is affected by the
stochastic part of the equation, and the other part is influenced by the
nonlinear term $b^{i}uu_{x^{i}}$. When we deal with the noise-dominating part
of the solution, the dominating function of the diffusion coefficient provides
a uniform $L_{p}$ bound for the noise-dominating part of the local solutions
(see Assumption 3.3 (p) (iii) and Lemma 4.5). The other part is controlled by
employing a version of the chain rule and Grönwall inequality (see Lemmas 4.6
and 4.8 and Theorem 4.7).

First, we introduce the uniqueness and existence theorem for semilinear SPDEs.

######  Assumption 4.1 ($\tau$).

  1. (i)

The functions $f(t,x,u)$ and $g^{k}(t,x,u)$ are
$\mathcal{P}\times\mathcal{B}(\mathbb{R}^{d})\times\mathcal{B}(\mathbb{R})$-measurable
functions satisfying the following:

$f(t,x,0)\in\mathbb{H}_{p}^{\gamma}(\tau)\quad\text{and}\quad
g(t,x,0)=(g^{1}(t,x,0),g^{2}(t,x,0),\dots)\in\mathbb{H}_{p}^{\gamma+1}(\tau,l_{2}).$

  2. (ii)

For any $\varepsilon>0$, there exists a constant $N_{\varepsilon}$ such that
for any $u,v\in\mathbb{H}_{p}^{\gamma}(\tau)$,

$\|f(u)-f(v)\|^{p}_{\mathbb{H}_{p}^{\gamma-2}(\tau)}+\|g(u)-g(v)\|^{p}_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}\leq\varepsilon\|u-v\|^{p}_{\mathbb{H}_{p}^{\gamma}(\tau)}+N_{\varepsilon}\|u-v\|^{p}_{\mathbb{H}_{p}^{\gamma-2}(\tau)},$

where $c_{0}$ is the constant introduced in (2.3).

######  Lemma 4.2.

Let $\tau\leq T$ be a bounded stopping time. Suppose Assumption 4.1 $(\tau)$
hold. Then, for initial data $u_{0}\in U_{p}^{\gamma}$, the following
equation:

$\partial_{t}^{\alpha}u=Lu+f(u)+\partial_{t}^{\beta}\int_{0}^{t}g^{k}(u)dw_{t}^{k},\quad
0<t\leq\tau;\quad u(0,\cdot)=u_{0}$ (4.1)

has a unique solution $u\in\mathcal{H}_{p}^{\gamma}(\tau)$. Moreover,

$\|u\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau)}\leq
N\left(\|u_{0}\|^{p}_{U_{p}^{\gamma}}+\|f(0)\|^{p}_{\mathbb{H}_{p}^{\gamma-2}(\tau)}+\|g(0)\|^{p}_{\mathbb{H}^{\gamma-2+c_{0}}_{p}(\tau,l_{2})}\right),$
(4.2)

where $N=N(\alpha,\beta,\gamma,d,p,K,T)$ and $c_{0}$ is the constant
introduced in (2.3).

###### Proof.

Theorem 5.1 of [26] is the motivation of the proof. The case $\tau\equiv T$ is
obtained by [25, Theorem 2.18]; thus, we only consider the case $\tau\leq T$.

(Step 1). (Existence) Set

$\bar{f}(t,u):=1_{t\leq\tau}f(t,u)\quad\text{and}\quad\bar{g}(t,u):=1_{t\leq\tau}g(t,u).$

Additionally, $\bar{f}(u)$ and $\bar{g}(u)$ satisfy Assumption 4.1 ($T$).
Then, by [25, Theorem 2.18], there exists a unique solution
$u\in\mathcal{H}_{p}^{\gamma}(T)$ such that $u$ satisfies equation (4.1) with
$\bar{f}$ and $\bar{g}$, instead of $f$ and $g$, respectively. As $\tau\leq
T$, we have $u\in\mathcal{H}_{p}^{\gamma}(\tau)$ and $u$ satisfies equation
(4.1) and estimate (4.2) with $f$ and $g$.

(Step 2). (Uniqueness) Let $u,v\in\mathcal{H}_{p}^{\gamma}(\tau)$ be two
solutions of equation (4.1). Then, [25, Theorem 2.18] yields there exists a
unique solution $\bar{v}\in\mathcal{H}_{p}^{\gamma}(T)$ satisfying

$\partial_{t}^{\alpha}\bar{v}=L\bar{v}+\bar{f}(v)+\sum_{k=1}^{\infty}\partial^{\beta}_{t}\int_{0}^{t}\bar{g}^{k}(v)dw^{k}_{t},\quad
0<t\leq T\,;\quad\bar{v}(0,\cdot)=u_{0}.$ (4.3)

Notice that in (4.3), $\bar{f}(v)$ and $\bar{g}(v)$ are used instead of
$\bar{f}(\bar{v})$ and $\bar{g}(\bar{v})$, respectively. Set
$\tilde{v}:=v-\bar{v}$. Then, for fixed $\omega\in\Omega$, we have

$\partial_{t}^{\alpha}\tilde{v}=L\tilde{v},\quad
0<t\leq\tau\,;\quad\tilde{v}(0,\cdot)=0.$

By the deterministic version of [25, Theorem 2.18], we have $\tilde{v}=0$ in
$L_{p}((0,\tau]\times\mathbb{R}^{d})$ almost surely. Additionally, it implies
$v(t,\cdot)=\bar{v}(t,\cdot)$ in $L_{p}((0,\tau]\times\mathbb{R}^{d})$ almost
surely. Thus, in equation (4.3), we can replace $\bar{f}(v)$ and $\bar{g}(v)$
with $\bar{f}(\bar{v})$ and $\bar{g}(\bar{v})$. Therefore,
$\bar{v}\in\mathcal{H}_{p}^{\gamma}(T)$ satisfies equation (4.1) on $(0,T]$
with $\bar{f},\bar{g}$ instead of $f,g$, respectively. Similarly, by following
word for word, there exists $\bar{u}\in\mathcal{H}_{p}^{\gamma}(T)$ such that
$\bar{u}$ satisfies equation (4.1) on $(0,T]$ with $\bar{f}$ and $\bar{g}$
instead of $f$ and $g$. Thus, by the uniqueness result in
$\mathcal{H}_{p}^{\gamma}(T)$, we have $\bar{u}=\bar{v}$ in
$\mathcal{H}_{p}^{\gamma}(T)$, which implies $u=v$ in
$\mathcal{H}_{p}^{\gamma}(\tau)$. Thus, the lemma is proved. ∎

Next, we provide the uniqueness and existence of a local solution to equation
(3.1). As an auxiliary function, we choose $\rho(\cdot)\in
C_{c}^{\infty}(\mathbb{R})$ such that $\rho(z)\geq 0$ on
$z\in(-\infty,\infty)$, $\rho(z)=1$ on $|z|\leq 1$, $\rho(z)=0$ on $|z|\geq
2$, and $\frac{d}{dz}\rho(z)\leq 0$ on $z\geq 0$. We define the following:

$\rho_{m}(z):=\rho(z/m).$ (4.4)

######  Lemma 4.3.

Let $\tau\leq T$ be a bounded stopping time. For $m\in\mathbb{N}$, there
exists $u_{m}\in\mathcal{H}_{p}^{\gamma}(\tau)$ such that

$\partial_{t}^{\alpha}u_{m}=Lu_{m}+\bar{b}^{i}\left(u_{m}^{2}\rho_{m}(u_{m})\right)_{x^{i}}+\partial_{t}^{\beta}\int_{0}^{t}\sigma(t,x,u_{m})\eta^{k}(x)dw^{k}_{t},\,\,0<t\leq\tau;\,\,u_{m}(0,\cdot)=u_{0},$

where $\rho_{m}$ is the function introduced in (4.4). Furthermore, $u_{m}\in
C([0,\tau];C(\mathbb{R}^{d}))$ almost surely and

$\mathbb{E}\sup_{t\leq\tau}\sup_{x\in\mathbb{R}^{d}}|u_{m}(t,x)|^{p}\leq
N\|u_{m}\|_{\mathcal{H}_{p}^{\gamma}(\tau)}^{p}<\infty$ (4.5)

almost surely.

###### Proof.

Due to Lemma 4.2 and Corollary 2.18, it suffices to show that Assumption 4.1
$(\tau)$ holds. Because $\sigma(t,x,0)\leq h(t,x)$ for all $\omega,t,x$ and
$h\in\mathbb{L}_{p}$, Assumption 4.1 (i) is satisfied.

In the case of Assumption 4.1 (ii), notice that for $u,v\in\mathbb{R}$, we
have

$\left|u^{2}\rho_{m}(u)-v^{2}\rho_{m}(v)\right|\leq N_{m}|u-v|.$

Then, for $u,v\in\mathbb{H}_{p}^{\gamma}(\tau)$, by Remark 2.8 and Lemmas 2.10
(viii) and (iii), we have

$\displaystyle\left\|\bar{b}^{i}\left((u(t,\cdot))^{2}\rho_{m}(u(t,\cdot))-(v(t,\cdot))^{2}\rho_{m}(v(t,\cdot))\right)_{x^{i}}\right\|^{p}_{H_{p}^{\gamma-2}}$
(4.6) $\displaystyle\leq
N\left\|(u(t,\cdot))^{2}\rho_{m}(u(t,\cdot))-(v(t,\cdot))^{2}\rho_{m}(v(t,\cdot))\right\|^{p}_{H^{\gamma-1}_{p}}$
$\displaystyle\leq
N\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\left|R_{1-\gamma}(x-y)\right|\left((u(\cdot))^{2}\rho_{m}(\cdot,u(\cdot))-(v(\cdot))^{2}\rho_{m}(\cdot,v(\cdot))\right)(t,y)dy\right)^{p}dx$
$\displaystyle\leq
N_{m}\left(\int_{\mathbb{R}^{d}}|R_{1-\gamma}(x)|dx\right)^{p}\int_{\mathbb{R}^{d}}|u(t,x)-v(t,x)|^{p}dx$

and

$\displaystyle\left\|\sigma(u)\bm{\eta}-\sigma(v)\bm{\eta}\right\|_{H_{p}^{\gamma-2+c_{0}}(l_{2})}^{p}$
(4.7)
$\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\sum_{k}\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(x-y)\right|(\sigma(\cdot,u(\cdot))-\sigma(\cdot,v(\cdot)))(t,y)\eta^{k}(y)dy\right)^{2}\right)^{p/2}dx$
$\displaystyle\leq\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(x-y)\right|^{2}\left(\sigma(t,y,u(t,y))-\sigma(t,y,v(t,y))\right)^{2}dy\right)^{p/2}dx$
$\displaystyle\leq
K^{p}\int_{\mathbb{R}^{d}}\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(y)\right|^{2}(u(t,x-y)-v(t,x-y))^{2}dy\right)^{p/2}dx$
$\displaystyle\leq
K^{p}\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(y)\right|^{2}dy\right)^{p/2}\int_{\mathbb{R}^{d}}|u(t,x)-v(t,x)|^{p}dx$

on almost every $(\omega,t)\in\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$}$.
Due to Remark 2.8, we have

$\int_{\mathbb{R}^{d}}\left|R_{1-\gamma}(y)\right|dy+\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(y)\right|^{2}dy<\infty.$

By integrating with respect to $(\omega,t)$ to (4.6) and (4.7), employing
Lemma 2.10 (vii), and Young’s inequality, we have

$\displaystyle\left\|\bar{b}^{i}\left(u^{2}\rho_{m}(u)-v^{2}\rho_{m}(v)\right)_{x^{i}}\right\|_{\mathbb{H}_{p}^{\gamma-2}(\tau)}^{p}+\left\|\sigma(u)\eta-\sigma(v)\eta\right\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}^{p}$
(4.8) $\displaystyle\quad\leq N_{m}\|u-v\|_{\mathbb{L}_{p}(\tau)}^{p}$
$\displaystyle\quad\leq\varepsilon\|u-v\|_{\mathbb{H}^{\gamma}_{p}(\tau)}^{p}+N_{m}\|u-v\|_{\mathbb{H}^{\gamma-2}_{p}(\tau)}^{p}.$

To obtain the second assertion, we employ Corollary 2.18. The lemma is proved.

∎

######  Remark 4.4.

We introduce a candidate for a global solution. Let $T<\infty$. For
$m\in\mathbb{N}$, let $u_{m}\in\mathcal{H}_{p}^{\gamma}(T)$ be the solution
introduced in Lemma 4.3. Then, for $R\in\\{1,2,\dots,m\\}$, define a stopping
time $\tau_{m}^{R}$

$\tau_{m}^{R}:=\inf\left\\{t\geq 0:\sup_{x\in\mathbb{R}}|u_{m}(t,x)|\geq
R\right\\}\wedge T.$ (4.9)

Observe that

$\tau_{R}^{R}\leq\tau_{m}^{m}$ (4.10)

Indeed, if $R=m$, (4.10) is obvious. If $R<m$, we have $u_{m}\wedge
m=u_{m}\wedge m\wedge R=u_{m}\wedge R$ for $t\leq\tau_{m}^{R}$. Therefore,
$u_{m}$ and $u_{R}$ are solutions to equation

$\partial_{t}^{\alpha}u=Lu+\bar{b}^{i}\left(u^{2}\rho_{R}(u)\right)_{x^{i}}+\sigma(u)\eta^{k}dw_{t}^{k},\quad
0<t\leq\tau_{m}^{R}\,;\quad u(0,\cdot)=u_{0}.$

In contrast, $u_{R}\wedge R=u_{R}\wedge R\wedge m=u_{R}\wedge m$ for
$t\leq\tau_{R}^{R}$. Thus, $u_{m}$ and $u_{R}$ are solutions to equation

$\partial_{t}^{\alpha}u=Lu+\bar{b}\left(u^{2}\rho_{m}(u)\right)_{x^{i}}+\sigma(u)\eta^{k}dw_{t}^{k},\quad
0<t\leq\tau_{R}^{R}\,;\quad u(0,\cdot)=u_{0}.$

Observe that the uniqueness and continuity results in Lemma 4.3 yields that
$u_{m}=u_{R}$ for all $t\leq(\tau_{m}^{R}\vee\tau_{R}^{R})$. Therefore, for
$t\leq\tau_{m}^{R}$,

$\sup_{s\leq t}\sup_{x\in\mathbb{R}}|u_{R}(s,x)|=\sup_{s\leq
t}\sup_{x\in\mathbb{R}}|u_{m}(s,x)|\leq R,$

and this implies $\tau_{m}^{R}\leq\tau_{R}^{R}$. Similarly,
$\tau_{m}^{R}\geq\tau_{R}^{R}$; thus,

$\tau_{R}^{R}=\tau_{m}^{R}$

almost surely. Moreover, we have $\tau_{m}^{R}\leq\tau_{m}^{m}$ because $m>R$.
Therefore, we have (4.10).

