# Global existence and blow-up for a stochastic transport equation with non-
local velocity

Diego Alonso-Orán  Departamento de Análisis Matemático, Universidad de La
Laguna, C/ Astrofísico Francisco Sánchez s/n, 38271, Spain
<EMAIL_ADDRESS>,  Yingting Miao  Department of Mathematics, South China
University of Technology, Guangzhou, Guangdong 510640, PR China
<EMAIL_ADDRESS>and  Hao Tang  Department of Mathematics,
University of Oslo, P.O. Box 1053
Blindern, N-0316 Oslo, Norway<EMAIL_ADDRESS>
###### Abstract.

In this paper we investigate a non-linear and non-local one dimensional
transport equation under random perturbations on the real line. We first
establish a local-in-time theory, i.e., existence, uniqueness and blow-up
criterion for pathwise solutions in Sobolev spaces $H^{s}$ with $s>3$.
Thereafter, we give a complete picture of the long time behavior of the
solutions based on the type of noise we consider. On one hand, we identify a
family of noises such that blow-up can be prevented with probability $1$,
guaranteeing the existence and uniqueness of global solutions almost surely.
On the other hand, in the particular linear noise case, we show that
singularities occur in finite time with positive probability, and we derive
lower bounds of these probabilities. To conclude, we introduce the notion of
stability of exiting times and show that one cannot improve the stability of
the exiting time and simultaneously improve the continuity of the dependence
on initial data.

###### Key words and phrases:

Stochastic evolution equations; Pathwise solution; Blow-up criterion; Noise
prevents blow-up; Weak instability.

###### 2020 Mathematics Subject Classification:

Primary: 60H15, 35Q51; Secondary: 35A01, 35B44.

D. Alonso-Orán is supported by the Spanish MINECO through Juan de la Cierva
fellowship FJC2020-046032-I. The major part of this work was carried out when
D. Alonso-Orán and H. Tang where supported by the Alexander von Humboldt
Foundation.

###### Contents

  1. 1 Introduction and main results
    1. 1.1 Noise versus finite time blow-up
    2. 1.2 Noise versus dependence on initial data
    3. 1.3 Notations, definitions and hypotheses
    4. 1.4 Main results and remarks
  2. 2 Preliminaries
  3. 3 Proof of Theorem 1.1
    1. 3.1 Approximation scheme and uniform estimates
    2. 3.2 Convergence of approximate solutions
    3. 3.3 Global pathwise solution to the cut-off problem
    4. 3.4 Concluding the proof of Theorem 1.1
  4. 4 Proof of Theorem 1.2
  5. 5 Proof of Theorem 1.3
    1. 5.1 An identity for the fractional Laplacian $\Lambda$
    2. 5.2 Proof of Theorem 1.3
  6. 6 Proof of Theorem 1.4
    1. 6.1 Approximate solutions and actual solutions
    2. 6.2 Estimates on the errors
      1. 6.2.1 Estimating the error $\mathcal{E}$
      2. 6.2.2 Estimating $u_{m,n}-u^{m,n}$
    3. 6.3 Concluding the proof of Theorem 1.4

##  1\. Introduction and main results

Consider the following one dimensional non-local transport equation

$u_{t}+(\mathcal{H}u)u_{x}=0,$ (1.1)

where $\mathcal{H}$ denotes the Hilbert transform. This equation first appears
in the literature due to its analogy with the Birkhoff-Rott equations
describing the evolution of vortex sheets with surface tension [4, 45].
Moreover, (1.1) can be also viewed as a toy-model of the two-dimensional
surface quasi-geostrophic equation (SQG) which describes the evolution of the
potential temperature in a rapidly rotating stratified fluid with uniform
potential vorticity [25, 47]. A striking result showing the finite time blow-
up of classical solutions to (1.1) for a generic class of smooth initial data
was first obtained by Córdoba, Córdoba and Fontelos in [11] by means of
complex analysis techniques. After that breakthrough, equation (1.1) is known
as the CCF equation. Subsequent works have shown finite blow-up avoiding
complex analysis approach (cf. [38, 53]). In particular, in the later
Silvestre and Vicol provided four elegant and simple real analysis proofs of
the blow-up phenomena.

In this paper, we are interested in stochastic variants of the CCF equation
(1.1). Indeed, the introduction of stochasticity into ideal fluid dynamics has
received special attention over the past two decades. The inclusion of
stochastic noise can be a way of representing model uncertainty and
turbulence. For example, in weather forecasting, phenomena as cloud formation
is to this day poorly understood and the inclusion of stochastic noise has
become an essential tool for gaining better understanding about it. Since the
pioneering work of Holm in [31], where a variational approach for introducing
noise in equations in a fashion that respects the geometry of the system is
developed, the literature regarding the analysis of non-linear stochastic
partial differential equations with transport type noise has increase
substantially (cf. [1, 2, 9, 10, 20] and the references therein). To the best
of the authors knowledge, there are very few results regarding the CCF model
under random perturbations. Only recently, by applying an abstract framework
for singular stochastic partial differential equations (SPDEs) derived by two
of the authors, cf. [3], the local existence, uniqueness and blow-up criterion
of pathwise solutions to (1.1) with transport noise has been addressed in the
periodic setting, i.e for $x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$.

To extend the theory developed in [3] to the real line case, i.e.,
$x\in\mathbb{R}$, and to study the noise effect, in this paper we will
consider the following stochastic CCF model

$\left\\{\begin{aligned} &{\rm d}u+(\mathcal{H}u)u_{x}{\rm d}t=h(t,u){\rm
d}\mathcal{W},\quad x\in\mathbb{R},\ t>0,\\\
&u(\omega,0,x)=u_{0}(\omega,x),\quad x\in\mathbb{R},\end{aligned}\right.$
(1.2)

where $\mathcal{W}$ is a cylindrical Wiener process and $h$ is a non-linear
function. In this work, under some natural assumptions collected in Assumption
(A), we obtain the local existence, uniqueness and a blow-up criterion of
pathwise solutions to (1.2). The detailed result is stated in Theorem 1.1.
Here we notice that classical probabilistic compactness arguments cannot be
directly applied to deal with the whole space $\mathbb{R}$ and new ideas are
in order, see Remark 1.2 for a more precise explanation.

It is well-known that noise effect is one of the crucial subjects in the study
of SPDEs. Indeed, regularization effects due to noise have been observed for
various models and different settings. For example, it is known that the well-
posedness of linear stochastic transport equation with noise can be
established under weaker hypotheses than its deterministic counterpart and
restore uniqueness of solutions, cf. [15, 18]. Regularization effects caused
by the noise on flux for stochastic scalar conservation laws have been studied
in [22]. Extensions of the previous works for the stochastic transport and
continuity equations to $k$-forms has been recently addressed in [5]. In terms
of numerical simulations, the regularization effects of noise can be found in
[40]. Moreover, for different fluid models with linear multiplicative noise,
we refer to [24, 37, 54] where noise provides a damping effect on the pathwise
behavior of solutions.

Compared to the deterministic counterpart of (1.2), i.e., (1.1), in this
artcicle we focus on the following issues for the problem (1.2) regarding the
noise effect:

  * •

Noise versus finite time blow-up;

  * •

Noise versus dependence on initial data.

###  1.1. Noise versus finite time blow-up

In this direction, we attempt to answer the following two important questions:

  1. (Q-1)

What kind of noise can prevent blow-up?

  2. (Q-2)

If blow-up may occur, what is the corresponding probability?

We remark here that most previous results in the literature on regularization
by noise are restricted to linear equations or linear noises. For instance, we
refer to [15, 18, 16, 37, 46] for linear transport equations, and to [19, 24,
50, 54] for linear noise. Therefore, for non-linear SPDEs, it is very natural
to analyze the validity of the regularization effects by non-linear noise.
Indeed, searching for nonlinear noise such that blow-up can be prevented is
important because it helps us to understand the regularization mechanisms of
noise, and this is the main motivation to study question (Q-1). Actually, even
in the case of non-linear equations with linear noise, the noise effects are
complicated because there are both, examples in positive direction, i.e.,
noises can regularize singularities, and negative direction, i.e., noises
cannot regularize singularities. For example, for the stochastic 2D Euler
equations, coalescence of vortices disappears (see [19]) but noise cannot
prevent the formation of shocks in the Burgers’ equation (see [1, 17]).

For simplicity, we set (Q-1) in the framework where $h(t,u)\,{\rm
d}\mathcal{W}=\alpha(t,u)\,{\rm d}W$, with $W$ a standard 1-D Brownian motion
and $\alpha$ a non-linear function. We then focus on the system

$\left\\{\begin{aligned} &{\rm d}u+(\mathcal{H}u)u_{x}\,{\rm
d}t=\alpha(t,u)\,{\rm d}W,\quad x\in\mathbb{R},\ t>0,\\\
&u(\omega,0,x)=u_{0}(\omega,x),\qquad x\in\mathbb{R}.\end{aligned}\right.$
(1.3)

We will show in Theorem 1.2 that if $\alpha(t,\cdot)$ grows fast enough, then
global existence of pathwise solutions holds true with probability $1$. This
is strongly in contrast with its deterministic counterpart where the breakdown
of classical solutions to (1.1) with generic smooth initial data occurs, cf.
[11]. Hence we justify the idea that fast growing non-linear noise (strong
noise) has regularization effects on the solutions in terms of preventing
singularities.

By Theorem 1.2, we have identified a family of noises that can prevent blow-
up, and this partially answers (Q-1). Next, we will pay our attention to the
case that blow-up may occur. Indeed, as a toy-model for the 2D surface quasi-
geostrophic equation (SQG)(and hence for the 3D incompressible Euler
equation), analyzing the possible blow-up of solutions is is one of the
central questions in the study of non-local transport type equations, cf. [11,
14, 38, 43, 53]. In this paper we are also interested in identifying the
possible formation of singularities in finite time and estimating its
probability, hence given a partial answer to (Q-2). Since Theorem 1.2 shows
that fast enough growing noises can prevent blow-up, it is natural to ponder
that singularities can only occur when the noise is somehow weak. Indeed, in
contrast to the fast growing noise, we will show that in the case of linear
noise (weak noise) given by equation

$\left\\{\begin{aligned} &{\rm d}u+(\mathcal{H}u)u_{x}{\rm d}t=b(t)u\,{\rm
d}W,\quad x\in\mathbb{R},\ t>0,\\\ &u(\omega,0,x)=u_{0}(\omega,x),\quad
x\in\mathbb{R},\end{aligned}\right.$ (1.4)

where $b$ is some continuous function and $W$ is a standard 1-D Brownian
motion, finite time blow-up cannot be prevented and finite time singularities
occur. The precise statement of this result is given in Theorem 1.3.

###  1.2. Noise versus dependence on initial data

Now we turn to the problem of noise effect on the initial-data dependence.
There are very few results concerning the noise effect in the direction of
dependence on initial data. In this work we will partially answer the
following question:

  1. (Q-3)

Whether and how noise can affect the initial-data dependence?

The main motivation to consider question (Q-3) relies on the following
observation. On the one hand, regularization provided by noise may look
related to regularization effects induced by an additional dissipative term (a
Laplacian). On the other hand, if one would add a real Laplacian to the
governing equations, parabolic techniques may be used to improve the
continuity of the initial-data dependence. For example, in the deterministic
incompressible Euler equations, the solution map $u_{0}\mapsto u$ cannot be
better than continuous [29] but for the deterministic incompressible Navier-
Stokes equations with sufficiently large viscosity, it is at least Lipschitz
continuous in sufficiently high Sobolev spaces (see pp. 79–81 in [26]). In the
deterministic setting, similar questions regarding the continuity map on the
initial-data have been widely investigated for various non-linear dispersive
and integrable equations of which we only mention a few related results. Koch
and Tzvetkov [39] proved that the solution map of the Benjamin–Ono equation
cannot be uniform continuous. For Camassa–Holm type equations, we refer to
[27, 28] for the non-uniform dependence on initial data in Sobolev spaces
$H^{s}$ with $s>3/2$. Similar results in Besov spaces $B^{s}_{p,q}$ first
appear in [57, 56], where the critical index $s$ can be also covered. In the
case of the SQG system, we refer the reader to [32]. In the stochastic
setting, the interplay between regularization provided by noise and the
dependence on initial conditions is first studied in [52, 55].

In this article we consider question (Q-3) for (1.2). More precisely, we first
recall the concept of stability of the exiting time as in [52, 55]. Roughly
speaking, this notion refers to the continuous changes of the point in time
with respect to the initial condition, where such point is defined as the time
when the solution leaves a certain range, see Definition 1.2 below. Later on,
in Theorem 1.4, we show that when $h(t,u)$ satisfies certain conditions (see
Assumption (D)), the multiplicative noise cannot improve the stability of the
exiting time, and, at the same time, improve the continuity of the map
$u_{0}\mapsto u$ defined by (1.2).

###  1.3. Notations, definitions and hypotheses

We now introduce some notations. $L^{2}(\mathbb{R})$ is the usual space of
square–integrable functions on $\mathbb{R}$. For $s\in\mathbb{R}$,
$D^{s}=(1-\partial_{xx}^{2})^{s/2}$ is defined by
$\widehat{D^{s}f}(\xi)=(1+\xi^{2})^{s/2}\widehat{f}(\xi)$, where $\widehat{g}$
is the Fourier transform of $g$. The Sobolev space $H^{s}(\mathbb{R})$ is
defined as

$\displaystyle H^{s}(\mathbb{R})\triangleq\left\\{f\in
L^{2}(\mathbb{R}):\|f\|_{H^{s}(\mathbb{R})}^{2}=\int_{\mathbb{R}}(1+\xi^{2})^{s}|\widehat{f}(\xi)|^{2}<+\infty\right\\},$

in which the inner product $(f,g)_{H^{s}}$ is given by

$(f,g)_{H^{s}}\triangleq\int_{\mathbb{R}}(1+\xi^{2})^{s}\widehat{f}(\xi)\cdot\overline{\widehat{g}}(\xi)=(D^{s}f,D^{s}g)_{L^{2}}.$

When the function space refers to $\mathbb{R}$, we will drop $\mathbb{R}$ if
there is no ambiguity. $x\lesssim y$ ($x\gtrsim y$) means that $x\leq cy$
($x\geq cy$) holds for some universal deterministic constant $c$. Such
constant may differ from line to line. For linear operators $A$ and $B$, the
commutator $[A,B]$ is defined by $[A,B]=AB-BA$.