Further, we define

$u(t,x):=u_{m}(t,x)\quad\text{on}\quad t\leq\tau_{m}^{m}$

and set

$\tau_{\infty}:=\limsup_{m\to\infty}\limsup_{T\to\infty}\tau_{m}^{m}.$ (4.11)

It should be remarked that $u(t,x)$ is well-defined on
$\Omega\times[0,\infty)\times\mathbb{R}^{d}$ and the nontrivial domain of $u$
is $\Omega\times[0,\tau_{\infty})\times\mathbb{R}^{d}$.

To obtain a uniform $L_{p}$ bound of the local solution $u_{m}$, we separate
$u_{m}$ into noise- and nonlinear-dominating parts. Lemma 4.5 provides the
existence, uniqueness, and estimate of noise-dominating parts of $u_{m}$.

######  Lemma 4.5.

Let $T<\infty$. Then, there exists $v\in\mathcal{H}_{p}^{\gamma}(T)$ such that

$\partial_{t}^{\alpha}v=Lv+\partial_{t}^{\beta}\int_{0}^{t}\sigma(s,x,u)\eta^{k}(x)dw^{k}_{s},\quad
0<t\leq T,\quad u(0,\cdot)=u_{0}$

Furthermore, $v\in C([0,T];C(\mathbb{R}^{d}))$ almost surely, and

$\mathbb{E}\sup_{t\leq
T}\sup_{x\in\mathbb{R}^{d}}|v(t,x)|^{p}+\mathbb{E}\sup_{t\leq
T}\|v(t,\cdot)\|_{L_{p}}^{p}\leq N\|v\|_{\mathcal{H}_{p}^{\gamma}(T)}^{p}\leq
N\|u_{0}\|_{U_{p}^{\gamma}}^{p}+N\|h\|_{\mathbb{L}_{p}(\tau)}^{p},$

where $N=N(\alpha,\beta,\gamma,d,p,K,T)$.

###### Proof.

Similar to the proof of Lemma 4.3, it is enough to show that Assumption 4.1
$(\tau)$ holds. Set $\bm{\eta}=(\eta^{1},\eta^{2},\dots)$. Then, by Remark
2.8, for $t\leq T$

$\displaystyle\|\sigma(t,\cdot,u(t,\cdot))\bm{\eta}\|^{p}_{H_{p}^{\gamma-2+c_{0}}(l_{2})}$
(4.12)
$\displaystyle=\int_{\mathbb{R}^{d}}\left(\sum_{k=1}^{\infty}\left(\int_{\mathbb{R}^{d}}R_{-\gamma+2-c_{0}}(x-y)\sigma(t,y,u(t,y))\eta^{k}(y)dy\right)^{2}\right)^{p/2}dx$
$\displaystyle=\left(\int_{\mathbb{R}^{d}}|R_{-\gamma+2-c_{0}}(x)|^{2}dx\right)^{p/2}\int_{\mathbb{R}^{d}}|\sigma(t,y,u(t,y))|^{p}dy$
$\displaystyle\leq\left(\int_{\mathbb{R}^{d}}|R_{-\gamma+2-c_{0}}(x)|^{2}dx\right)^{p/2}\int_{\mathbb{R}^{d}}|h(t,y)|^{p}dy$
$\displaystyle\leq N\|h(t,\cdot)\|_{L_{p}}^{p}.$

Therefore,

$\displaystyle\left\|\sigma(u)\bm{\eta}\right\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(T,l_{2})}^{p}$
$\displaystyle\leq\mathbb{E}\int_{0}^{T}\|\sigma(t,\cdot,u(t,\cdot))\bm{\eta}\|_{H_{p}^{\gamma-2+c_{0}}(l_{2})}^{p}dt\leq
N\|h\|_{\mathbb{L}_{p}}^{p}.$

Thus, the lemma is proved by Lemma 4.2. ∎

Next, we control the nonlinear-dominating parts of the local solutions. The
following two lemmas are crucial in obtaining uniform $L_{p}$ bounds. Lemma
4.6 functions as a chain rule, and Theorem 4.7 is a version of the Grönwall
inequality.

######  Lemma 4.6.

Suppose $\alpha\in(0,1)$ and $k\in\mathbb{N}$. For any $\psi\in
C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$, we have

$\partial_{t}^{\alpha}(\psi(\cdot,x))^{2^{k}}(t)\leq
2^{k}\psi(t,x)|\psi(t,x)|^{2^{k}-2}\partial_{t}^{\alpha}\psi(t,x),$ (4.13)

for all $(t,x)\in(0,\infty)\times\mathbb{R}^{d}$.

###### Proof.

We employ the mathematical induction. The results and proof are motivated by
(4.2) of [8].

(Step 1). First, we consider the case $k=1$. Although the proof is in the
proof of [8, Proposition 4.1], we include the proof for the completeness of
this paper.

Let $\psi\in C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$ and
$t\in(0,\infty)$ and $x\in\mathbb{R}^{d}$. For $s\in(0,t]$, set

$F_{1}(s):=\frac{1}{2}|\psi(s,x)|^{2},\quad F_{2}(s):=\psi(s,x)\psi(t,x),$

and

$\displaystyle F(s)$
$\displaystyle:=\frac{1}{2}\left(|\psi(s,x)|^{2}-|\psi(t,x)|^{2}\right)-(\psi(s,x)-\psi(t,x))\psi(t,x).$

Further,

$F(s)=\frac{1}{2}|\psi(s,x)-\psi(t,x)|^{2}\geq 0$

on $s\leq t$, and the equality holds for $s=t$. Notice that the integration by
parts implies that

$\displaystyle\int_{0}^{t}(t-s)^{-\alpha}(F_{1}^{\prime}(s)-F_{2}^{\prime}(s))ds=\int_{0}^{t}(t-s)^{-\alpha}F^{\prime}(s)ds\leq
0.$

Then, by the definition of $\partial_{t}^{\alpha}$ (Definition 2.5), we have
(4.13) with $k=1$.

(Step 2). Let $n\in\mathbb{N}$ and assume that the results hold for
$k=1,2,\dots,n-1$. Set $\tilde{\psi}(t,x):=(\psi(t,x))^{2}$. Since
$\tilde{\psi}(t,x)\in C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$, we have

$\displaystyle\partial_{t}^{\alpha}(\psi(\cdot,x))^{2^{n}}(t)$
$\displaystyle=\partial_{t}^{\alpha}(\tilde{\psi}(\cdot,x))^{2^{n-1}}(t)$
$\displaystyle\leq
2^{n-1}\tilde{\psi}(t,x)\left|\tilde{\psi}(t,x)\right|^{2^{n-1}-2}\partial_{t}^{\alpha}\tilde{\psi}(t,x)$
$\displaystyle=2^{n-1}\left|\psi(t,x)\right|^{2^{n}-2}\partial_{t}^{\alpha}(\psi(t,x))^{2}$
$\displaystyle\leq
2^{n}\psi(t,x)\left|\psi(t,x)\right|^{2^{n}-2}\partial_{t}^{\alpha}\psi(t,x).$

The lemma is proved.

∎

######  Theorem 4.7 (Theorem 8 of [4]).

Let $\psi(t)$ be a nonnegative integrable function on $[0,T]$. For a constant
$N_{1}$, if the function $\psi$ satisfies

$\psi(t)\leq\psi_{0}+N_{1}I_{t}^{\alpha}\psi$

on $t\in[0,T]$, then

$\psi(t)\leq\left(1+\sum_{k=0}^{\infty}\frac{N_{1}^{k}}{\Gamma(k\alpha)}\frac{(\Gamma(\alpha)t^{\alpha})^{k}}{k\alpha}\right)\psi_{0}$

on $t\in[0,T]$.

We consider following lemma to control the remainder of the local solution
$u_{m}$.

######  Lemma 4.8.

Let $u_{m}\in\mathcal{H}_{p}^{\gamma}(T)$ and
$v\in\mathcal{H}_{p}^{\gamma}(T)$ be functions introduced in Lemmas 4.3 and
4.5, and $\tau_{m}^{m}$ be the stopping time introduced in (4.9). Then,

$\displaystyle\|u_{m}(t,\cdot)-v(t,\cdot)\|_{L_{p}}^{p}$ $\displaystyle\leq
N\sup_{t\leq T}\sup_{x\in\mathbb{R}^{d}}|v(t,x)|^{p}\sup_{t\leq
T}\|v(s,\cdot)\|_{L_{p}}^{p}\left[1+\sum_{k=0}^{\infty}\frac{\left(1+\sup_{s\leq
t,x\in\mathbb{R}^{d}}|v(s,x)|^{2}\right)^{k}}{\Gamma(k\alpha)}\frac{(\Gamma(\alpha)T^{\alpha})^{k}}{k\alpha}\right]$

for all $t\leq\tau_{m}^{m}$ almost surely, where $N=N(p,K)$.

###### Proof.

Set

$w_{m}:=u_{m}-v\quad\text{and}\quad
f_{m}:=Lw_{m}+\bar{b}^{i}((u_{m})^{2}\rho_{m}(u_{m}))_{x^{i}}.$

Then, we have $f_{m}\in\mathbb{H}_{p}^{\gamma-2}(T)$ since
$w_{m},u_{m}\in\mathbb{H}_{p}^{\gamma}(T)$ and estimates similar to (4.8).
Additionally, $\partial_{t}^{\alpha}w_{m}=f_{m}$. Let
$(\omega,t)\in\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau_{m}^{m}\text{$]$\kern-1.49994pt$]$}$.
Due to [23, Remark 2.9], there exists $w_{m}^{n}\in
C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$ such that $w_{m}^{n}\to w_{m}$
in $L_{p}((0,t);H_{p}^{\gamma})$, and $\partial_{t}^{\alpha}w_{m}^{n}$ is a
Cauchy in $L_{p}((0,t);H_{p}^{\gamma-2})$. Define

$f_{m}^{n}:=\partial_{t}^{\alpha}w_{m}^{n}.$

Moreover, $f_{m}$ is the limit of $f_{m}^{n}$ as $n\to\infty$ in
$L_{p}((0,t);H_{p}^{\gamma-2})$ (see [23, Remark 2.9]). Choose a nonnegative
function $\zeta\in C_{c}^{\infty}(\mathbb{R}^{d})$ with a unit integral and
set $\zeta_{\varepsilon}(x):=\varepsilon^{-d}\zeta(x/\varepsilon)$ for
$\varepsilon>0$. For $h\in L_{1,loc}(\mathbb{R}^{d})$, set
$h^{(\varepsilon)}(x):=\int_{\mathbb{R}^{d}}h(y)\zeta_{\varepsilon}(x-y)dy.$

Next, let $\varepsilon>0$ and $x\in\mathbb{R}^{d}$. Since
$w_{m}^{n(\varepsilon)}\in C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$ and
$p=2^{k}$, Lemma 4.6 yields

$\displaystyle\frac{1}{p}\partial_{t}^{\alpha}\left(w_{m}^{n(\varepsilon)}(\cdot,x)\right)^{p}(t)$
$\displaystyle\leq
f_{m}^{n(\varepsilon)}(t,x)w_{m}^{n(\varepsilon)}(t,x)\left|w_{m}^{n(\varepsilon)}(t,x)\right|^{p-2}$
(4.14)

on $t\in(0,\infty)$. Additionally, as $w_{m}^{n}(0,x)=0$, we have

$w_{m}^{n(\varepsilon)}(0,x)=0.$ (4.15)

Thus, if we take stochastic integral $I_{t}^{\alpha}$ on both sides of (4.14),
we have

$\displaystyle\frac{1}{p}\left|w_{m}^{n(\varepsilon)}(t,x)\right|^{p}\leq
I_{t}^{\alpha}\left[f_{m}^{n(\varepsilon)}(\cdot,x)w_{m}^{n(\varepsilon)}(\cdot,x)\left|w_{m}^{n(\varepsilon)}(\cdot,x)\right|^{p-2}\right]$
(4.16)

due to $\left(w_{m}^{n(\varepsilon)}\right)^{p}\in
C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$, (4.15), and Remark 2.6.
Observe that (2.1) with $q=\infty$ and the Hölder inequality imply that

$\displaystyle\left\|I^{\alpha}_{\cdot}\left[f_{m}^{n(\varepsilon)}(\cdot,x)w^{n(\varepsilon)}_{m}(\cdot,x)\left|w^{n(\varepsilon)}_{m}(\cdot,x)\right|^{p-2}-f_{m}^{(\varepsilon)}(\cdot,x)w^{(\varepsilon)}_{m}(\cdot,x)\left|w^{(\varepsilon)}_{m}(\cdot,x)\right|^{p-2}\right]\right\|_{L_{1}((0,t))}$
(4.17)
$\displaystyle\leq\int_{0}^{t}\left|f_{m}^{n(\varepsilon)}(s,x)w^{n(\varepsilon)}_{m}(s,x)\left|w^{n(\varepsilon)}_{m}(s,x)\right|^{p-2}-f_{m}^{(\varepsilon)}(s,x)w^{(\varepsilon)}_{m}(s,x)\left|w^{(\varepsilon)}_{m}(s,x)\right|^{p-2}\right|ds$
$\displaystyle\leq
N\int_{0}^{t}\left|f_{m}^{n(\varepsilon)}(s,x)-f_{m}^{(\varepsilon)}(s,x)\right|\left|w^{n(\varepsilon)}_{m}(s,x)\right|^{p-1}ds$
$\displaystyle\quad+N\int_{0}^{t}\left|f_{m}^{(\varepsilon)}(s,x)\left[w^{n(\varepsilon)}_{m}(s,x)\left|w^{n(\varepsilon)}_{m}(s,x)\right|^{p-2}-w^{(\varepsilon)}_{m}(s,x)\left|w^{(\varepsilon)}_{m}(s,x)\right|^{p-2}\right]\right|ds$
$\displaystyle\leq
N\bigg{[}A_{n}\left\|w^{n(\varepsilon)}_{m}(\cdot,x)\right\|_{L_{p}(0,t)}^{2}+B_{n}C_{n}\left\|f_{m}^{(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}\bigg{]}\left\|w_{m}^{n(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}^{p-3},$

where

$\displaystyle
A_{n}=\left\|f_{m}^{n(\varepsilon)}(\cdot,x)-f_{m}^{(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}\quad
B_{n}=\left\|w_{m}^{n(\varepsilon)}(\cdot,x)-w_{m}^{(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)},$

and

$C_{n}=\left\|w_{m}^{n(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}+\left\|w_{m}^{(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}.$