The triplet $(\Omega,\mathcal{F},\mathbb{P})$ denotes a complete probability
space, where $\mathbb{P}$ is a probability measure on $\Omega$ and
$\mathcal{F}$ is a $\sigma$-algebra. $\mathbb{E}X$ is the mathematical
expectation of $X$ with respect to $\mathbb{P}$. Let
$\mathcal{W}(t)=\mathcal{W}(\omega,t),\omega\in\Omega$ be a cylindrical Wiener
process. More precisely, we consider a separable Hilbert space $\mathcal{U}$
and let $\\{e_{k}\\}$ be a complete orthonormal basis of $\mathcal{U}$. Then
we define

$\mathcal{W}\triangleq\sum_{k=1}^{\infty}W_{k}e_{k},$

where $\\{W_{k}\\}_{k\geq 1}$ is a sequence of mutually independent standard
one-dimensional Brownian motions. We call
$\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ a stochastic basis, where $\\{\mathcal{F}_{t}\\}_{t\geq 0}$
is a right-continuous filtration endowed on $(\Omega,\mathcal{F},\mathbb{P})$
such that $\\{\mathcal{F}_{0}\\}$ contains all the $\mathbb{P}$-negligible
subsets.

$\mathcal{L}_{2}(\mathcal{U};\mathcal{X})$ stands for the set of Hilbert-
Schmidt operators from $\mathcal{U}$ to another Hilbert space $\mathcal{X}$.
For a predictable process $Z\in\mathcal{L}_{2}(\mathcal{U};\mathcal{X})$,

$\int_{0}^{t}Z{\rm
d}\mathcal{W}\triangleq\sum_{k=1}^{\infty}\int_{0}^{t}Ze_{k}{\rm d}W_{k}$

is a well-defined $\mathcal{X}$-valued continuous square integrable
martingale, see [12, 48] for more details. In the sequel of the paper, when a
stopping time is defined, we set $\inf\emptyset\triangleq\infty$ by
convention.

We now give the precise notion of a pathwise solution to (1.2).

######  Definition 1.1 (Pathwise solutions).

Let $\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ be a fixed stochastic basis. Let $s>3/2$ and $u_{0}$ be an
$H^{s}$-valued $\mathcal{F}_{0}$-measurable random variable.

  1. (1)

A local pathwise solution to (1.2) is a pair $(u,\tau)$, where $\tau$ is a
stopping time satisfying $\mathbb{P}\\{\tau>0\\}=1$ and
$u:\Omega\times[0,\infty]\rightarrow H^{s}$ is an
$\mathcal{F}_{t}$-predictable $H^{s}$-valued process satisfying

$u(\cdot\wedge\tau)\in C([0,\infty);H^{s})\ \ \mathbb{P}-a.s.,$

and for all $t>0$,

$u(t\wedge\tau)-u(0)+\int_{0}^{t\wedge\tau}(\mathcal{H}u)u_{x}{\rm
d}t^{\prime}=\int_{0}^{t\wedge\tau}h(t^{\prime},u){\rm d}\mathcal{W}\ \
\mathbb{P}-a.s.$

  2. (2)

The local pathwise solutions are said to be pathwise unique, if given any two
pairs of local pathwise solutions $(u_{1},\tau_{1})$ and $(u_{2},\tau_{2})$
with $\mathbb{P}\left\\{u_{1}(0)=u_{2}(0)\right\\}=1,$ we have

$\mathbb{P}\left\\{u_{1}(t,x)=u_{2}(t,x),\ \forall\
(t,x)\in[0,\tau_{1}\wedge\tau_{2}]\times\mathbb{R}\right\\}=1.$

  3. (3)

Additionally, $(u,\tau^{*})$ is called a maximal pathwise solution to (1.2) if
$\tau^{*}>0$ almost surely and if there is an increasing sequence
$\tau_{n}\rightarrow\tau^{*}$ such that for any $n\in\mathbb{N}$,
$(u,\tau_{n})$ is a pathwise solution to (1.2) and on the set
$\\{\tau^{*}<\infty\\}$,

$\sup_{t\in[0,\tau_{n}]}\|u\|_{H^{s}}\geq n.$

  4. (4)

If $(u,\tau^{*})$ is a maximal pathwise solution and $\tau^{*}=\infty$ almost
surely, then we say that the pathwise solution exists globally.

Inspired by [52, 55], we introduce the concept on stability of exiting time in
Sobolev spaces. Exiting time, as the name suggests, describes the first time
that the solution leaves a given range. More precisely,

######  Definition 1.2 (Stability of exiting time).

Let $s>3/2$ and $\mathcal{S}=(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P},\mathcal{W})$ be a fixed stochastic basis. Let $u_{0}\in
L^{2}(\Omega;H^{s})$ and $\\{u_{0,n}\\}\subset L^{2}(\Omega;H^{s})$ be
$\mathcal{F}_{0}$-measurable. For each $n$, let $u$ and $u_{n}$ be the unique
solutions to (1.2), as in Definition 1.1, with initial values $u_{0}$ and
$u_{0,n}$, respectively. For any $R>0$, define the $R$-exiting times as

$\tau_{n}^{R}\triangleq\inf\\{t\geq 0:\|u_{n}\|_{H^{s}}>R\\},\ \
\tau^{R}\triangleq\inf\\{t\geq 0:\|u\|_{H^{s}}>R\\}.$ (1.5)

Then we define the following properties on stability:

  1. (1)

If $u_{0,n}\rightarrow u_{0}$ in $H^{s}$ $\mathbb{P}-a.s.$ implies that

$\displaystyle\lim_{n\rightarrow\infty}\tau^{R}_{n}=\tau^{R}\ \
\mathbb{P}-a.s.,$ (1.6)

then the $R$-exiting time of $u$ is said to be stable.

  2. (2)

If $u_{0,n}\rightarrow u_{0}$ in $H^{s^{\prime}}$ for all $s^{\prime}<s$
almost surely, implies that (1.6) holds true, the $R$-exiting time of $u$ is
said to be strongly stable.

To study the existence of pathwise solutions to (1.2), we need the following
assumptions on the $h$:

######  Assumption (A).

We assume that when $s>3/2$, $h:[0,\infty)\times H^{s}\ni(t,u)\mapsto
h(t,u)\in\mathcal{L}_{2}(\mathcal{U};H^{s})$ is continuous. Furthermore, we
assume that there are two non-decreasing locally bounded functions
$f,q:[0,+\infty)\rightarrow[0,+\infty)$ such that

  * •

For any $t>0$ and $u\in H^{s}$,

$\displaystyle\|h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}\leq
f(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})(1+\|u\|_{H^{s}}).$
(1.7)

  * •

For any $t>0$,

$\sup_{\|u\|_{H^{s}},\|v\|_{H^{s}}\leq N}\left\\{{\bf 1}_{\\{u\neq
v\\}}\frac{\|h(t,u)-h(t,v)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}}{\|u-v\|_{H^{s}}}\right\\}\leq
q(N),\ \ N\geq 1.$ (1.8)

Example. Now we give an example of noise coefficient satisfying Assumption
(A). For simplicity, we only consider the 1-D case, i.e., $\mathcal{W}=W$, and
$h:(t,u)\mapsto h(t,u)$ is a map from $[0,\infty)\times H^{s}$ to $H^{s}$. It
is easy to verify that if that $h$ can be constructed, the extension of this
example to the general $\mathcal{L}_{2}(\mathcal{U};H^{s})$-valued $h$ can be
carried out by considering $h(e_{k})$, where $\\{e_{k}\\}_{k\geq 1}$ is a
complete orthonormal basis of $\mathcal{U}$ (as 1-D case with suitable
coefficient) such that $\sum_{k\geq 1}\|h(e_{k})\|^{2}_{H^{s}}<\infty$. To
that purpose, let $G=\frac{1}{2}{\rm e}^{-|x|}$. Then in 1-D case,
$(1-\partial_{xx}^{2})^{-1}$ can be understood as

$\displaystyle\left[(1-\partial_{xx}^{2})^{-1}f\right](x)=[G\star f](x)\
\text{for}\ f\in L^{2}(\mathbb{R}),$ (1.9)

where $\star$ stands for the convolution. Now we let

$h(t,u)=q(t)(1-\partial_{xx}^{2})^{-1}\partial_{x}[(u_{x})^{k}+(\mathcal{H}u_{x})^{n}],\
\ k,n\geq 1.$

If $q(\cdot)$ is smooth with both upper and lower bounds, then it is easy to
see that $h$ satisfies Assumption (A).