Moreover,

$A_{n},B_{n}\to 0\quad\text{and}\quad C_{n}\to
2\left\|w_{m}^{(\varepsilon)}(\cdot,x)\right\|_{L_{p}(0,t)}\quad\text{as}\quad
n\to\infty$ (4.18)

since $w_{m}^{n}\to w_{m}$ and $f_{m}^{n}\to f_{m}$ in
$L_{p}((0,t);H_{p}^{\gamma})$. Then, by applying (4.18) to (4.17), we have

$\left\|I^{\alpha}_{\cdot}\left[f_{m}^{n(\varepsilon)}(\cdot,x)w^{n(\varepsilon)}_{m}(\cdot,x)\left|w^{n(\varepsilon)}_{m}(\cdot,x)\right|^{p-2}-f_{m}^{(\varepsilon)}(\cdot,x)w^{(\varepsilon)}_{m}(\cdot,x)\left|w^{(\varepsilon)}_{m}(\cdot,x)\right|^{p-2}\right]\right\|_{L_{1}((0,t))}\to
0$

as $n\to\infty$. Therefore, there exists a sequence $n_{l}$ such that
$w^{n_{l}(\varepsilon)}_{m}(\cdot,x)\to w^{(\varepsilon)}_{m}(\cdot,x)$ and
$I_{\cdot}^{\alpha}\left[f_{m}^{n_{l}(\varepsilon)}w^{n_{l}(\varepsilon)}_{m}\left|w^{n_{l}(\varepsilon)}_{m}\right|^{p-2}\right]\to
I_{\cdot}^{\alpha}\left[f_{m}^{(\varepsilon)}w^{(\varepsilon)}_{m}\left|w^{(\varepsilon)}_{m}\right|^{p-2}\right]$
almost everywhere on $[0,t]$. Furthermore, the convergence holds everywhere on
$[0,t]$ due to the continuity in $t$. Then, by considering sequence $n_{l}$
instead of $n$ and letting $l\to\infty$ for (4.16), we have

$\displaystyle\frac{1}{p}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p}\leq
I_{t}^{\alpha}\left[f_{m}^{(\varepsilon)}(\cdot,x)w^{(\varepsilon)}_{m}(\cdot,x)\left|w^{(\varepsilon)}_{m}(\cdot,x)\right|^{p-2}\right].$

Since $t\leq\tau_{m}^{m}$, $\rho_{m}(u_{m})=1$. By integrating with respect to
$x$, we have

$\displaystyle\frac{\Gamma(\alpha)}{p}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p}dx$
(4.19)
$\displaystyle\leq\int_{0}^{t}(t-s)^{\alpha-1}\int_{\mathbb{R}^{d}}(Lw_{m}(s,\cdot))^{(\varepsilon)}(x)w_{m}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dxds$
$\displaystyle\quad+\int_{0}^{t}(t-s)^{\alpha-1}\int_{\mathbb{R}^{d}}\left[\bar{b}^{i}(s,\bar{x}^{i})\left(|w_{m}(s,\cdot)+v(s,\cdot)|^{2}\right)_{x^{i}}^{{(\varepsilon)}}(x)\right]w_{m}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dxds.$

Furthermore, by integration by parts, we obtain

$\displaystyle\int_{\mathbb{R}^{d}}\left[(Lw_{m}(s,\cdot))^{(\varepsilon)}(x)+\bar{b}^{i}\left(|w_{m}(s,\cdot)+v(s,\cdot)|^{2}\right)_{x^{i}}^{{(\varepsilon)}}(x)\right]w_{m}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dx$
(4.20)
$\displaystyle\leq-(p-1)\int_{\mathbb{R}^{d}}\left(a^{ij}w_{m}\right)_{x^{j}}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w^{(\varepsilon)}_{mx^{i}}(s,x)dx$
$\displaystyle\quad+(p-1)\int_{\mathbb{R}^{d}}\left(\left(2a^{ij}_{x^{j}}-b^{i}\right)w_{m}\right)^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w^{(\varepsilon)}_{mx^{i}}(s,x)dx$
$\displaystyle\quad+\int_{\mathbb{R}^{d}}\left(\left(a^{ij}_{x^{i}x^{j}}-b^{i}_{x^{i}}+c\right)w_{m}\right)^{(\varepsilon)}(s,x)w_{m}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dx$
$\displaystyle\quad-(p-1)\int_{\mathbb{R}^{d}}\bar{b}^{i}(s,\bar{x}^{i})\left((w_{m}(s,\cdot)+v(s,\cdot))^{2}\right)^{(\varepsilon)}(x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w_{mx^{i}}^{(\varepsilon)}(s,x)dx.$

Additionally, observe that

$\displaystyle\left|\left(a^{ij}w_{m}\right)_{x^{j}}^{(\varepsilon)}(s,x)-a^{ij}(s,x)w_{mx^{j}}^{(\varepsilon)}(s,x)\right|$
(4.21)
$\displaystyle\quad=\varepsilon^{-1}\left|\int_{\mathbb{R}^{d}}\left(a^{ij}(s,x-\varepsilon
y)-a^{ij}(s,x)\right)w_{m}(s,x-\varepsilon y)\zeta_{y^{j}}(y)dy\right|$
$\displaystyle\quad\leq N(K)\int_{\mathbb{R}^{d}}|w_{m}(s,x-\varepsilon
y)||y||\zeta_{y}(y)|dy,$

and by (3.2),

$\displaystyle-\int_{\mathbb{R}^{d}}a^{ij}(s,x)w_{mx^{i}}^{(\varepsilon)}(s,x)w_{mx^{j}}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dx$
(4.22)
$\displaystyle\quad\leq-K^{-1}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left|w_{mx}^{(\varepsilon)}(s,x)\right|^{2}dx.$

Thus, by combining (4.21) and (4.22)

$\displaystyle-\int_{\mathbb{R}^{d}}\left(a^{ij}w_{m}\right)_{x^{j}}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w_{mx^{i}}^{(\varepsilon)}(s,x)dx$
(4.23)
$\displaystyle\quad=-\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w_{mx^{i}}^{(\varepsilon)}(s,x)\left[\left(aw_{m}\right)_{x^{j}}^{(\varepsilon)}(s,x)-a(s,x)w_{mx^{j}}^{(\varepsilon)}(s,x)\right]dx$
$\displaystyle\quad\quad-\int_{\mathbb{R}^{d}}a^{ij}(s,x)w_{mx^{i}}^{(\varepsilon)}(s,x)w_{mx^{j}}^{(\varepsilon)}(s,x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dx$
$\displaystyle\quad\leq
N\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left|w_{mx^{i}}^{(\varepsilon)}(s,x)\right|\int_{\mathbb{R}^{d}}|w_{m}(s,x-\varepsilon
y)||y||\zeta_{y}(y)|dydx$
$\displaystyle\quad\quad-K^{-1}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left|w_{mx}^{(\varepsilon)}(s,x)\right|^{2}dx$
$\displaystyle\quad\leq
N\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left(\int_{\mathbb{R}^{d}}|w_{m}(s,x-\varepsilon
y)||y||\zeta_{y}(y)|dy\right)^{2}dx$
$\displaystyle\quad\quad-\frac{1}{2}K^{-1}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left|w_{mx}^{(\varepsilon)}(s,x)\right|^{2}dx,$

where $N=N(K)$. Moreover,

$\displaystyle\left|\left(\left(2a^{ij}_{x^{j}}-b^{i}\right)w_{m}\right)^{(\varepsilon)}(s,x)\right|$
$\displaystyle=\left|\int_{\mathbb{R}^{d}}\left(2a^{ij}_{y^{j}}(s,y)-b^{i}(s,y)\right)w_{m}(s,y)\zeta_{\varepsilon}(x-y)dy\right|$
(4.24) $\displaystyle\leq
K\int_{\mathbb{R}^{d}}|w_{m}(s,y)|\zeta_{\varepsilon}(x-y)dy$
$\displaystyle=K(|w_{m}(s,\cdot)|)^{(\varepsilon)}(x)$

and

$\left|\left(\left(a^{ij}_{x^{i}x^{j}}-b^{i}_{x^{i}}+c\right)w_{m}\right)^{(\varepsilon)}(s,x)\right|\leq
K(|w_{m}(s,\cdot)|)^{(\varepsilon)}(x).$ (4.25)

Thus, by applying Hölder’s inequality, (4.23), (4.24), and (4.25) to (4.20),
we have

$\displaystyle\int_{\mathbb{R}^{d}}\left[(Lw_{m}(s,\cdot))^{(\varepsilon)}(x)+\bar{b}^{i}\left(|w_{m}(s,\cdot)+v(s,\cdot)|^{2}\right)_{x^{i}}^{{(\varepsilon)}}(x)\right]w^{(\varepsilon)}_{m}(s,x)\left|w^{(\varepsilon)}_{m}(s,x)\right|^{p-2}dx$
(4.26) $\displaystyle\quad\leq
N\int_{\mathbb{R}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left(\int_{\mathbb{R}}|w_{m}(s,x-\varepsilon
y)||y||\zeta_{y}(y)|dy\right)^{2}dx$
$\displaystyle\quad\quad-\frac{p-1}{4K}\sum_{i}\int_{\mathbb{R}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left|w_{mx^{i}}^{(\varepsilon)}(s,x)\right|^{2}dx$
$\displaystyle\quad\quad+N\sum_{i}\int_{\mathbb{R}^{d}}\left((|w_{m}(s,\cdot)|)^{(\varepsilon)}(x)\right)^{2}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}dx$
$\displaystyle\quad\quad+N\int_{\mathbb{R}^{d}}(|w_{m}(s,\cdot)|)^{(\varepsilon)}(x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-1}dx.$
$\displaystyle\quad\quad-(p-1)\sum_{i}\int_{\mathbb{R}^{d}}\bar{b}^{i}(s,\bar{x}^{i})\left((w_{m}(s,\cdot)+v(s,\cdot))^{2}\right)^{(\varepsilon)}(x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}w_{mx^{i}}^{(\varepsilon)}(s,x)dx,$

where $N=N(K)$. Furthermore, note that

$\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p}w_{mx^{i}}^{(\varepsilon)}(s,x)dx=0\quad\text{for}\quad
s\leq t.$ (4.27)

Indeed, take a nonnegative smooth function $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$ such that $\phi(x)=1$ on $|x|<1$, $\phi(x)=0$
on $|x|>2$, and $\sup_{x\in\mathbb{R}^{d}}|\phi^{\prime}(x)|\leq 2$. Then,
integration by parts yields

$\displaystyle\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p}w_{mx^{i}}^{(\varepsilon)}(s,x)\phi(x/n)dx$
$\displaystyle\quad=-p\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p}w_{mx^{i}}^{(\varepsilon)}(s,x)\phi(x/n)dx-\frac{1}{n}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p}w_{m}^{(\varepsilon)}(s,x)\phi^{\prime}(x/n)dx.$

Thus, we have

$\limsup_{n\to\infty}\left|\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p}w_{mx^{i}}^{(\varepsilon)}(s,x)\phi(x/n)dx\right|\leq\limsup_{n\to\infty}\frac{2}{n(p+1)}\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p+1}dx=0$
(4.28)

and (4.28) yields (4.27). Then, from the last term of (4.26), by applying
(4.27) and the Hölder’s inequality, we have

$\displaystyle\left|\sum_{i}\int_{\mathbb{R}^{d-1}}\bar{b}^{i}(s,\bar{x}^{i})\int_{\mathbb{R}}\left(\left|w_{m}(t,\cdot)+v(t,\cdot)\right|^{2}\right)^{(\varepsilon)}(x)\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p-2}w_{mx^{i}}^{(\varepsilon)}(t,x)dx^{i}d\bar{x}^{i}\right|$
(4.29) $\displaystyle\quad\leq
N\sum_{i}\int_{\mathbb{R}^{d}}\left|\left(\left|w_{m}(t,\cdot)+v(t,\cdot)\right|^{2}\right)^{(\varepsilon)}(x)-\left|w_{m}^{(\varepsilon)}(t,x)\right|^{2}\right|\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p-2}\left|w_{mx^{i}}^{(\varepsilon)}(t,x)\right|dx$
$\displaystyle\quad\leq
N\int_{\mathbb{R}^{d}}\left(\left(\left|w_{m}(t,\cdot)+v(t,\cdot)\right|^{2}\right)^{(\varepsilon)}(x)-\left|w_{m}^{(\varepsilon)}(t,x)\right|^{2}\right)^{2}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p-2}dx$
$\displaystyle\quad\quad+\frac{1}{8KN}\sum_{i}\int_{\mathbb{R}^{d}}\left|w_{mx^{i}}^{(\varepsilon)}(t,x)\right|^{2}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p-2}dx,$

where $N=N(K)$. Then, by applying (4.26) and (4.29) to (4.19), we have

$\displaystyle\int_{\mathbb{R}^{d}}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p}dx$
$\displaystyle\quad\leq
NI^{\alpha}_{t}\int_{\mathbb{R}}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}\left(\int_{\mathbb{R}}|w_{m}(s,x-\varepsilon
y)||y||\zeta_{y}(y)|dy\right)^{2}dx$
$\displaystyle\quad\quad+NI^{\alpha}_{t}\int_{\mathbb{R}^{d}}\left|(|w_{m}(s,\cdot)|)^{(\varepsilon)}(x)\right|^{2}\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-2}+(|w_{m}(s,\cdot)|)^{(\varepsilon)}(x)\left|w_{m}^{(\varepsilon)}(s,x)\right|^{p-1}dx$
$\displaystyle\quad\quad+NI^{\alpha}_{t}\int_{\mathbb{R}^{d}}\left(\left(\left|w_{m}(t,\cdot)+v(t,\cdot)\right|^{2}\right)^{(\varepsilon)}(x)-\left|w_{m}^{(\varepsilon)}(t,x)\right|^{2}\right)^{2}\left|w_{m}^{(\varepsilon)}(t,x)\right|^{p-2}dx,$

where $N=N(p,K)$. By letting $\varepsilon\downarrow 0$, we have

$\displaystyle\|w_{m}(t,\cdot)\|_{L_{p}}^{p}$ $\displaystyle\leq
NI^{\alpha}_{t}\int_{\mathbb{R}}\left|w_{m}(\cdot,x)\right|^{p}dx+NI^{\alpha}_{t}\int_{\mathbb{R}^{d}}\left(\left|w_{m}(\cdot,x)+v(\cdot,x)\right|^{2}-\left|w_{m}(\cdot,x)\right|^{2}\right)^{2}\left|w_{m}(\cdot,x)\right|^{p-2}dx$
$\displaystyle\leq
NI^{\alpha}_{t}\int_{\mathbb{R}}\left|w_{m}(\cdot,x)\right|^{p}dx+NI^{\alpha}_{t}\int_{\mathbb{R}^{d}}\left|v(\cdot,x)\right|^{2}\left|w_{m}(\cdot,x)\right|^{p}+\left|v(\cdot,x)\right|^{4}\left|w_{m}(\cdot,x)\right|^{p-2}dx$
$\displaystyle\leq N\left(1+\sup_{s\leq
t,x\in\mathbb{R}^{d}}|v(s,x)|^{2}\right)I_{t}^{\alpha}\left\|w_{m}(\cdot,\cdot)\right\|_{L_{p}}^{p}+N\sup_{s\leq
t}\|v(s,\cdot)\|_{L_{2p}}^{2p}.$

for all $t\leq\tau_{m}^{m}$. Then, by Theorem 4.7, we obtain

$\|w_{m}(t,\cdot)\|_{L_{p}}^{p}\leq N\sup_{s\leq
t}\|v(s,\cdot)\|^{2p}_{L_{2p}}\left[1+\sum_{k=0}^{\infty}\frac{\left(1+\sup_{s\leq
t,x\in\mathbb{R}^{d}}|v(s,x)|^{2}\right)^{k}}{\Gamma(k\alpha)}\frac{(\Gamma(\alpha)T^{\alpha})^{k}}{k\alpha}\right]$

for all $t\leq\tau_{m}^{m}$. The lemma is proved.

∎

Finally, we demonstrate that the global solution candidate does not explode in
a finite time.

######  Lemma 4.9.

For any $T<\infty$, we have

$\lim_{R\to\infty}P\left(\left\\{\omega\in\Omega:\sup_{t\leq
T,x\in\mathbb{R}^{d}}|u(t,x)|>R\right\\}\right)=0.$

###### Proof.

Let $v$ be the function introduced in Lemma 4.5. Define

$\displaystyle\tau^{1}(S)$ $\displaystyle:=\inf\left\\{t\geq
0:\|v(t,\cdot)\|_{L_{p}}\geq S\right\\}\wedge
T,\quad\tau^{2}(S):=\inf\left\\{t\geq 0:\sup_{x\in\mathbb{R}^{d}}|v(t,x)|\geq
S\right\\}\wedge T,$

and

$\tau_{m}^{0}(S):=\tau_{m}^{m}\wedge\tau^{1}(S)\wedge\tau^{2}(S),$

where $\tau_{m}^{m}$ is the stopping time introduced in (4.9). Set
$r:=\frac{p}{p-1}$. Then, by Lemmas 4.2 and 2.10, (viii), Hölder inequality,
and Minkowski inequality, we have

$\displaystyle\|u_{m}\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau_{m}^{0}(S))}-N\|u_{0}\|^{p}_{U_{p}^{\gamma}}$
(4.30) $\displaystyle\quad\leq
N\left\|\sum_{i}\bar{b}^{i}(u_{m}^{2}\rho_{m}(u_{m}))_{x^{i}}\right\|_{\mathbb{H}_{p}^{\gamma-2}(\tau_{m}^{0}(S))}^{p}+N\|\sigma(u_{m})\bm{\eta}\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau_{m}^{0}(S),l_{2})}^{p}$
$\displaystyle\quad\leq
N\left\|u_{m}^{2}\right\|_{\mathbb{H}_{p}^{\gamma-1}(\tau_{m}^{0}(S))}^{p}+N\|\sigma(u_{m})\bm{\eta}\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau_{m}^{0}(S),l_{2})}^{p}$
$\displaystyle\quad\leq
N\mathbb{E}\int_{0}^{\tau_{m}^{0}(S)}\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}R_{1-\gamma}(y)|u_{m}(s,x-y)|^{2}dy\right|^{p}dxds$
$\displaystyle\quad\quad+N\mathbb{E}\int_{0}^{\tau_{m}^{0}(S)}\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(y)\right|^{2}|u_{m}(s,x-y)|^{2}dy\right|^{p/2}dxds$
$\displaystyle\quad\leq
N\mathbb{E}\int_{0}^{\tau_{m}^{0}(S)}\int_{\mathbb{R}^{d}}\left|\int_{\mathbb{R}^{d}}\left|R_{1-\gamma}(y)\right|^{r}|u_{m}(s,x-y)|^{r}dy\right|^{p/r}dx\int_{\mathbb{R}^{d}}|u_{m}(s,x)|^{p}dxds$
$\displaystyle\quad\quad+N\mathbb{E}\int_{0}^{\tau_{m}^{0}(S)}\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(x)\right|^{2}dx\right)^{p/2}\int_{\mathbb{R}^{d}}|u_{m}(s,x)|^{p}dxds$
$\displaystyle\quad\leq
N_{0}\mathbb{E}\int_{0}^{\tau_{m}^{0}(S)}\left[1+\int_{\mathbb{R}^{d}}|u_{m}(s,x)|^{p}dx\right]\int_{\mathbb{R}^{d}}|u_{m}(s,x)|^{p}dxds,$

where
$N_{0}=N(\alpha,\beta,\gamma,d,p,K,T)\left[\left(\int_{\mathbb{R}^{d}}\left|R_{1-\gamma}(x)\right|^{r}dx\right)^{p/r}+\left(\int_{\mathbb{R}^{d}}\left|R_{-\gamma+2-c_{0}}(x)\right|^{2}dx\right)^{p/2}\right]$.
Note that $N_{0}<\infty$ due to $r<\frac{d}{d+\gamma-1}$ and Remark 2.8. Then,
by Lemma 4.8 and the definitions of $\tau_{1}(S)$ and $\tau_{2}(S)$,

$\displaystyle\int_{\mathbb{R}^{d}}|u_{m}(t,x)|^{p}dx$ (4.31)
$\displaystyle\leq
N\int_{\mathbb{R}^{d}}|u_{m}(t,x)-v(t,x)|^{p}+|v(t,x)|^{p}dx$
$\displaystyle\leq N\sup_{s\leq t}\sup_{x\in\mathbb{R}^{d}}|v(t,x)|\sup_{s\leq
t}\|v(s,\cdot)\|_{p}^{p}\left[1+\sum_{k=0}^{\infty}\frac{\left(1+\sup_{s\leq
t,x\in\mathbb{R}^{d}}|v(t,x)|^{2}\right)^{k}}{\Gamma(k\alpha)}\frac{(\Gamma(\alpha)T^{\alpha})^{k}}{k\alpha}\right]$
$\displaystyle\quad+\int_{\mathbb{R}^{d}}|v(t,x)|^{p}dx$
$\displaystyle<N(p,S,K).$

Therefore, by combining (4.30) and (4.31), we have

$\|u_{m}\|^{p}_{\mathcal{H}_{p}^{\gamma}(\tau_{m}^{0}(S))}\leq
N+N\|u_{0}\|^{p}_{U_{p}^{\gamma}},$ (4.32)

where $N=N(\alpha,\beta,\gamma,d,p,S,K,T)$. It should be noted that the right-
hand side of (4.32) is independent of $m$. Therefore, by Chebyshev’s
inequality and Lemma 4.3, we have

$\displaystyle
P\left(\sup_{t\leq\tau^{0}_{m}(S)}\sup_{x\in\mathbb{R}^{d}}|u(t,x)|>R\right)$
$\displaystyle\leq\frac{1}{R^{p}}\mathbb{E}\sup_{t\leq\tau^{0}_{m},x\in\mathbb{R}^{d}}|u(t,x)|^{p}$
$\displaystyle\leq\frac{1}{R^{p}}\mathbb{E}\sup_{t\leq\tau^{0}_{m},x\in\mathbb{R}^{d}}|u_{m}(t,x)|^{p}$
$\displaystyle\leq\frac{1}{R^{p}}\|u_{m}\|_{\mathcal{H}_{p}^{\gamma}(\tau^{0}_{m})}^{p}$
$\displaystyle\leq\frac{N}{R^{p}},$

where $N=N(u_{0},\alpha,\beta,\gamma,d,p,S,K,T).$ In contrast, by Lemma 4.5,

$\displaystyle P\left(\tau^{1}(S)<T\right)+P\left(\tau^{2}(S)<T\right)$
$\displaystyle\leq P\left(\sup_{t\leq
T}\|v(t,\cdot)\|_{L_{p}}>S\right)+P\left(\sup_{t\leq
T}\sup_{x\in\mathbb{R}^{d}}|v(t,x)|>S\right)$
$\displaystyle\leq\frac{1}{S^{p}}\mathbb{E}\sup_{t\leq
T}\|v(t,\cdot)\|_{L_{p}}^{p}+\frac{1}{S^{p}}\mathbb{E}\sup_{t\leq
T,x\in\mathbb{R}^{d}}|v(t,x)|^{p}$
$\displaystyle\leq\frac{1}{S^{p}}N(u_{0},h,\alpha,\beta,\gamma,d,p,K,T).$

Thus,

$\displaystyle P\left(\sup_{t\leq T,x\in\mathbb{R}^{d}}|u(t,x)|>R\right)$
$\displaystyle\leq\liminf_{m\to\infty}P\left(\sup_{t\leq\tau_{m}^{0}(S),x\in\mathbb{R}^{d}}|u(t,x)|>R\right)+P\left(\tau^{1}(S)<T\right)+P\left(\tau^{2}(S)<T\right)$
$\displaystyle\leq\frac{N_{1}}{R^{p}}+\frac{N_{2}}{S^{p}},$

where $N_{1}=N_{1}(u_{0},\alpha,\beta,\gamma,d,p,S,K,T)$ and
$N_{2}=N_{2}(u_{0},h,\alpha,\beta,\gamma,d,p,K,T)$. The lemma is proved by
letting $R\to\infty$ and $S\to\infty$ in order.

∎

###### Proof of Theorem 3.5.

Step 1. (Uniqueness).  Suppose $u,\bar{u}\in\mathcal{H}_{p,loc}^{\gamma}$ are
nonnegative solutions of equation (3.1). By Definition 2.13, there are bounded
stopping times $\tau_{n}$ $(n=1,2,\cdots)$ such that

$\tau_{n}\uparrow\infty\quad\mbox{and}\quad
u,\bar{u}\in\mathcal{H}_{p}^{\gamma}(\tau_{n}).$

Fix $n\in\mathbb{N}$. Note that $u,\bar{u}\in
C([0,\tau_{n}];C(\mathbb{R}^{d}))$ almost surely and

$\mathbb{E}\sup_{t\leq\tau_{n}}\sup_{x\in\mathbb{R}^{d}}|u(t,x)|^{p}+\mathbb{E}\sup_{t\leq\tau_{n}}\sup_{x\in\mathbb{R}^{d}}|\bar{u}(t,x)|^{p}<\infty.$
(4.33)

Then, for $m\in\mathbb{N}$, define

$\begin{gathered}\tau_{m,n}^{1}:=\inf\left\\{t\geq
0:\sup_{x\in\mathbb{R}^{d}}|u(t,x)|>m\right\\}\wedge\tau_{n},\\\
\tau_{m,n}^{2}:=\inf\left\\{t\geq
0:\sup_{x\in\mathbb{R}^{d}}|\bar{u}(t,x)|>m\right\\}\wedge\tau_{n},\end{gathered}$

and

$\tau_{m,n}:=\tau_{m,n}^{1}\wedge\tau_{m,n}^{2}.$ (4.34)

Due to (4.33), $\tau_{m,n}^{1}$ and $\tau_{m,n}^{2}$ are well-defined stopping
times; thus, $\tau_{m,n}$ is a stopping time. Observe that
$u,\bar{u}\in\mathcal{H}_{p}^{\gamma}(\tau_{m,n})$ and
$\tau_{m,n}\uparrow\tau_{n}$ as $m\to\infty$ almost surely. Fix
$m\in\mathbb{N}$. Notice that
$u,\bar{u}\in\mathcal{H}_{p}^{\gamma}(\tau_{m,n})$ are solutions to equation

$\partial_{t}^{\alpha}v=Lv+\bar{b}^{i}\left(v^{2}\rho_{m}(v)\right)_{x^{i}}+\partial_{t}^{\beta}\int_{0}^{t}\sigma(v)dW_{t},\quad
0<t\leq\tau_{m,n};\quad v(0,\cdot)=u_{0},$

where $Lv=a^{ij}v_{x^{i}x^{j}}+b^{i}v_{x^{i}}+cv$. By the uniqueness result in
Lemma 4.3, we conclude that $u=\bar{u}$ in
$\mathcal{H}_{p}^{\gamma}(\tau_{m,n})$ for each $m\in\mathbb{N}$. The monotone
convergence theorem yields $u=\bar{u}$ in
$\mathcal{H}_{p}^{\gamma}(\tau_{n})$, which implies $u=\bar{u}$ in
$\mathcal{H}_{p,loc}^{\gamma}$.

Step 2 (Existence.). Let $T<\infty$. For $m\in\mathbb{N}$, define
$\tau_{m}^{m}$ and $u$ as in Remark 4.4. Observe that

$\displaystyle P\left(\tau_{m}^{m}<T\right)\leq P\left(\sup_{t\leq
T,x\in\mathbb{R}^{d}}|u(t,x)|\geq m\right).$

Indeed, if $\tau_{m}^{m}<T$, then
$\sup_{t\leq\tau_{m}^{m},x\in\mathbb{R}^{d}}|u(t,x)|=\sup_{t\leq\tau_{m}^{m},x\in\mathbb{R}^{d}}|u_{m}(t,x)|=m$
almost surely. Then, by Lemma 4.9, we have

$\displaystyle\limsup_{m\to\infty}P\left(\tau_{m}^{m}<T\right)$
$\displaystyle\leq\limsup_{m\to\infty}P\left(\sup_{t\leq
T,x\in\mathbb{R}^{d}}|u(t,x)|\geq m\right)$ $\displaystyle=0$

Since $T<\infty$ is arbitrary, $\tau_{m}^{m}\to\infty$ in probability. In
addition, we conclude that $\tau_{m}^{m}\uparrow\infty$ almost surely, because
$\tau_{m}^{m}$ is increasing in $m$.

Last, set $\tau_{m}:=\tau_{m}^{m}\wedge m$. Note that (see Remark 4.4)

$u(t,x)=u_{m}(t,x)\quad\text{for}\quad t\in[0,\tau_{m}].$

Observe that $\sup_{x\in\mathbb{R}^{d}}|u_{m}(t,x)|\leq m$ for
$t\in[0,\tau_{m}]$; thus, $u_{m}$ satisfies (3.1) almost everywhere
$t\in[0,\tau_{m}]$ almost surely. Because $u=u_{m}$ for $t\in[0,\tau_{m}]$ and
$u_{m}\in\mathcal{H}_{p}^{\gamma}(\tau_{m})$, it follows that
$u\in\mathcal{H}_{p}^{\gamma}(\tau_{m})$ and $u$ satisfies (3.1) for all
$t\leq\tau_{m}$ almost surely. We have $u\in\mathcal{H}_{p,loc}^{\gamma}$
because $\tau_{m}\uparrow\infty$ as $m\to\infty$ almost surely.