To find global existence, we need some stronger condition on the noise
coefficient $h$ and we make the following assumption:

######  Assumption (B).

We assume that when $s>\frac{3}{2}$, $\alpha:[0,\infty)\times
H^{s}\ni(t,u)\mapsto\alpha(t,u)\in H^{s}$ is continuous. Moreover, we assume
the following properties hold true:

  * •

$\alpha(\cdot,u)$ is bounded for all $u\in H^{s}$ and there is a non-
decreasing locally bounded function $l(\cdot):[0,\infty)\rightarrow[0,\infty)$
such that for any $t\geq 0$,

$\sup_{\|u\|_{H^{s}},\|v\|_{H^{s}}\leq N}\left\\{{\bf 1}_{\\{u\neq
v\\}}\frac{\|\alpha(t,u)-\alpha(t,v)\|_{H^{s}}}{\|u-v\|_{H^{s}}}\right\\}\leq
l(N),\ \ N\geq 1,\ s>3/2.$ (1.10)

  * •

Define

$\mathfrak{G}=\left\\{\mathcal{G}\in
C^{2}([0,\infty);[0,\infty)):\mathcal{G}(0)=0,\ \mathcal{G}^{\prime}(x)>0,\
\mathcal{G}^{\prime\prime}(x)\leq 0\ \text{and}\
\lim_{x\to\infty}\mathcal{G}(x)=\infty\right\\},$

and we assume that there is a function $\mathcal{G}\in\mathfrak{G}$ and
constants $K_{1},K_{2}>0$ such that for all $(t,u)\in[0,\infty)\times H^{s}$
with $s>5/2$,

$\displaystyle\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\mathcal{M}(t)+2\mathcal{G}^{\prime\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|^{2}\leq
K_{1}-K_{2}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})},$
(1.11)

where

$\displaystyle\mathcal{M}(t)=2Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})\|u\|^{2}_{H^{s-1}}+\|\alpha(t,u)\|^{2}_{H^{s-1}}$

and $Q$ is the constant given in Lemma 2.3.

Example. Let $q(t)$ be a continuous function such that $q_{*}<q^{2}(t)<q^{*}$
for all $t$ and let $Q$ be the constant given in Lemma 2.3 below. Then it is
easy to check that

$\displaystyle\alpha(t,u)\triangleq
q(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{\theta}u$
(1.12)

with

$\text{either}\ \theta>\frac{1}{2},\ q^{*}>q_{*}>0\ \text{or}\
\theta=\frac{1}{2},\ q^{*}>q_{*}>2Q$ (1.13)

satisfies Assumption (B) with $\mathcal{G}(x)=\log(1+x)\in\mathfrak{G}$. For
simplicity we only prove that (1.11). Indeed, we observe that

$\displaystyle\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\mathcal{M}(t)+2\mathcal{G}^{\prime\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|^{2}+K_{2}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}$
$\displaystyle=\,$
$\displaystyle\frac{2Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})\|u\|^{2}_{H^{s-1}}+q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}\|u\|^{2}_{H^{s-1}}}{1+\|u\|^{2}_{H^{s-1}}}$
$\displaystyle-\frac{2q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}\|u\|^{4}_{H^{s-1}}}{\left(1+\|u\|^{2}_{H^{s-1}}\right)^{2}}+K_{2}\frac{q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}\|u\|^{4}_{H^{s-1}}}{\left(1+\|u\|^{2}_{H^{s-1}}\right)^{2}\left(1+\log(1+\|u\|^{2}_{H^{s-1}})\right)}$
$\displaystyle\leq\,$ $\displaystyle
2Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})+q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}$
$\displaystyle-\frac{2q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}\|u\|^{4}_{H^{s-1}}}{\left(1+\|u\|^{2}_{H^{s-1}}\right)^{2}}+K_{2}\frac{q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}}{\left(1+\log(1+\|u\|^{2}_{H^{s-1}})\right)}:=\mathfrak{J}$

If $\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}$ is bounded, then
$\mathfrak{J}$ is also bounded, and hence can be controlled by some constant
$K_{1}>0$. To prove (1.11), we only need to check that $\mathfrak{J}$ can be
also controlled by $K_{1}$ when
$\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow\infty$.
Let
$\mathcal{P}=q^{2}(t)\left(1+\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)^{2\theta}$.
Due to the embedding $H^{s-1}\hookrightarrow W^{1,\infty}$, when
$\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow\infty$,
$\|u\|_{H^{s-1}}\rightarrow\infty$. Hence, we have that

$\displaystyle\limsup_{\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow+\infty}\mathfrak{J}$
$\displaystyle\leq\,\limsup_{\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow+\infty}\left\\{\frac{2Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})}{\mathcal{P}}+1-\frac{2\|u\|^{4}_{H^{s-1}}}{\left(1+\|u\|^{2}_{H^{s-1}}\right)^{2}}\right.$
$\displaystyle\hskip
142.26378pt\left.+K_{2}\frac{\|u\|^{2}_{H^{s-1}}}{\left(1+\|u\|^{2}_{H^{s-1}}\right)^{2}\left(1+\log(1+\|u\|^{2}_{H^{s-1}})\right)}\right\\}\mathcal{P}.$

Moreover, if (1.13) is satisfied,

$\limsup_{\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow+\infty}\frac{2Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})}{\mathcal{P}}<1$

and consequently
$\limsup_{\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\rightarrow+\infty}\mathfrak{J}\leq
0$. Therefore $\alpha$ given by (1.12) and $\mathcal{G}(\cdot)=\log(1+\cdot)$
satisfies (1.11).

In the linear noise case (1.4), we make the following assumption on $b$:

######  Assumption (C).

We assume $b(t)$ in (1.4) satisfies that $b(t)\in
C\left([0,\infty);[0,\infty)\right)$ and there exists some $b^{*}>0$ such that
$b^{2}(t)<b^{*}$ for all $t\geq 0$.

Finally, we assume the following conditions to study the question (Q-3):

######  Assumption (D).

When considering (1.2) in Section 6, we assume that for $s>3/2$,
$h:[0,\infty)\times H^{s}\ni(t,u)\mapsto
h(t,u)\in\mathcal{L}_{2}(\mathcal{U};H^{s})$ is continuous. Moreover, we
assume the following:

  * •

There exists a non-decreasing and locally bounded functions
$l(\cdot):[0,+\infty)\rightarrow[0,+\infty)$ such that for any $t\geq 0$ and
$u\in H^{s}$ with $s>3$, we have that

$\displaystyle\|h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}\leq
l(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})\|u\|_{H^{s}}.$

  * •

There exists a non-decreasing and locally bounded functions
$g(\cdot):[0,+\infty)\rightarrow[0,+\infty)$ and a real number
$\sigma_{0}\in(3/2,7/4)$ such that for all $N\geq 1$,

$\displaystyle\sup_{t\geq 0,\|u\|_{H^{s}}\leq
N}\|h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}\leq g(N){\rm
e}^{-\frac{1}{\|u\|_{H^{\sigma_{0}}}}}.$

  * •

Property (1.8) holds true.

Example. Let us here give an example of noise structure satisfying Assumption
(D). As above, we only consider the case where $h(t,u)\ {\rm
d}\mathcal{W}=b(t,u)\ {\rm d}W$ with $W$ a standard 1-D Brownian motion. Let
$k\geq 1$ and $q(\cdot)$ be a continuous and bounded function, then for any
$k,n\geq 1$,

$b(t,u)=q(t){\rm
e}^{-\frac{1}{\|u\|_{H^{\sigma_{0}}}}}(1-\partial_{xx}^{2})^{-1}\partial_{x}[(u_{x})^{k}+(\mathcal{H}u_{x})^{n}]$

satisfies Assumption (D), where $(1-\partial_{xx}^{2})^{-1}$ is defined in
(1.9).

###  1.4. Main results and remarks

In this subsection, we present the precise statements of the different results
shown in this article. The first result reads

######  Theorem 1.1.

Let $s>3$ and let $h(t,u)$ satisfy Assumption (A). If $u_{0}$ is an
$H^{s}$-valued $\mathcal{F}_{0}$-measurable random variable satisfying
$\mathbb{E}\|u_{0}\|^{2}_{H^{s}}<\infty$, then there is a local unique
pathwise solution $(u,\tau)$ to (1.2) in the sense of Definition 1.1 with

$u(\cdot\wedge\tau)\in
L^{2}\left(\Omega;C\left([0,\infty);H^{s}\right)\right).$ (1.14)

Moreover, $(u,\tau)$ can be extended to a unique maximal pathwise solution
$(u,\tau^{*})$ with the following blow-up criterion:

$\textbf{1}_{\left\\{\limsup_{t\rightarrow\tau^{*}}\|u(t)\|_{H^{s}}=\infty\right\\}}=\textbf{1}_{\left\\{\limsup_{t\rightarrow\tau^{*}}\|u_{x}(t)\|_{L^{\infty}}+\|\mathcal{H}u_{x}(t)\|_{L^{\infty}}=\infty\right\\}}\
\mathbb{P}-a.s.$ (1.15)

######  Remark 1.1.

Before we explain the ideas and difficulties regarding the existence of local
pathwise solutions,let us first stress some important remarks on the blow-up
criterion (1.15).

  * •

The blow-up criterion (1.15) implies that the $H^{s^{\prime}}$ norm of the
solution within the range $s^{\prime}\in(\frac{3}{2},s]$ blows up at the same
time $\tau^{*}$. Indeed, for any fixed $s$ and $s^{\prime}\in(\frac{3}{2},s)$,
since

$\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\lesssim\|u\|_{H^{s^{\prime}}}\leq\|u\|_{H^{s}},$

we can conclude that $\|u\|_{H^{s^{\prime}}}$ blows up no later than the time
$\|u_{x}(t)\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}$ blows up but no
earlier than the time $\|u\|_{H^{s}}$ blows up. Therefore, equality (1.15)
shows that all of the $H^{s^{\prime}}$ norms have the same blow-up time. This
fact will be used to prove Theorem 1.2 (see (4.1)).

  * •

Invoking the well-known logarithmic Sobolev inequality involving the Hilbert
transform $\mathcal{H}$

$\|\mathcal{H}u_{x}\|_{L^{\infty}}\lesssim\,\left(1+\|u_{x}\|_{L^{\infty}}\log\left({\rm
e}+\|u_{x}\|_{H^{1}}\right)+\|u_{x}\|_{L^{2}}\right),$ (1.16)

one can show in deterministic case, by using (1.16), that the blow-up
criterion (1.15) can be improved into (cf. [14])

$\limsup_{t\rightarrow\tau^{*}}\|u(t)\|_{H^{s}}=\infty\Longleftrightarrow\limsup_{t\rightarrow\tau^{*}}\|u_{x}(t)\|_{L^{\infty}}=\infty.$

However, it is still not clear how to achieve this in the stochastic setting.
Technically, because
$\mathbb{E}\left[\left(1+\|u_{x}\|_{L^{\infty}}\log\left({\rm
e}+\|u_{x}\|_{H^{1}}\right)+\|u_{x}\|_{L^{2}}\right)\|u\|_{H^{s}}\right]$ is
involved, we have not been able to close the estimate for
$\mathbb{E}\|u\|^{2}_{H^{s}}$.

######  Remark 1.2.

Now we give a remark regarding the existence part in Theorem 1.1. Since we
focus on a Cauchy problem defined on the whole space $\mathbb{R}$, the
classical probabilistic compactness argument for non-linear SPDEs in bounded
domain seems inapplicable in this work. Indeed, we have the following
essential difficulties:

  * •

For smooth $u$, in the a priori $L^{2}(\Omega;H^{s})$ estimate for $u$, after
using the Itô formula for $\|u\|^{2}_{H^{s}}$, we will have to deal with
$\mathbb{E}\left(f^{2}(\|u_{x}\|_{L^{\infty}})(1+\|u\|^{2}_{H^{s}})\right)$
and
$\mathbb{E}\left(\left(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)\|u\|^{2}_{H^{s}}\right)$,
coming from $h(t,u)$ and $(\mathcal{H}u)u_{x}$, respectively. To close the
estimate, these two terms should be controlled in terms of
$\mathbb{E}\left(1+\|u\|^{2}_{H^{s}}\right)$. Since we cannot split the
mathematical expectation, we add a cut-off function to the original problem to
cut the non-linear parts in terms of $\|\cdot\|_{H^{r}}$ (see (3.1) below)
with suitable $r$ such that $H^{s}\hookrightarrow H^{r}\hookrightarrow
W^{1,\infty}$. This enables us to close the a priori estimate. In this work,
$s>3$ and we let $r=s-3/2$.

  * •

As a second step, we construct the approximation scheme and obtain certain
uniform estimates. Our next aim, is to pass to the limit to obtain the
existence of a solution. If the target problem is defined on bounded domain,
usually one can find $q\in(r,s)$ such that
$H^{s}\hookrightarrow\hookrightarrow H^{q}\hookrightarrow H^{r}$ (here
$\hookrightarrow\hookrightarrow$ means the embedding is compact). By the
uniform estimates, one can follow the compactness argument (cf. Prokhorov’s
Theorem and Skorokhod’s Theorem) to obtain the convergence in $H^{q}$, which
enables us to pass to the limit to find a martingale solution to the cut-off
version of the target SPDE. We can a posteriori introduce a stopping time to
remove the cut-off. We refer to [6, 8, 13, 30, 24, 54, 55] for different
examples. On the contrary, in unbounded domains, the compact embedding only
holds true for the “local” space $H^{s}_{\rm
loc}\hookrightarrow\hookrightarrow H^{q}_{\rm loc}$. Because the cut-off
$\|\cdot\|_{H^{r}}$ appears in the problem itself, even though we obtain the
convergence in $H^{q}_{\rm loc}$ (cf. Prokhorov’s Theorem and Skorokhod’s
Theorem), it is still not enough to guarantee the convergence of
$\|\cdot\|_{H^{r}}$ because $\|\cdot\|_{H^{r}}$ is a global object (for space
variable), which cannot be controlled by a local condition. As mentioned
above, because the cut-off is needed, we have to establish a convergence
result in some topology no weaker than $H^{r}$.