To obtain Hölder regularity, we employ Corollary 2.18. The theorem is proved.

∎

###### Proof of Theorem 3.9.

The proof of Theorem 3.9 is motivated by [26, Corollarly 5.11]. Since $q>p$,
by Theorem 3.5, there exists a unique solution
$\bar{u}\in\mathcal{H}_{q,loc}^{\gamma}$ satisfying equation (3.1). By
Definition 2.13, there exists $\tau_{n}$ such that $\tau_{n}\to\infty$ almost
surely as $n\to\infty$, $u\in\mathcal{H}_{p}^{\gamma}(\tau_{n})$ and
$\bar{u}\in\mathcal{H}_{q}^{\gamma}(\tau_{n})$. Fix $n\in\mathbb{N}$. Because
$\frac{2+\alpha d}{\alpha\gamma}<p<q$, we can define $\tau_{m,n}$
$(m\in\mathbb{N})$ as in (4.34). Notice that for any $p_{0}>p$, we have

$u\in\mathbb{L}_{p_{0}}(\tau_{m,n})$

since

$\mathbb{E}\int_{0}^{\tau_{m,n}}\int_{\mathbb{R}}|u(t,x)|^{p_{0}}dxdt\leq
m^{p_{0}-p}\mathbb{E}\int_{0}^{\tau_{m,n}}\int_{\mathbb{R}}|u(t,x)|^{p}dxdt<\infty.$

Observe that
$\bar{b}^{i}uu_{x^{i}}\in\mathbb{H}_{q}^{\gamma-2}(\tau_{m,n})\subset\mathbb{H}_{q}^{-2}(\tau_{m,n})$.
Indeed, similar to (4.6),

$\displaystyle\mathbb{E}\int_{0}^{\tau_{m,n}}\left\|\frac{1}{2}\bar{b}^{i}(s,\cdot)\left((u(s,\cdot))^{2}\right)_{x^{i}}\right\|_{H_{q}^{\gamma-2}}^{q}ds$
$\displaystyle\leq
N\mathbb{E}\int_{0}^{\tau_{m,n}}\int_{\mathbb{R}}|u(s,x)|^{2q}dxds<\infty.$

Additionally, we have

$\displaystyle au_{xx}\in\mathbb{H}_{q}^{-2}(\tau_{m,n}),\quad
bu_{x}\in\mathbb{H}_{q}^{-1}(\tau_{m,n}),\quad\text{and}\quad
cu\in\mathbb{L}_{q}(\tau_{m,n}).$

Therefore, because
$\mathbb{L}_{q}(\tau_{m,n})\subset\mathbb{H}_{q}^{-1}(\tau_{m,n})\subset\mathbb{H}_{q}^{-2}(\tau_{m,n})$,

$a^{ij}u_{x^{i}x^{j}}+b^{i}u_{x^{i}}+cu+\bar{b}^{i}uu_{x^{i}}\in\mathbb{H}_{q}^{-2}(\tau_{m,n}).$
(4.35)

Similar to (4.12), we have

$\|\sigma(u)\bm{\eta}\|^{q}_{\mathbb{H}^{\gamma-2+c_{0}}_{q}(\tau_{m,n},l_{2})}\leq
N\int_{0}^{\tau_{m,n}}\left\|h(t,\cdot)\right\|_{L_{q}}^{q}dt<\infty.$ (4.36)

Thus, we have

$\sigma(u)\bm{\eta}\in\mathbb{H}_{q}^{\gamma-2+c_{0}}(\tau_{m,n},l_{2})\subset\mathbb{H}_{q}^{-2+c_{0}}(\tau_{m,n},l_{2}).$
(4.37)

Due to (4.35), (4.37), and Lemma 4.2, $u$ is in $\mathcal{L}_{q}(\tau_{m,n})$
and $u$ satisfies (3.1) for almost everywhere $t\leq\tau_{m,n}$ almost surely.
On the other hand, since
$\bar{b}^{i}uu_{x^{i}}\in\mathbb{H}_{q}^{\gamma-2}(\tau_{m,n})$ and
$\sigma(u)\bm{\eta}\in\mathbb{H}^{\gamma-2+c_{0}}_{q}(\tau_{m,n},l_{2})$,
Lemma 4.2 implies that there exists $v\in\mathcal{H}_{q}^{\gamma}(\tau_{m,n})$
satisfying

$\partial_{t}^{\alpha}v=Lv+\bar{b}^{i}uu_{x^{i}}+\partial_{t}^{\alpha}\int_{0}^{t}\sigma(u)\eta_{k}dw^{k}_{t},\quad
0<t\leq\tau_{m,n}\,;\quad v(0,\cdot)=u_{0},$ (4.38)

where $Lv=a^{ij}v_{x^{i}x^{j}}+b^{i}v_{x^{i}}+cv$. In (4.38), note that
$\bar{b}^{i}uu_{x^{i}}$ and $\sigma^{k}(u)$ are used instead of
$\bar{b}^{i}vv_{x^{i}}$ and $\sigma^{k}(v)$. Moreover, because
$u\in\mathcal{L}_{q}(\tau_{m,n})$ satisfies equation (4.38),
$\bar{v}:=u-v\in\mathcal{L}_{q}(\tau_{m,n})$ satisfies

$\partial_{t}^{\alpha}\bar{v}=a^{ij}\bar{v}_{x^{i}x^{j}}+b^{i}\bar{v}_{x^{i}}+c\bar{v},\quad
0<t\leq\tau_{m,n}\,;\quad\bar{v}(0,\cdot)=0.$

By the deterministic version of Lemma 4.2, we have $\bar{v}=0$ in
$\mathcal{L}_{q}(\tau_{m,n})$; thus, $u=v$ in $\mathcal{L}_{q}(\tau_{m,n})$.
Therefore, $u$ is in $\mathcal{H}_{q}^{\gamma}(\tau_{m,n})$. As
$\bar{u}\in\mathcal{H}_{q}^{\gamma}(\tau_{m,n})$ and $\bar{u}$ satisfies
equation (3.1), by Lemma 4.3, we have $u=\bar{u}$ in
$\mathcal{H}_{q}^{\gamma}(\tau_{m,n})$. The theorem is proved. ∎

##  5\.  Proof of Theorem 2.16

This section provides a proof of the embedding theorem for solution spaces
$\mathcal{H}_{p}^{\gamma}(\tau)$. Consider the following fractional diffusion
equation

$\partial_{t}^{\alpha}u=\Delta u\quad t>0\,;\quad u(0,\cdot)=u_{0}(\cdot),$
(5.1)

where $\alpha\in(0,1)$ and $u_{0}(\cdot)\in C_{c}^{\infty}(\mathbb{R}^{d})$.
It turns out that a fundamental solution $p(t,x)$ exists such that

$p(t,\cdot)\in
L_{1}(\mathbb{R}^{d})\quad\mbox{and}\quad\mathcal{F}(p(t,\cdot))(\xi)=E_{\alpha}(-t^{\alpha}|\xi|^{2})$

(e.g., [24, Theorem 2.1]), and the solution of (5.1) is given by

$\displaystyle u(t,x)$ $\displaystyle=(u_{0}\ast
p(t,\cdot))(x)=\int_{\mathbb{R}^{d}}u_{0}(y)p(t,x-y)dy.$

For convenience, define

$q_{\alpha,\beta}(t,x):=\begin{cases}I^{\alpha-\beta}p(t,x)\quad\quad&\text{if}\quad\quad\alpha\geq\beta\\\
D^{\beta-\alpha}p(t,x)\quad\quad&\text{if}\quad\quad\alpha<\beta.\end{cases}$

We gather some facts related to $p(t,x)$ and $q_{\alpha,\beta}(t,x)$ (for more
information, see [22, 23, 24]).

######  Lemma 5.1.

Let $d\in\mathbb{N}$, $\alpha\in(0,1)$, $\beta<\alpha+1/2$, $\gamma\in[0,2)$,
and $\sigma\in\mathbb{R}$.

  1. (i)

For all $t\neq 0$ and $x\neq 0$,

$\partial_{t}^{\alpha}p(t,x)=\Delta
p(t,x)\quad\text{and}\quad\partial_{t}p(t,x)=\Delta q_{\alpha,1}(t,x).$

Additionally, for each $x\neq 0$, $\frac{\partial}{\partial t}p(t,x)\to 0$ as
$t\downarrow 0$. Moreover, $\frac{\partial}{\partial t}p(t,x)$ is integrable
in $\mathbb{R}^{d}$ uniformly on $t\in[\delta,T]$ for any $\delta>0$.

  2. (ii)

There exist constants $c=c(\alpha,d)$ and $N=N(\alpha,d)$ such that if
$|x|^{2}\geq t^{\alpha}$,

$|p(t,x)|\leq
N|x|^{-d}\exp\left\\{-c|x|^{\frac{2}{2-\alpha}}t^{-\frac{\alpha}{2-\alpha}}\right\\}.$

  3. (iii)

Let $n\in\mathbb{N}$. Then, there exists $N=N(\alpha,\gamma,n)$ such that

$\left|D_{t}^{\sigma}D_{x}^{n}(-\Delta)^{\gamma/2}q_{\alpha,\beta}(1,x)\right|\leq
N\left(|x|^{-d+2-\gamma-n}\wedge|x|^{-d-\gamma-n}\right).$

  4. (iv)

The scaling properties hold. In other words,

$\begin{gathered}q_{\alpha,\beta}(t,x)=t^{-\frac{\alpha
d}{2}+\alpha-\beta}q_{\alpha,\beta}(1,xt^{-\frac{\alpha}{2}}),\\\
(-\Delta)^{\gamma/2}q_{\alpha,\beta}(t,x)=t^{-\frac{\alpha(d+\gamma)}{2}+\alpha-\beta}(-\Delta)^{\gamma/2}q_{\alpha,\beta}(1,xt^{-\frac{\alpha}{2}}).\end{gathered}$

###### Proof.

To see (i), (ii), and (iii) follow from Theorems 2.1 and 2.3 of [23]. For
(iv), see (5.2) in [23]. ∎

######  Remark 5.2.

To prove Theorem 2.16, we define the operators. Let $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$ and $f\in
C_{c}^{\infty}((0,\infty)\times\mathbb{R}^{d})$. Take a function
$g=(g^{1},g^{2},\dots)$ satisfying the form

$g^{k}(t,x)=\begin{cases}\sum_{i=1}^{n}1_{\text{\,\raise
0.60275pt\hbox{${\scriptstyle|}$}\kern-2.37997pt$($}\tau_{i-1},\tau_{i}\text{$]$\kern-1.04996pt$]$}}(t)g^{ik}(x)&\quad\mbox{for}\quad
k=1,2,\dots,n,\\\ \quad\quad\quad\quad 0&\quad\mbox{for}\quad
k=n+1,\dots\end{cases}$ (5.2)

for some $n\in\mathbb{N}$, where $\tau_{i}$ is the bounded stopping time and
$g^{ik}\in C_{c}^{\infty}(\mathbb{R}^{d})$. Further, we set

$\displaystyle T^{1}_{t}\phi(x):=\int_{\mathbb{R}^{d}}p(t,x-y)\phi(y)dy,$
(5.3) $\displaystyle
T^{2}_{t}f(t,x):=\int_{0}^{t}\int_{\mathbb{R}^{d}}q_{\alpha,1}(t-r,x-y)f(r,y)dyds,$
(5.4) $\displaystyle
T^{3}_{t}g(t,x):=\sum_{k}\int_{0}^{t}\int_{\mathbb{R}^{d}}q_{\alpha,\beta}(t-r,x-y)g^{k}(r,y)dydw_{s}^{k}.$
(5.5)

It is well-known that $T_{t}^{1}\phi$, $T_{t}^{2}f$, and $T_{t}^{3}g$ are
solutions to

$\partial_{t}^{\alpha}u^{1}=\Delta u^{1};\quad u^{1}(0,\cdot)=\phi,$ (5.6)
$\displaystyle\partial_{t}^{\alpha}u^{2}=\Delta u^{2}+f;\quad
u^{2}(0,\cdot)=0,$ $\displaystyle\partial_{t}^{\alpha}u^{3}=\Delta
u^{3}+\partial^{\beta}_{t}\int_{0}^{t}g^{k}dw_{s}^{k};\quad u^{3}(0,\cdot)=0,$

respectively (for more information, see [22, 23, 24]).

First, we provide a smoothing effect of $T_{t}^{1}$, which implies the
continuity of $T_{t}^{1}\phi$ in $t$.

######  Lemma 5.3.

Let $\tau\leq T$ be a bounded stopping time and $\alpha\in(0,1)$. If
$p\in(1,\infty)$, $\theta\in[0,\alpha)$, $\phi\in
C_{c}^{\infty}(\mathbb{R}^{d})$, and $t\in(0,T)$, we have

$e^{-t}\left\|T^{1}_{t}\phi\right\|_{H^{\frac{2\theta}{\alpha}}_{p}}\leq
Nt^{-\theta}\|\phi\|_{L_{p}}$ (5.7)

where $N=N(\alpha,\theta,d,p,T)$.