  * •

In this paper, motivated by [42], by a careful analysis on the differences
between any two approximate solutions, we will show that there exists a sub-
sequence of the approximate solutions converging in $C([0,T];H^{r})$ almost
surely. We also remark that our analysis is also available for the torus case,
i.e. $x\in\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$, since all the estimates can
be perform in the same way (up to some obvious modifications) for
$x\in\mathbb{T}$. In the deterministic case where the cut-off is no longer
needed, convergence in local Sobolev spaces is sufficient to take limit to
find a solution.

Hereafter, we consider the noise effect versus finite time blow-up. Our second
result gives a partial answer to (Q-1):

######  Theorem 1.2 (Noise preventing blow-up).

Let $s>3$ and $u_{0}\in H^{s}$ be an $H^{s}$-valued
$\mathcal{F}_{0}$-measurable random variable with
$\mathbb{E}\|u_{0}\|^{2}_{H^{s}}<\infty$. If Assumption (B) holds true, then
the corresponding maximal pathwise solution $(u,\tau^{*})$ to (1.3) satisfies

$\mathbb{P}\left\\{\tau^{*}=\infty\right\\}=1.$

######  Remark 1.3.

Let us first recall that in the deterministic counterpart of (1.3), the blow-
up of regular enough solutions actually cannot be prevented, cf. [11, 53, 43].
Therefore, Theorem 1.2 justifies the idea that fast growing non-linear noise
(strong noise) can regularize the solutions in terms of preventing
singularities. Here we use the terminology “fast growing” condition, described
by (1.11) in Assumption (B), to cancel (notice that
$\mathcal{G}^{\prime\prime}<0$ in (1.11)) the growth of the non-local
transport term $(\mathcal{H}u)u$ such that
$\mathbb{E}\mathcal{G}(\|u\|^{2}_{H^{s}})$ can be controlled with a Lyapunov
type function $\mathcal{G}$. The idea of using a Lyapunov type function is
motivated by the works [7, 36, 49, 51].

Complementary, in the case of linear noise, we can show that singularities
occur in finite time with positive probability which yields a partial answers
to (Q-2). The precise statement reads

######  Theorem 1.3 (Blow-up linear noise).

Let $0<K<1$, $s>3$ and $b(t)$ satisfy Assumption (C). Assume that
$u_{0}=u_{0}(\omega,x)$ is an $H^{s}$ valued $\mathcal{F}_{0}$ measurable
random variable satisfying $\mathbb{E}\left\lVert
u_{0}\right\rVert^{2}_{H^{s}}<\infty$ and attaining a global maximum at
$x_{0}$. If

$\Lambda u_{0}(\omega,x_{0})>\frac{b^{*}}{K}\ \ \mathbb{P}-a.s.,$ (1.17)

where $b^{*}$ is given in Assumption (C), then the unique maximal pathwise
solution $(u,\tau^{*})$ to (1.4) satisfies that

$\mathbb{P}\\{\tau^{*}<\infty\\}\geq\mathbb{P}\\{\limsup_{t\rightarrow\tau^{*}}\max_{x\in\mathbb{R}}\Lambda
u(t)=+\infty\\}\geq\mathbb{P}\\{e^{\int_{0}^{t}b(t^{\prime})\,{\rm
d}\mathcal{W}_{t^{\prime}}}>K\ \forall t\\}>0.$ (1.18)

######  Remark 1.4.

We make the following remarks regarding Theorem 1.3:

  * •

Motivated by previous works [24, 50, 51], to study (1.4), we make use of
Girsanov type transform to obtain a PDE with random coefficient instead of a
SPDE to study blow-up. In this linear noise case, the blow-up criterion (1.15)
also holds true. By direct computation, we have $\mathcal{H}u_{x}=\Lambda u$
for the fractional Laplace operator $\Lambda=(-\Delta)^{\frac{1}{2}}$. Hence
the blow-up criterion (1.15) becomes

$\textbf{1}_{\left\\{\limsup_{t\rightarrow\tau^{*}}\|u(t)\|_{H^{s}}=\infty\right\\}}=\textbf{1}_{\left\\{\limsup_{t\rightarrow\tau^{*}}\|u_{x}(t)\|_{L^{\infty}}+\|\Lambda
u(t)\|_{L^{\infty}}=\infty\right\\}}\ \mathbb{P}-a.s.$

This motivates us to consider blow-up for the particular case
$\limsup_{t\rightarrow\tau^{*}}\|\Lambda u(t)\|_{L^{\infty}}=\infty$. Actually
the case we obtain, as shown in (1.18), is
$\limsup_{t\rightarrow\tau^{*}}\max_{x\in\mathbb{R}}\Lambda u(t)=+\infty$. So
far it is still not clear if
$\limsup_{t\rightarrow\tau^{*}}\min_{x\in\mathbb{R}}\Lambda u(t)=-\infty$
holds for blow-up.

  * •

The key identity used in our analysis on blow-up is originally introduced in
[53, Proposition 3.5], where the authors show that $C^{1}$ global solutions to
(1.4) cannot exists. The precise statement of this identity in our stochastic
context is given in (5.9). Although the idea towards the proof of (5.9) is
similar to [53], we cannot assume without loss of generality (as in [53,
Proposition 3.5]) that $\tilde{v}(z_{0})=0$ which simplifies the proof of
(5.9). The main reason relies on the fact that in the stochastic setting
$z_{0}$ depends on $t$ and $\omega\in\Omega$. As a consequence, for different
$\omega$, the time $t$ such that $\tilde{v}(z_{0})=0$ may be different, and
therefore we cannot assume that $\tilde{v}(z_{0})=0$. On the other hand, if we
fix $t$ such that $\tilde{v}(z_{0})=0$, then $\tilde{v}(z_{0})=0$ may not hold
almost surely, which brings further obstacles in proving (5.9).

  * •

For trivial $b^{*}=0$, which means $b(t)\equiv 0$, Theorem 1.3 recovers the
deterministic result [53, Theorem 3.7]. Due to the presence of the noise term
$b(t)u{\rm d}W$, the condition (1.17) is more restrictive compared to the
deterministic case where $\Lambda u_{0}(x_{0})>0$ suffices to show the finite
time blow-up, [53, Theorem 3.7]. Therefore, although finite time singularities
can be shown in the stochastic case it is somehow restrictive and strongly
depends on the choice of the coefficient $b(t)$.

Regarding question (Q-3), we have the following partial (negative) answer:

######  Theorem 1.4 (Weak instability).

Let $\mathcal{S}=(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P},\mathcal{W})$ be a fixed stochastic basis and $s>3$. If $h$
satisfies Assumption (D), then at least one of the following properties holds
true.