###### Proof.

In the case of $\theta=0$, by Minköwski’s inequality, we have

$\displaystyle\|T^{1}_{t}\phi\|_{L_{p}}\leq\|p(t,\cdot)\ast\phi\|_{L_{p}}\leq\|p(t,\cdot)\|_{L_{1}}\|\phi\|_{L_{p}}\leq\|\phi\|_{L_{p}}.$
(5.8)

Thus, we have (5.7). For $\theta\in(0,\alpha)$, observe that

$\|e^{-t}T^{1}_{t}\phi\|_{H_{p}^{2\theta/\alpha}}=\|(1-\Delta)^{\theta/\alpha}(e^{-t}T^{1}_{t}\phi)\|_{L_{p}}\leq\|e^{-t}T^{1}_{t}\phi\|_{L_{p}}+\|(-\Delta)^{\theta/\alpha}(e^{-t}T^{1}_{t}\phi)\|_{L_{p}},$

where the last inequality follows from Lemma 2.10 (v). As
$e^{-t}\leq(Nt^{-\theta}\wedge 1)$, we have

$\left\|e^{-t}T^{1}_{t}\phi\right\|_{H_{p}^{2\theta/\alpha}}\leq
N\left(t^{-\theta}\left\|T^{1}_{t}\phi\right\|_{L_{p}}+\left\|(-\Delta)^{\theta/\alpha}T^{1}_{t}\phi\right\|_{L_{p}}\right).$
(5.9)

By inequality (5.8), we have

$t^{-\theta}\|T^{1}_{t}\phi\|_{L_{p}}\leq t^{-\theta}\|\phi\|_{L_{p}}.$ (5.10)

On the other hand, Minkowski’s inequality yields

$\|(-\Delta)^{\theta/\alpha}T^{1}_{t}h\|_{L_{p}}=\|(-\Delta)^{\theta/\alpha}(p(t,\cdot)\ast\phi)\|_{L_{p}}\leq\|((-\Delta)^{\theta/\alpha}p)(t,\cdot)\|_{L_{1}}\|\phi\|_{L_{p}}.$
(5.11)

Additionally Lemma 5.1 (iv), (ii), and (iii) imply

$\displaystyle\|((-\Delta)^{\theta/\alpha}p)(t,\cdot)\|_{L_{1}}$
$\displaystyle=\int_{\mathbb{R}^{d}}|((-\Delta)^{\theta/\alpha}p)(t,x)|dx$
(5.12) $\displaystyle\leq
t^{-\theta}\int_{\mathbb{R}^{d}}|((-\Delta)^{\theta/\alpha}p)(1,x)|dx$
$\displaystyle\leq N(\alpha,\theta,d,p)t^{-\theta}.$

Then, by applying (5.12) to (5.11), we have

$\|(-\Delta)^{\theta/\alpha}T^{1}_{t}\phi\|_{L_{p}}\leq
Nt^{-\theta}\|\phi\|_{L_{p}}.$ (5.13)

Thus, by plugging in (5.10) and (5.13) into (5.9), we have (5.7). The lemma is
proved. ∎

To proceed further, we introduce one of the embedding theorems for
Slobodetskii’s spaces.

######  Lemma 5.4.

If $\mu p>1$ and $p\geq 1$, for any continuous $L_{p}$-valued function $\phi$
and $\gamma\leq\rho$, we have the following:

$\displaystyle\|\phi(\rho)-\phi(\gamma)\|_{L_{p}}^{p}\leq N(\rho-\gamma)^{\mu
p-1}\int_{\gamma}^{\rho}\int_{\gamma}^{\rho}1_{t>s}\frac{\|\phi(t)-\phi(s)\|_{L_{p}}^{p}}{|t-s|^{1+\mu
p}}dsdt\quad\left(\frac{0}{0}:=0\right),$ (5.14)

where $N=N(\mu,p)$. In particular,

$\displaystyle\mathbb{E}\sup_{0\leq s<t\leq
T}\frac{\|\phi(t)-\phi(s)\|_{L_{p}}^{p}}{|t-s|^{\mu p-1}}$ $\displaystyle\leq
N\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\mathbb{E}\left\|\phi(t)-\phi(s)\right\|_{L_{p}}^{p}}{|t-s|^{1+\mu
p}}dsdt.$ (5.15)

With the help of Lemma 5.4, we obtain the Hölder continuity of
$T_{t}^{1}\phi$, $T_{t}^{2}f$, and $T_{t}^{3}g$ on $t\in[0,T]$.

######  Lemma 5.5.

Let $\alpha\in(0,1)$.

  1. (i)

If $p\in(1,\infty)$, $\theta\in(0,\alpha)$, and $\mu\in(0,1)$ satisfy

$p>\frac{1}{\alpha-\theta},\quad\mu\in\left(\frac{1}{\alpha
p},\frac{\alpha-\theta}{\alpha}\right)$ (5.16)

and $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$, then for $t,s\in(0,T)$,

$\left\|T^{1}_{t}f-T^{1}_{s}f\right\|_{H^{\frac{2\theta}{\alpha}-2}_{p}}^{p}\leq
N|t-s|^{\alpha\mu p-1}\left\|\phi\right\|_{L_{p}}^{p},$ (5.17)

where $N=N(\alpha,\theta,d,p,T)$. Additionally,

$\sup_{0\leq s<t\leq T}\frac{\left\|T_{t}^{1}\phi-
T_{s}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}^{p}}{|t-s|^{\alpha\mu
p-1}}\leq N\left\|\phi\right\|_{L_{p}}^{p}$ (5.18)

where $N=N(\alpha,\theta,d,p,T)$.

  2. (ii)

If $p\in(1,\infty)$, $\theta\in(0,\alpha)$, and $\mu\in(0,1)$ satisfy (5.16)
and $f\in C_{c}^{\infty}((0,T)\times\mathbb{R}^{d})$, then for $t,s\in(0,T)$,

$\left\|T^{2}_{t}f-T^{2}_{s}f\right\|_{H^{2\theta/\alpha}_{p}}^{p}\leq
N|t-s|^{\alpha\mu p-1}\int_{0}^{T}\left\|f(r,\cdot)\right\|_{L_{p}}^{p}dr$
(5.19)

where $N=N(\alpha,\theta,d,p,T)$. Additionally,

$\sup_{0\leq s<t\leq
T}\frac{\left\|T^{2}_{t}f-T^{2}_{s}f\right\|_{H^{2\theta/\alpha}_{p}}^{p}}{|t-s|^{\alpha\mu
p-1}}\leq N\int_{0}^{T}\left\|f(r,\cdot)\right\|_{L_{p}}^{p}dr,$ (5.20)

where $N=N(\alpha,\theta,d,p,T)$.

  3. (iii)

Let $\tau\leq T$ be a bounded stopping time. If $\beta<\alpha+\frac{1}{2}$,
$p\in[2,\infty)$, $\theta\in(0,\alpha-\beta+1/2)$, and $\mu\in(0,1)$ satisfy

$p>\frac{1}{(\alpha-\beta-\theta)\wedge
1/2+1/2},\quad\mu\in\left(\frac{1}{\alpha p},\frac{(\alpha-\beta-\theta)\wedge
1/2+1/2}{\alpha}\right),$

and $g=(g^{1},g^{2},\dots)$ is a function of type (5.2), then, for
$t,s\in(0,T)$, we have

$\mathbb{E}\left\|T_{t\wedge\tau}^{3}g-T_{s\wedge\tau}^{3}g\right\|_{H^{2\theta/\alpha}_{p}(l_{2})}^{p}\leq
N|t-s|^{\alpha\mu p-1}\|g\|_{\mathbb{L}_{p}(\tau,l_{2})}^{p},$ (5.21)

where $N=N(\alpha,\beta,\theta,d,p,T)$. Additionally,

$\mathbb{E}\sup_{0\leq
s<t\leq\tau}\frac{\left\|T_{t}^{3}g-T_{s}^{3}g\right\|_{H^{2\theta/\alpha}_{p}(l_{2})}^{p}}{|t-s|^{\alpha\mu
p-1}}\leq N\|g\|_{\mathbb{L}_{p}(\tau,l_{2})}^{p},$ (5.22)

where $N=N(\alpha,\beta,\theta,d,p,T)$.

###### Proof of (i).

By (5.14), we have

$\displaystyle\left\|T_{\rho}^{1}\phi-
T_{\gamma}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}^{p}$
$\displaystyle\leq N|\rho-\gamma|^{\alpha\mu
p-1}\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\left\|T_{t}^{1}\phi-
T_{s}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}^{p}}{|t-s|^{1+\alpha\mu
p}}dsdt$ (5.23) $\displaystyle\leq N|\rho-\gamma|^{\alpha\mu
p-1}\int_{0}^{T}\int_{0}^{T-t}t^{-1-\alpha\mu p}\left\|T_{t+s}^{1}\phi-
T_{s}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}^{p}dsdt.$

Since $T_{t}^{1}\phi$ is a solution to (5.6), Lemma 5.3 yields

$\displaystyle\left\|T_{t+s}^{1}\phi-
T_{s}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}$ (5.24)
$\displaystyle\quad\leq\left\|(T_{t+s}^{1}\phi-\phi)-(T_{s}^{1}\phi-\phi)\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}$
$\displaystyle\quad\leq\left\|\int_{0}^{t+s}(t+s-r)^{\alpha-1}\Delta
T_{r}^{1}\phi dr-\int_{0}^{s}(s-r)^{\alpha-1}\Delta T_{r}^{1}\phi
dr\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}$
$\displaystyle\quad\leq\int_{s}^{t+s}(t+s-r)^{\alpha-1}\left\|T_{r}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}}}dr$
$\displaystyle\quad\quad+\int_{0}^{s}\left|(t+s-r)^{\alpha-1}-(s-r)^{\alpha-1}\right|\left\|T_{r}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}}}dr$
$\displaystyle\quad\leq
N(2t^{\alpha}+s^{\alpha}-(t+s)^{\alpha})t^{-\theta}\|\phi\|_{L_{p}}.$

Since $0\leq 2t^{\alpha}+s^{\alpha}-(t+s)^{\alpha}\leq 2t^{\alpha}$, by
applying (5.24) to (5.23),

$\displaystyle\left\|T_{t}^{1}\phi-
T_{s}^{1}\phi\right\|_{H_{p}^{\frac{2\theta}{\alpha}-2}}^{p}$
$\displaystyle\leq N|\rho-\gamma|^{\alpha\mu p-1}\int_{0}^{T}t^{-1-\alpha\mu
p+\alpha p-\theta p}dt\|\phi\|_{L_{p}}^{p}$ $\displaystyle\leq
N|\rho-\gamma|^{\alpha\mu p-1}\|\phi\|_{L_{p}}^{p}.$

Thus, we have (5.17).

To obtain (5.18), use (5.15) instead of (5.14) and repeat the proof.

Proof of (ii). Let $\rho>0$ and $\gamma>0$. Notice that (5.14) yields

$\displaystyle\left\|T_{\rho}^{2}f-T_{\gamma}^{2}f\right\|_{H^{2\theta/\alpha}_{p}}^{p}$
$\displaystyle\leq N|\rho-\gamma|^{\alpha\mu
p-1}\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\left\|T_{t}^{2}f-T_{s}^{2}f\right\|_{H^{2\theta/\alpha}_{p}}^{p}}{|t-s|^{1+\alpha\mu
p}}dsdt.$ (5.25)

Then, by definition of $T_{t}^{2}f$ (see (5.4)),

$\displaystyle\left\|T^{2}_{t}f-T^{2}_{s}f\right\|_{H_{p}^{2\theta/\alpha}}$
(5.26)
$\displaystyle\quad\leq\left\|\int_{s}^{t}\int_{\mathbb{R}^{d}}q_{\alpha,1}(t-r,y)f(r,\cdot-y)dydr\right\|_{H_{p}^{2\theta/\alpha}}$
$\displaystyle\quad\quad+\left\|\int_{0}^{s}\int_{\mathbb{R}^{d}}\left(q_{\alpha,1}(t-r,y)-q_{\alpha,1}(s-r,y)\right)f(r,\cdot-y)dydr\right\|_{H_{p}^{2\theta/\alpha}}.$

Set

$\displaystyle I_{1}$
$\displaystyle:=\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\left\|\int_{s}^{t}\int_{\mathbb{R}^{d}}q_{\alpha,1}(t-r,y)f(r,\cdot-y)dydr\right\|_{H_{p}^{2\theta/\alpha}}^{p}}{|t-s|^{1+\alpha\mu
p}}dsdt,$ (5.27) $\displaystyle I_{2}$
$\displaystyle:=\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\left\|\int_{0}^{s}\int_{\mathbb{R}^{d}}\left(q_{\alpha,1}(t-r,y)-q_{\alpha,1}(s-r,y)\right)f(r,\cdot-y)dydr\right\|_{H_{p}^{2\theta/\alpha}}^{p}}{|t-s|^{1+\alpha\mu
p}}dsdt.$

Then, apply (5.26) and (5.27) to (5.25),

$\displaystyle\left\|T_{t}^{2}f-T_{s}^{2}f\right\|_{H^{2\theta/\alpha}_{p}}^{p}$
$\displaystyle\leq|t-s|^{\alpha\mu p-1}(I_{1}+I_{2}).$ (5.28)

To deal with $I_{1}$, we employ Minkowski’s inequality, the change of
variable, and Lemma 2.10 (v). Then,

$\displaystyle I_{1}$ $\displaystyle\leq\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|\left((1-\Delta)^{\theta/\alpha}q_{\alpha,1}\right)(r,y)\right|dydr\right)^{p}dt\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds$
(5.29) $\displaystyle\leq I_{11}+I_{12},$

where

$\displaystyle I_{11}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|q_{\alpha,1}(r,y)\right|dydr\right)^{p}dt\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds,$
$\displaystyle I_{12}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\int_{\mathbb{R}^{d}}\left|\left(\Delta^{\theta/\alpha}q_{\alpha,1}\right)(r,y)\right|dydr\right)^{p}dt\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds.$

Because $\mu<1$, Lemma 5.1 (iv) and (iii) yield

$\displaystyle I_{11}$ $\displaystyle=\int_{0}^{T}t^{-1-\alpha\mu p+\alpha
p}dt\left(\int_{\mathbb{R}^{d}}\left|q_{\alpha,1}(1,y)\right|dy\right)^{p}\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds$
(5.30) $\displaystyle\leq
N\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds.$

Similarly, since $\mu<1-\theta/\alpha$,

$\displaystyle I_{12}$ $\displaystyle=\int_{0}^{T}t^{-1-\alpha\mu p+\alpha
p-\theta
p}dt\left(\int_{\mathbb{R}^{d}}\left|\left(\Delta^{\theta/\alpha}q_{\alpha,1}\right)(1,y)\right|dy\right)^{p}\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds$
(5.31) $\displaystyle\leq
N\int_{0}^{T}\left\|f(s,\cdot)\right\|_{L_{p}}^{p}ds.$

Thus, by applying (5.30) and (5.31) to (5.29), we have

$I_{1}\leq N\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}}^{p}ds.$ (5.32)

Next, we address $I_{2}$. Similar to the case for $I_{1}$, we have

$\displaystyle I_{2}$
$\displaystyle\leq\int_{0}^{T}\frac{\int_{0}^{T-t}\left(\int_{0}^{s}\int_{\mathbb{R}^{d}}\left|(1-\Delta)^{\theta/\alpha}\left(q_{\alpha,1}(t+r,y)-q_{\alpha,1}(r,y)\right)\right|dy\|f(s-r,\cdot)\|_{L_{p}}dr\right)^{p}ds}{t^{1+\alpha\mu
p}}dt$ (5.33) $\displaystyle\leq I_{21}+I_{22},$

where

$\displaystyle I_{21}$
$\displaystyle:=\int_{0}^{T}\frac{\int_{0}^{T-t}\left(\int_{0}^{s}\int_{\mathbb{R}^{d}}\left|q_{\alpha,1}(t+r,y)-q_{\alpha,1}(r,y)\right|dy\|f(s-r,\cdot)\|_{L_{p}}dr\right)^{p}ds}{t^{1+\alpha\mu
p}}dt,$ $\displaystyle I_{22}$
$\displaystyle:=\int_{0}^{T}\frac{\int_{0}^{T-t}\left(\int_{0}^{s}\int_{\mathbb{R}^{d}}\left|\Delta^{\theta/\alpha}\left(q_{\alpha,1}(t+r,y)-q_{\alpha,1}(r,y)\right)\right|dy\|f(s-r,\cdot)\|_{L_{p}}dr\right)^{p}ds}{t^{1+\alpha\mu
p}}dt.$

Since $\mu<1$, by Minkowski’s inequality and the fundamental theorem of
calculus, we have

$\displaystyle I_{21}$ $\displaystyle\leq\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\int_{\mathbb{R}^{d}}\left|q_{\alpha,1}(t+r,y)-q_{\alpha,1}(r,y)\right|dydr\right)^{p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{p}ds$
(5.34) $\displaystyle\leq\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\int_{r}^{t+r}\int_{\mathbb{R}^{d}}\left|q_{\alpha,2}(s,y)\right|dydsdr\right)^{p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{p}ds$
$\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}r^{\alpha-1}-(t+r)^{\alpha-1}dr\right)^{p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{p}ds$
$\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu p+\alpha
p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{p}ds$
$\displaystyle\leq N\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}(\mathbb{R}^{d})}^{p}ds.$

Additionally, since $\mu<1-\theta/\alpha$,

$\displaystyle I_{22}$ $\displaystyle\leq\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\int_{\mathbb{R}^{d}}\int_{r}^{t+r}\left|(\Delta^{\theta/\alpha}q_{\alpha,2}(s,y)\right|dsdydr\right)^{p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}}^{p}ds$
(5.35) $\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu p+\alpha p-\theta
p}dt\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}}^{p}ds$ $\displaystyle\leq
N\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}}^{p}ds.$

Therefore, by employing (5.34) and (5.35) to (5.33), we have

$I_{2}\leq N\int_{0}^{T}\|f(s,\cdot)\|_{L_{p}}^{p}ds,$ (5.36)

and thus by combining (5.32) and (5.36) to (5.28), we have (5.19).