  1. (1)

For any $R\gg 1$, the $R$-exiting time is not strongly stable for the zero
solution to (1.2) in the sense of Definition 1.2;

  2. (2)

There is a $T>0$ such that solution map $u_{0}\mapsto u$ defined by (1.2) is
not uniformly continuous as a map from $L^{\infty}(\Omega,H^{s})$ into
$L^{1}(\Omega;C([0,T],H^{s}))$. More precisely, there exist two sequences of
solutions $u^{1,n}(t,x)$ and $u^{2,n}(t,x)$, and two sequences of stopping
time $\tau_{1,n}$ and $\tau_{2,n}$, such that

     * •

For $i=1,2$, $\mathbb{P}\\{\tau_{i,n}>0\\}=1$ for each $n>1$. Besides,

$\lim_{n\rightarrow\infty}\tau_{1,n}=\lim_{n\rightarrow\infty}\tau_{2,n}=\infty\
\ \mathbb{P}-a.s.$ (1.19)

     * •

For $i=1,2$, $u^{i,n}\in C([0,\tau_{i,n}],H^{s})$ $\mathbb{P}-a.s.$, and

$\mathbb{E}\left(\sup_{t\in[0,\tau_{1,n}]}\|u^{1,n}(t)\|_{H^{s}}+\sup_{t\in[0,\tau_{2,n}]}\|u^{2,n}(t)\|_{H^{s}}\right)\lesssim
1.$ (1.20)

     * •

At initial time $t=0$,

$\lim_{n\rightarrow\infty}\mathbb{E}\|u^{1,n}(0)-u^{2,n}(0)\|_{H^{s}}^{2}=0.$
(1.21)

     * •

When $t>0$,

$\liminf_{n\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T\wedge\tau_{1,n}\wedge\tau_{2,n}]}\|u^{1,n}(t)-u^{2,n}(t)\|_{H^{s}}\gtrsim\sup_{t\in[0,T]}|\sin(t)|.$
(1.22)

######  Remark 1.5.

The idea behind Theorem 1.4 can be understood as follows: one cannot improve
the continuity of the map $u_{0}\mapsto u$, and simultaneously, the stability
of the exiting time at $u\equiv 0$. Hereafter, we briefly outline the main
difficulties encountered in the proof of Theorem 1.4 and the main strategies
used to tackle them.

  * •

For system (1.2), we do not know how to obtain any explicit expression of the
solutions. Therefore, to establish (1.22), the idea relies on finding two
sequences of approximate solutions $\\{u_{m,n}\\}$ ($m\in\\{1,2\\}$) such that
the difference between $u_{m,n}$ and the actual solution $u^{m,n}$ tends to
zero as $n\to\infty$, from which one can prove (1.22) by using $\\{u_{m,n}\\}$
rather than $\\{u^{m,n}\\}$. In this way, the first difficulty lies in the
construction of such approximate solutions $\\{u_{m,n}\\}$. In this work, by
some delicate calculation, we are able to construct two sequences of
approximate solutions $\\{u_{m,n}\\}$ such that the actual solutions
$\\{u^{m,n}\\}$ satisfy

$\displaystyle u^{m,n}(0)=u_{m,n}(0),\ \
\lim_{n\rightarrow\infty}\mathbb{E}\sup_{[0,\tau_{m,n}]}\|u^{m,n}-u_{m,n}\|_{H^{s}}=0,$
(1.23)

where $u^{m,n}$ exists at least on $[0,\tau_{m,n}]$. Technically, since the
equation involves the non-local Hilbert transform $\mathcal{H}$ and the
problem is defined on $\mathbb{R}$, the construction of approximate solutions
$u_{m,n}$ is much more elaborated than the constructions in [44, 52, 55].

  * •

The second difficulty we have to surpass is that we need to guarantee that
$\inf_{n}\tau_{m,n}>0$ almost surely when dealing with (1.23). This obstacle
comes from the lack of lifespan estimates which we believe is quite a common
issue in SPDEs. Indeed, in deterministic cases, one can easily obtain the
lifespan estimate, from which it is not difficult to find a common interval
$[0,T]$ such that all actual solutions exist on $[0,T]$ (see for example Lemma
6.1). In the stochastic setting, we are not able to show the precise estimate.
The key observation to surpass this difficulty is that, the property
$\inf_{n}\tau_{m,n}>0$ can be connected to the stability property of the
exiting time (see Definition 1.2). The condition comprising the fact that the
$R_{0}$-exiting time is strongly stable at the zero solution will be used to
provide a common existence time $T>0$ such that for all $n$, $u^{m,n}$ exists
up to $T$ (see Lemma 6.4 below). Therefore, to prove Theorem 1.4, we will show
that, if the $R_{0}$-exiting time is strongly stable at the zero solution for
some $R_{0}\gg 1$, then the solution map $u_{0}\mapsto u$ defined by (1.2)
cannot be uniformly continuous.

### Plan of the paper

We outline the structure of the paper. In Section 2 we provide some relevant
preliminaries and recall well-known estimates that will be employed throughout
the paper. In Section 3 we show the existence and uniqueness of local pathwise
solutions and derived a blow-up criterion proving Theorem 1.1. We will divide
the proof into several subsections. First we use an approximation scheme and
perform uniform bounds in Subsection 3.1. Next, in Subsection 3.2 we show the
convergence of the approximated solutions and afterwards in Subsection 3.3 we
prove the global pathwise solutions to the cur-off problem. We conclude the
proof of Theorem 1.1 in Subsection 1.1. Section 4 is devoted to study the
effect of strong noise and prove Theorem 1.2. In Section 5 we show that finite
time singularity occur in the case of linear noise thus proving Theorem 1.3.
To conclude the article, in Section 6, we prove Theorem 1.4. The proof is
divided into Subsection 6.1 \- Subsection 6.3.

##  2\. Preliminaries

For any $\varepsilon\in(0,1)$, $J_{\varepsilon}$ is the Friedrichs mollifier
defined by $J_{\varepsilon}f(x)=j_{\varepsilon}\star f(x)$, where $\star$
stands for the convolution,
$j_{\varepsilon}(x)=\frac{1}{\varepsilon}j(\frac{x}{\varepsilon})$ and $j(x)$
is a Schwartz function satisfying $0\leq\widehat{j}(\xi)\leq 1$ for all
$\xi\in\mathbb{R}$ and $\widehat{j}(\xi)=1$ for any $|\xi|\leq 1$, where
$\widehat{f}$ denotes the Fourier transform of $f$. It is obvious that
$\widehat{j}_{\varepsilon}(\xi)=\widehat{j}(\varepsilon\xi)$. Moreover, for
any $u\in H^{s}$, we have, cf. [54, 55],

$\displaystyle\|u-J_{\varepsilon}u\|_{H^{r}}$ $\displaystyle\sim
o(\varepsilon^{s-r}),\ \ r\leq s,$ (2.1)
$\displaystyle\|J_{\varepsilon}u\|_{H^{r}}$ $\displaystyle\lesssim
O(\varepsilon^{s-r})\|u\|_{H^{s}},\ \ r>s,$ (2.2)
$\displaystyle(J_{\varepsilon}f,g)_{L^{2}}$
$\displaystyle=(f,J_{\varepsilon}g)_{L^{2}}.$ (2.3)

The Hilbert transform is defined as

$\mathcal{H}f(x)=\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{f(y)}{y-x}\,{\rm
d}y.$ (2.4)

The fractional differential operator
$\Lambda^{\alpha}=(-\Delta)^{\frac{\alpha}{2}}$ is defined as the following
singular integral operator

$\Lambda^{\alpha}f(x)=c_{\alpha}\mbox{p.v.}\int_{\mathbb{R}}\frac{f(x)-f(y)}{|x-y|^{1+\alpha}}\,{\rm
d}y,$ (2.5)

where the constant
$c_{\alpha}=\frac{4^{\frac{\alpha}{2}}\Gamma(\frac{1}{2}+\alpha/2)}{\sqrt{\pi}\lvert\Gamma(-\alpha/2)\rvert}$
is a normalization constant and $\Gamma$ represents the classical gamma
function. In particular when $\alpha=1$, $\mathcal{H}f_{x}=-\Lambda f.$ Since
$J_{\varepsilon}$, $\mathcal{H}$ and $D^{s}$ can be characterized by their
Fourier multipliers, it is easy to see

$[D^{s},\mathcal{H}]=[D^{s},J_{\varepsilon}]=[\partial_{x},\mathcal{H}]=[\partial_{x},J_{\varepsilon}]=[J_{\varepsilon},\mathcal{H}]=0,$
(2.6)

and for any $s\geq 0$,

$\displaystyle\|\mathcal{H}u\|_{H^{s}}\leq\|u\|_{H^{s}},\
\|J_{\varepsilon}u\|_{H^{s}}\leq\|u\|_{H^{s}}.$ (2.7)

######  Lemma 2.1 (Page 3 in [58]).

Let $J_{\varepsilon}$ be defined as in the above. Assume $g\in W^{1,\infty}$
and $f\in L^{2}$. Then for some $C>0$,

$\displaystyle\|[J_{\varepsilon},g]\partial_{x}f\|_{L^{2}}\leq
C\|\partial_{x}g\|_{L^{\infty}}\|f\|_{L^{2}}.$

We also recall the following well-known estimates.