To obtain (5.20), employ (5.15) instead of (5.14) and repeat the proof word
for word.

Proof of (iii). By (5.14), we have

$\displaystyle\mathbb{E}\left\|T_{\rho}^{3}g-T_{\gamma}^{3}g\right\|_{H^{2\theta/\alpha}_{p}(l_{2})}^{p}$
$\displaystyle\leq N|\rho-\gamma|^{\alpha\mu
p-1}\int_{0}^{T}\int_{0}^{T}1_{t>s}\frac{\mathbb{E}\left\|T_{t}^{3}g-T_{s}^{3}g\right\|_{H^{2\theta/\alpha}_{p}(l_{2})}^{p}}{|t-s|^{1+\alpha\mu
p}}dsdt.$

Notice that the Burkholder-Davis-Gundy and Minkowski’s inequalities imply that

$\displaystyle\mathbb{E}\left\|T^{3}_{t}g-T^{3}_{s}g\right\|_{H_{p}^{2\theta/\alpha}(l_{2})}^{p}$
$\displaystyle\leq
N\int_{\mathbb{R}^{d}}\mathbb{E}\left(\sum_{k}\left|\int_{s}^{t}\int_{\mathbb{R}^{d}}(\left(1-\Delta\right)^{\theta/\alpha}q_{\alpha,\beta})(t-r,y)g^{k}(r,x-y)dydw_{r}^{k}\right|^{2}\right)^{p/2}$
$\displaystyle\quad+\left(\sum_{k}\left|\int_{0}^{s}\int_{\mathbb{R}^{d}}\left[\left(1-\Delta\right)^{\theta/\alpha}\left(q_{\alpha,\beta}(t-r,y)-q_{\alpha,\beta}(s-r,y)\right)\right]g^{k}(r,x-y)dydw_{r}^{k}\right|^{2}\right)^{p/2}dx$
$\displaystyle\leq
N\int_{\mathbb{R}^{d}}\mathbb{E}\left(\int_{0}^{t-s}\left(\int_{\mathbb{R}^{d}}\left|(\left(1-\Delta\right)^{\theta/\alpha}q_{\alpha,\beta})(t-s-r,y)\right||g(s+r,x-y)|_{l_{2}}dy\right)^{2}dr\right)^{p/2}$
$\displaystyle\quad+\left(\int_{0}^{s}\left(\int_{\mathbb{R}^{d}}\left|\left(1-\Delta\right)^{\theta/\alpha}\left(q_{\alpha,\beta}(t-r,y)-q_{\alpha,\beta}(s-r,y)\right)\right||g(r,x-y)|_{l_{2}}dy\right)^{2}dr\right)^{p/2}dx$
$\displaystyle\leq
N\mathbb{E}\left(\int_{0}^{t-s}\left(\int_{\mathbb{R}^{d}}\left|(\left(1-\Delta\right)^{\theta/\alpha}q_{\alpha,\beta})(t-s-r,y)\right|dy\right)^{2}\|g(s+r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}$
$\displaystyle\quad+N\mathbb{E}\left(\int_{0}^{s}\left(\int_{\mathbb{R}^{d}}\left|\left(1-\Delta\right)^{\theta/\alpha}\left(q_{\alpha,\beta}(t-r,y)-q_{\alpha,\beta}(s-r,y)\right)\right|dy\right)^{2}\|g(r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}.$

Then, set

$\displaystyle I_{3}:=$
$\displaystyle\int_{0}^{T}\int_{0}^{T}1_{t>s}|t-s|^{-1-\alpha\mu
p}\mathbb{E}\left(\int_{0}^{t-s}A(t,s,r)\|g(s+r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}dsdt,$
$\displaystyle I_{4}:=$
$\displaystyle\int_{0}^{T}\int_{0}^{T}1_{t>s}|t-s|^{-1-\alpha\mu
p}\mathbb{E}\left(\int_{0}^{s}B(t,s,r)\|g(r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}dsdt,$

where

$\displaystyle A(t,s,r)$
$\displaystyle=\left(\int_{\mathbb{R}^{d}}|\left(1-\Delta\right)^{\theta/\alpha}q_{\alpha,\beta}(t-s-r,y)|dy\right)^{2},$
$\displaystyle B(t,s,r)$
$\displaystyle=\left(\int_{\mathbb{R}^{d}}\left|\left(1-\Delta\right)^{\theta/\alpha}\left(q_{\alpha,\beta}(t-r,y)-q_{\alpha,\beta}(s-r,y)\right)\right|dy\right)^{2}.$

Note that Minkowski’s inequality and Lemma 2.10 (v) imply that

$\displaystyle I_{3}$ $\displaystyle\leq\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\left(\int_{\mathbb{R}^{d}}\left|(\left(1-\Delta\right)^{\theta/\alpha}q_{\alpha,\beta})(t-r,y)\right|dy\right)^{2}dr\right)^{p/2}dt\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}$
(5.37) $\displaystyle\leq I_{31}+I_{32},$

where

$\displaystyle I_{31}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\left(\int_{\mathbb{R}^{d}}|q_{\alpha,\beta}(r,y)|dy\right)^{2}dr\right)^{p/2}dt\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p},$
$\displaystyle I_{32}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\left(\int_{\mathbb{R}^{d}}\left|\left(\Delta^{\theta/\alpha}q_{\alpha,\beta}\right)(r,y)\right|dy\right)^{2}dr\right)^{p/2}dt\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}.$

Since $\frac{1}{\alpha}\left(\alpha-\beta+\frac{1}{2}\right)>\mu$, by Lemma
5.1 (iii), we have

$\displaystyle I_{31}$ $\displaystyle=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}\left(\int_{\mathbb{R}^{d}}|q_{\alpha,\beta}(r,y)|dy\right)^{2}dr\right)^{p/2}dt\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}$
(5.38) $\displaystyle=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{t}r^{2(\alpha-\beta)}dr\right)^{p/2}dt\left(\int_{\mathbb{R}^{d}}|q_{\alpha,\beta}(1,y)|dy\right)^{p}\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}$
$\displaystyle=\int_{0}^{T}t^{-1+(\alpha-\beta+1/2-\alpha\mu)p}dt\left(\int_{\mathbb{R}^{d}}|q_{\alpha,\beta}(1,y)|dy\right)^{p}\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}$
$\displaystyle\leq N\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}.$

Similarly, as
$\frac{1}{\alpha}\left(\alpha-\beta-\theta+\frac{1}{2}\right)>\mu$,

$\displaystyle I_{32}$
$\displaystyle\leq\int_{0}^{T}t^{-1+(-\alpha\mu+\alpha-\beta-\theta+1/2)p}dt\left(\int_{\mathbb{R}^{d}}|(\Delta^{\theta/\alpha}q_{\alpha,\beta})(1,y)|dy\right)^{p}\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}$
(5.39) $\displaystyle\leq N\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}.$

Therefore, by employing (5.38) and (5.39) to (5.37), we have

$I_{3}\leq N\|g\|_{\mathbb{L}_{p}(T,l_{2})}^{p}.$ (5.40)

In the case of $I_{4}$, by Minkowski’s inequality and Lemma 2.10 (v), we have

$\displaystyle I_{4}$ $\displaystyle\leq I_{41}+I_{42}.$

Further,

$\displaystyle I_{41}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\mathbb{E}\int_{0}^{T-t}\left(\int_{0}^{s}\left(\int_{\mathbb{R}^{d}}|C(t,r,y)|dy\right)^{2}\|g(s-r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}dsdt,$
$\displaystyle I_{42}$ $\displaystyle:=\int_{0}^{T}t^{-1-\alpha\mu
p}\mathbb{E}\int_{0}^{T-t}\left(\int_{0}^{s}\left(\int_{\mathbb{R}^{d}}|\Delta^{\theta/\alpha}C(t,r,y)|dy\right)^{2}\|g(s-r,\cdot)\|_{L_{p}(l_{2})}^{2}dr\right)^{p/2}dsdt,$

where

$C(t,r,y)=q_{\alpha,\beta}(t+r,y)-q_{\alpha,\beta}(r,y).$

We address $I_{41}$. By Minkowski’s inequality, the fundamental theorem of
calculus, and Lemma 5.1, we have

$\displaystyle I_{41}$ $\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{\mathbb{R}^{d}}|q_{\alpha,\beta}(t+r,y)-q_{\alpha,\beta}(r,y)|dy\right)^{2}dr\right)^{p/2}dt\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$
(5.41) $\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{r}^{t+r}\int_{\mathbb{R}^{d}}|q_{\alpha,\beta+1}(s,y)|dyds\right)^{2}dr\right)^{p/2}dt\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$
$\displaystyle\leq NH_{1}\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})},$

where

$H_{1}:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{r}^{t+r}s^{\alpha-\beta-1}ds\right)^{2}dr\right)^{p/2}dt$
(5.42)

and $N=N(\alpha,\beta,d,p,T)$. Next, we claim that

$H_{1}<\infty$ (5.43)

To demonstrate (5.43), set

$\displaystyle\chi(t):=\int_{0}^{T-t}\left(\int_{r}^{t+r}s^{\alpha-\beta-1}ds\right)^{2}dr.$

Furthermore,

$H_{1}=\int_{0}^{T}t^{-1-\alpha\mu p}(\chi(t))^{p/2}dt$ (5.44)

Depending on the range of $\alpha-\beta$, we consider the following five
cases.

(Case 1.) $-1/2<\alpha-\beta<0$

For $t\in(0,T)$, we have

$\displaystyle 0\leq\chi(t)\leq
N\int_{0}^{T-t}r^{2(\alpha-\beta)}-(t+r)^{2(\alpha-\beta)}dr\leq
Nt^{2(\alpha-\beta)+1},$ (5.45)

where $N=N(\alpha,\beta)$. Then, since
$\frac{1}{\alpha}(\alpha-\beta+1/2)>\mu$, by combining (5.44) and (5.45),

$\displaystyle H_{1}\leq
N(\alpha,\beta,p)\int_{0}^{T}t^{-1+p(-\alpha\mu+\alpha-\beta+1/2)}dt<\infty.$

(Case 2.) $\alpha-\beta=0$

Notice that

$\displaystyle\chi(t)=\int_{0}^{T-t}\left(\int_{r}^{t+r}s^{-1}ds\right)^{2}dr=\int_{0}^{T-t}\left(\log\left(\frac{t+r}{r}\right)\right)^{2}dr.$

Obviously, $\chi(0)=0$. Note that

$\displaystyle\chi^{\prime}(t)\leq
2\int_{0}^{\infty}\frac{1}{1+r}\log\left(\frac{1+r}{r}\right)dr=2\int_{0}^{\infty}x(e^{x}-1)^{-1}dx=\pi^{2}/3$

on $t\in(0,T/2)$. Thus,

$\displaystyle
0\leq\chi(t)=\chi(t)-\chi(0)=\int_{0}^{t}\chi^{\prime}(s)ds\leq\frac{\pi^{2}}{3}t$
(5.46)

on $t\in(0,T/2)$. Additionally, $\chi(t)\leq N$ on $t\in(T/2,T)$. Therefore,

$H_{1}\leq N\int_{0}^{T/2}t^{-1+(-\alpha\mu+1/2)p}dt+N<\infty,$

where $N=N(\alpha,\beta,p)$.

(Case 3.) $0<\alpha-\beta<1/2$

Observe that $\chi$ is twice continuously differentiable, and

$\displaystyle\chi^{\prime}(t)$
$\displaystyle=(\alpha-\beta)^{-2}\left[-T^{2(\alpha-\beta)}-(T-t)^{2(\alpha-\beta)}+2T^{\alpha-\beta}(T-t)^{\alpha-\beta}\right]$
$\displaystyle\quad+2(\alpha-\beta)^{-1}\int_{0}^{T-t}(t+r)^{2(\alpha-\beta)-1}-(t+r)^{\alpha-\beta-1}r^{\alpha-\beta}dr,$

and

$\displaystyle\chi^{\prime\prime}(t)$
$\displaystyle=2(\alpha-\beta)^{-1}(T-t)^{2(\alpha-\beta)-1}-2(\alpha-\beta)^{-1}T^{\alpha-\beta}(T-t)^{\alpha-\beta-1}$
(5.47)
$\displaystyle\quad+2(\alpha-\beta)^{-1}T^{\alpha-\beta-1}(T-t)^{\alpha-\beta}-2(\alpha-\beta)^{-1}t^{2(\alpha-\beta)-1}$
$\displaystyle\quad-2(\alpha-\beta)^{-1}(\alpha-\beta-1)\int_{0}^{T-t}(t+r)^{\alpha-\beta-2}r^{\alpha-\beta}dr.$

In addition, $\chi(0)=\chi^{\prime}(0)=0$. Then, by using the fundamental
theorem of calculus and $\alpha-\beta\in(0,1/2)$, we obtain

$\displaystyle\chi(t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\chi^{\prime\prime}(\rho)d\rho ds$
(5.48) $\displaystyle\leq
2(\alpha-\beta)^{-1}\int_{0}^{t}\int_{0}^{s}(T-\rho)^{2(\alpha-\beta)-1}d\rho
ds$
$\displaystyle\quad-2(\alpha-\beta)^{-1}(\alpha-\beta-1)\int_{0}^{t}\int_{0}^{s}\int_{0}^{T-\rho}(\rho+r)^{\alpha-\beta-2}r^{\alpha-\beta}drd\rho
ds$ $\displaystyle\leq
N\int_{0}^{t}\left(T^{2(\alpha-\beta)}-(T-s)^{2(\alpha-\beta)}\right)ds+N\int_{0}^{t}\int_{0}^{s}\int_{0}^{T-\rho}(\rho+r)^{2(\alpha-\beta)-2}drd\rho
ds$ $\displaystyle\leq
N\int_{0}^{t}s^{2(\alpha-\beta)}ds+N\int_{0}^{t}\int_{0}^{s}T^{2(\alpha-\beta)-1}drd\rho
ds$ $\displaystyle\leq Nt^{2(\alpha-\beta)+1},$

where $N=N(\alpha,\beta,T)$. Thus,

$H_{1}\leq
N(\alpha,\beta,p,T)\int_{0}^{T}t^{-1+p(-\alpha\mu+\alpha-\beta+1/2)}dt<\infty.$

(Case 4.) $\alpha-\beta=1/2$

Because $\chi(0)=\chi^{\prime}(0)=0$, by the fundamental theorem of calculus,
we have

$\displaystyle\chi(t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\chi^{\prime\prime}(\rho)d\rho ds$
$\displaystyle\leq
N\int_{0}^{t}\int_{0}^{s}(T-\rho)^{1/2}+N\int_{0}^{T-\rho}(\rho+r)^{-3/2}r^{1/2}drd\rho
ds$ $\displaystyle\leq Nt^{2}(1+|\log t|),$

where $N=N(T)$. Therefore,

$H_{1}\leq N(p,T)\int_{0}^{T}t^{-1+p(-\alpha\mu+1)}(1+|\log
t|)^{p/2}dt<\infty.$

(Case 5.) $\alpha-\beta>1/2$

Similar to before, $\chi(0)=\chi^{\prime}(0)=0$. Additionally, as in (5.48),
we have

$\displaystyle\chi(t)$
$\displaystyle=\int_{0}^{t}\int_{0}^{s}\chi^{\prime\prime}(r)drds$
$\displaystyle\leq
N\int_{0}^{t}\left(T^{2(\alpha-\beta)}-(T-s)^{2(\alpha-\beta)}\right)ds+N\int_{0}^{t}\int_{0}^{s}\int_{0}^{T-\rho}(\rho+r)^{2(\alpha-\beta)-2}drd\rho
ds$ $\displaystyle\leq N\int_{0}^{t}sds+N\int_{0}^{t}\int_{0}^{s}d\rho ds$
$\displaystyle\leq Nt^{2},$

where $N=N(\alpha,\beta,T)$. Therefore,

$H_{1}\leq N(\alpha,\beta,p,T)\int_{0}^{T}t^{-1+p(-\alpha\mu+1)}dt<\infty.$

Thus, we have (5.43). Then, by combining (5.43) and (5.41), we have
$I_{41}\leq N\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$.

Next, we deal with $I_{42}$. Minkowski’s inequality, the fundamental theorem
of calculus, and Lemma 5.1 yield

$\displaystyle I_{42}$ $\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{\mathbb{R}^{d}}|\Delta^{\theta/\alpha}q_{\alpha,\beta}(t+r,y)-\Delta^{\theta/\alpha}q_{\alpha,\beta}(r,y)|dy\right)^{2}dr\right)^{p/2}dt\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$
$\displaystyle\leq N\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{r}^{t+r}\int_{\mathbb{R}^{d}}|\partial_{s}\Delta^{\theta/\alpha}q_{\alpha,\beta}(s,y)|dyds\right)^{2}dr\right)^{p/2}dt\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$
$\displaystyle\leq NH_{2}\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})},$

where $N=N(\alpha,\beta,d,p,T)$ and $H_{2}:=\int_{0}^{T}t^{-1-\alpha\mu
p}\left(\int_{0}^{T-t}\left(\int_{r}^{t+r}s^{\alpha-\beta-\theta-1}ds\right)^{2}dr\right)^{p/2}dt$.
Similar to the case of $H_{1}$ (see (5.42) and (5.43)), we demonstrate that

$H_{2}<\infty$

by considering five cases for $\alpha-\beta-\theta$ instead of $\alpha-\beta$.
Then, we have $I_{42}\leq N\|g\|^{p}_{\mathbb{L}_{p}(T,l_{2})}$. The lemma is
proved.

∎

###### Proof of Theorem 2.16.

It suffices to show that the assertion holds for $\tau=T$. Indeed, assume the
results holds for $\mathcal{H}_{p}^{\gamma}(T)$, and let $\tau\leq T$ be a
bounded stopping time and $u\in\mathcal{H}_{p}^{\gamma}(\tau)$. Then, by
Definition 2.4 for $\varepsilon>0$, there exists
$(f,g)\in\mathbb{H}_{p}^{\gamma-2}(\tau)\times\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})$
such that

$\partial_{t}^{\alpha}u=f+\partial_{t}^{\beta}\int_{0}^{t}g^{k}dw_{t}^{k}\,;\quad
u(0,\cdot)=u_{0}(\cdot)$

and

$\|u\|_{\mathbb{H}_{p}^{\gamma}(\tau)}+\|u_{0}\|_{U_{p}^{\gamma}}+\|f\|_{\mathbb{H}_{p}^{\gamma-2}(\tau)}+\|g\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}\leq\|u\|_{\mathcal{H}_{p}^{\gamma}(\tau)}+\varepsilon.$

Set $\bar{f}:=(f-\Delta u)1_{t\leq\tau}$ and $\bar{g}:=g1_{t\leq\tau}$; thus,
$u$ satisfies

$\partial_{t}^{\alpha}u=\Delta
u+\bar{f}+\partial_{t}^{\beta}\int_{0}^{t}\bar{g}^{k}dw_{t}^{k},\quad
0<t\leq\tau\,;\quad u(0,\cdot)=u_{0}(\cdot).$ (5.49)

In contrast, by [25, Theorem 2.18], there exists
$v\in\mathcal{H}_{p}^{\gamma}(T)$ such that $v$ satisfies

$\partial_{t}^{\alpha}v=\Delta
v+\bar{f}+\partial_{t}^{\beta}\int_{0}^{t}\bar{g}^{k}dw_{t}^{k},\quad
0<t\leq\tau\,;\quad v(0,\cdot)=u_{0}(\cdot).$

Additionally,

$\displaystyle\|v\|_{\mathcal{H}_{p}^{\gamma}(T)}$ $\displaystyle\leq
N\left(\left\|\bar{f}\right\|_{\mathbb{H}_{p}^{\gamma-2}(T)}+\left\|\bar{g}\right\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(T,l_{2})}+\left\|u_{0}\right\|_{U_{p}^{\gamma}}\right)$
(5.50) $\displaystyle\leq
N\left(\|u\|_{\mathbb{H}_{p}^{\gamma}(\tau)}+\|u_{0}\|_{U_{p}^{\gamma}}+\|f\|_{\mathbb{H}_{p}^{\gamma-2}(\tau)}+\|g\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}\right)$
$\displaystyle\leq N\|u\|_{\mathcal{H}_{p}^{\gamma}(\tau)}+N\varepsilon,$

where $N$ is independent of $\varepsilon$. Because
$v\in\mathcal{H}_{p}^{\gamma}(T)$, $v\in
C^{\alpha\mu-1/p}\left([0,T];H_{p}^{\gamma-2\nu}\right)$ almost surely and

$\mathbb{E}\|v\|_{C^{\alpha\mu-1/p}([0,T];H_{p}^{\gamma-2\nu})}^{p}\leq
N\|v\|_{\mathcal{H}_{p}^{\gamma}(T)}^{p}$ (5.51)

by the hypothesis. Therefore, due to $\tau\leq T$, (5.50) and (5.51) yield

$\mathbb{E}\|u\|_{C^{\alpha\mu-1/p}([0,\tau];H_{p}^{\gamma-2\nu})}^{p}\leq\mathbb{E}\|v\|_{C^{\alpha\mu-1/p}([0,T];H_{p}^{\gamma-2\nu})}^{p}\leq
N\|v\|_{\mathcal{H}_{p}^{\gamma}(T)}^{p}\leq
N\|u\|_{\mathcal{H}_{p}^{\gamma}(\tau)}^{p}+N\varepsilon,$ (5.52)

where $N$ is independent of $\varepsilon$. Note that $\bar{u}:=u-v$ satisfies

$\partial_{t}^{\alpha}\bar{u}=\Delta\bar{u},\quad
0<t\leq\tau;\quad\bar{u}(0,\cdot)=0.$

Then, by the deterministic version of [25, Theorem 2.18], we have
$u(t,\cdot)=v(t,\cdot)$ for almost every $(\omega,t)\in\Omega\times[0,\tau]$.
Thus, $v$ is an $H_{p}^{\gamma-2\nu}$-valued continuous version of $u$.
Additionally, from (5.52)

$\mathbb{E}\|u\|_{C^{\alpha\mu-1/p}([0,\tau];H_{p}^{\gamma-2\nu})}^{p}\leq
N\|u\|_{\mathcal{H}_{p}^{\gamma}(\tau)}^{p}+N\varepsilon.$

In addition, as $\varepsilon>0$ is arbitrary and $N$ is independent of
$\varepsilon$, we have (2.8). Additionally, notice that (5.50) allows us to
prove the assertions with
$\left\|f\right\|_{\mathbb{H}_{p}^{\gamma-2}(T)}+\left\|g\right\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(T,l_{2})}+\left\|u_{0}\right\|_{U_{p}^{\gamma}}$
instead of $\|u\|_{\mathcal{H}_{p}^{\gamma}(T)}$.

Due to Lemma 2.10 (vi), we only consider the case $\gamma=2\nu$, where
$\nu\in(0,1)$ satisfies (2.7). Moreover, by using the approximation to the
identity, we may assume that $u_{0}$ is infinitely differentiable and
compactly supported in $x$. Furthermore, we also assume that $f$ and
$g=(g^{1},g^{2},\dots)$ denotes the function of the form satisfying (5.2) (see
[26, Theorem 3.10]). Additionally, it should be remarked that $u$ can be
written as

$u(t,x)=T^{1}_{t}u_{0}+T^{2}_{t}f+T^{3}_{t}g$

since $u$ satisfies

$\partial_{t}^{\alpha}u=\Delta
u+f+\partial_{t}^{\beta}\int_{0}^{t}g^{k}dw_{t}^{k}\,;\quad
u(0,\cdot)=u_{0}(\cdot)$

for almost every $(\omega,t)\in\text{\,\raise
0.86108pt\hbox{${\scriptstyle|}$}\kern-3.39996pt$($}0,\tau\text{$]$\kern-1.49994pt$]$}$
(see [22, 23, 24]).

Now, observe that $u\in C^{\alpha\mu-1/p}([0,\tau];H_{p}^{\gamma-2\nu})$
almost surely. Indeed, by Lemma 5.5, we have

$\displaystyle\mathbb{E}\left\|T^{1}_{t}f-T^{1}_{s}f\right\|_{H^{\gamma-2\nu}_{p}}^{p}$
$\displaystyle\leq N|t-s|^{\alpha\mu
p-1}\left\|\phi\right\|_{H^{\gamma}_{p}}^{p},$
$\displaystyle\mathbb{E}\left\|T^{2}_{t}f-T^{2}_{s}f\right\|_{H^{\gamma-2\nu}_{p}}^{p}$
$\displaystyle\leq N|t-s|^{\alpha\mu
p-1}\left\|f\right\|_{\mathbb{H}^{\gamma-2}_{p}(\tau)}^{p}dr,$
$\displaystyle\mathbb{E}\left\|T_{t\wedge\tau}^{3}g-T_{s\wedge\tau}^{3}g\right\|_{H^{\gamma-2\nu}_{p}(l_{2})}^{p}$
$\displaystyle\leq N|t-s|^{\alpha\mu
p-1}\|g\|_{\mathbb{H}^{\gamma-2+c_{0}}_{p}(\tau,l_{2})}^{p}.$

Thus, Kolmogorov continuity theorem (e.g., [28, Theorem 1.4.8]) and the above
inequalities again, we have the continuity of $u$.

Next we prove (2.8). Choose $\mu$ and $\nu$ satisfy (2.7). Observe that Lemma
2.10 (vi) and (5.17) with $\theta=\alpha-\alpha\nu$ imply that

$\displaystyle\mathbb{E}\sup_{0\leq s<t\leq
T}\frac{\|T_{t}^{1}u_{0}-T_{s}^{1}u_{0}\|_{H^{\gamma-2\nu}_{p}}^{p}}{|t-s|^{\alpha\mu
p-1}}\leq N\mathbb{E}\left\|u_{0}\right\|_{H^{\gamma}_{p}}^{p}.$

As (2.7) is assumed, Lemma 2.10 (vi) and (5.20) with $\theta=\alpha-\alpha\nu$
yield the following:

$\displaystyle\mathbb{E}\sup_{0\leq
s<t\leq\tau}\frac{\|T^{2}_{t}f-T^{2}_{2}f\|_{H^{\gamma-2\nu}_{p}}^{p}}{|t-s|^{\alpha\mu
p-1}}$ $\displaystyle\leq N\|f\|_{\mathbb{H}_{p}^{\gamma-2}(\tau)}^{p}.$

Furthermore, by (2.7), Lemma 2.10 (vi), and (5.22) with
$\theta=\alpha(1-c_{0}/2)-\alpha\nu$, we have

$\displaystyle\mathbb{E}\sup_{0\leq s<t\leq
T}\frac{\left\|T_{t}^{3}g-T_{s}^{3}g\right\|_{H^{\gamma-2\nu}_{p}(l_{2})}^{p}}{|t-s|^{\alpha\mu
p-1}}$ $\displaystyle\leq
N\|g\|_{\mathbb{H}_{p}^{\gamma-2+c_{0}}(\tau,l_{2})}^{p}.$

The theorem is proved.

∎

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