######  Lemma 2.2 ([34, 35]).

If $f,g\in H^{s}\bigcap W^{1,\infty}$ with $s>0$, then for
$p,p_{i}\in(1,\infty)$ with $i=2,3$ and
$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}$,
we have

$\|\left[D^{s},f\right]g\|_{L^{p}}\leq C_{s,p}(\|\nabla
f\|_{L^{p_{1}}}\|D^{s-1}g\|_{L^{p_{2}}}+\|D^{s}f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}}),$

and

$\|D^{s}(fg)\|_{L^{p}}\leq
C_{s,p}(\|f\|_{L^{p_{1}}}\|D^{s}g\|_{L^{p_{2}}}+\|D^{s}f\|_{L^{p_{3}}}\|g\|_{L^{p_{4}}}).$

######  Lemma 2.3.

Let $s>3$. Let $J_{\varepsilon}$ be the Friedrichs mollifier defined before.
There is a constant $Q=Q(s)>0$ such that

$\left|\left((\mathcal{H}u)u_{x},u\right)_{H^{s-1}}\right|\leq
Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})\|u\|^{2}_{H^{s-1}},$
(2.8)

and for all $\varepsilon>0$,

$\left|\left(J_{\varepsilon}\left[(\mathcal{H}u)u_{x}\right],J_{\varepsilon}u\right)_{H^{s}}\right|\leq
Q(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}})\|u\|^{2}_{H^{s}}.$
(2.9)

###### Proof.

We first prove (2.9). Due to (2.6) and (2.3), we commute the operator to
derive

$\displaystyle\left(D^{s}J_{\varepsilon}\left[(\mathcal{H}u)u_{x}\right],D^{s}J_{\varepsilon}u\right)_{L^{2}}$
$\displaystyle=$
$\displaystyle\left(\left[D^{s},\mathcal{H}u\right]u_{x},D^{s}J^{2}_{\varepsilon}u\right)_{L^{2}}+\left([J_{\varepsilon},\mathcal{H}u]D^{s}u_{x},D^{s}J_{\varepsilon}u\right)_{L^{2}}+\left((\mathcal{H}u)D^{s}J_{\varepsilon}u_{x},D^{s}J_{\varepsilon}u\right)_{L^{2}}.$

Then it follows from Lemmas 2.1 and 2.2, integration by parts, (2.7) and
$H^{s}\hookrightarrow W^{1,\infty}$ that

$\left|\left(\left[D^{s},\mathcal{H}u\right]u_{x},D^{s}J^{2}_{\varepsilon}u\right)_{L^{2}}\right|\lesssim\left(\|u_{x}\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)\|u\|^{2}_{H^{s}},$
$\left|\left([J_{\varepsilon},\mathcal{H}u]D^{s}u_{x},D^{s}J_{\varepsilon}u\right)_{L^{2}}\right|\lesssim\
\|\mathcal{H}u_{x}\|_{L^{\infty}}\|u\|^{2}_{H^{s}},$

and

$\left|\left((\mathcal{H}u)D^{s}J_{\varepsilon}u_{x},D^{s}J_{\varepsilon}u\right)_{L^{2}}\right|\lesssim\
\|\mathcal{H}u_{x}\|_{L^{\infty}}\|u\|^{2}_{H^{s}}.$

Combining the above inequalities gives rise to (2.9). For (2.8), since
$\left((\mathcal{H}u)u_{x},u\right)_{H^{s-1}}$ is well-defined in this case,
by repeating the above analysis, one can easily obtain the (2.8) and the proof
is therefore completed. ∎

######  Remark 2.1.

We remark that (2.9) will be used in the proof of blow-up criterion (1.15)
(see (3.26) below) and (2.8) will be used in the proof of Theorem 1.2 (see
(4.3) below). But it is worthwhile pointing out that (2.8) is also used
implicitly in the proof of Theorem 1.2. We refer to Remark 4.1 for more
details.

Let us collect some identities and formulas regarding the Hilbert transform
and the fractional Laplacian operator, cf. [53, 11]. The first one is the so-
called Cotlar’s identity

$2\mathcal{H}\left(f(\mathcal{H}f)\right)=(\mathcal{H}f)^{2}-f^{2}.$ (2.10)

The second one is the following equality:

######  Lemma 2.4 (Corollary 3.3, [53]).

For any $f\in L^{1}(\mathbb{R};\mathbb{R})$, we have that for $g(x)=xf(x)$,

$\Lambda g(x)=x\Lambda f(x)-\mathcal{H}f(x)$

Finally, we recall the following estimate on the product of a Schwartz
function and a trigonometric function.

######  Lemma 2.5 ([27]).

Let $\delta>0$ and $\alpha\in\mathbb{R}$. Then for any $r\geq 0$ and any
Schwartz function $\psi$, the following equation holds true:

$\lim_{n\rightarrow\infty}n^{-\frac{\delta}{2}-r}\left\|\psi\left(\frac{x}{n^{\delta}}\right)\cos(nx-\alpha)\right\|_{H^{r}}=\frac{1}{\sqrt{2}}\|\psi\|_{L^{2}}.$
(2.11)

Moreover, (2.11) also holds true when $\cos$ is replaced by $\sin$.

##  3\. Proof of Theorem 1.1

For the sake of clarity, the proof is divided into several subsections.

###  3.1. Approximation scheme and uniform estimates

The first step is to construct a suitable approximation scheme. For any $R>1$,
we let $\chi_{R}(x):[0,\infty)\rightarrow[0,1]$ be a $C^{\infty}$ function
such that $\chi_{R}(x)=1$ for $x\in[0,R]$ and $\chi_{R}(x)=0$ for $x>2R$. Then
we consider the following cut-off problem on $\mathbb{R}$,

$\left\\{\begin{aligned} &{\rm
d}u+\chi_{R}(\|u\|_{H^{s-3/2}})(\mathcal{H}u)u_{x}{\rm
d}t=\chi_{R}(\|u\|_{H^{s-3/2}})h(t,u){\rm d}\mathcal{W},\\\
&u(\omega,0,x)=u_{0}(\omega,x)\in H^{s},\ \ s>3.\end{aligned}\right.$ (3.1)

To apply the theory of SDEs in Hilbert space to (3.1), we will have to mollify
the transport term $(\mathcal{H}u)u_{x}$ since the product
$(\mathcal{H}u)u_{x}$ loses regularity. To this end, we consider the following
approximation scheme:

$\left\\{\begin{aligned} {\rm d}u+G_{1,\varepsilon}(u){\rm d}t&=G_{2}(t,u){\rm
d}\mathcal{W},\\\
G_{1,\varepsilon}(u)&=\chi_{R}(\|u\|_{H^{s-3/2}})J_{\varepsilon}[\left(\mathcal{H}J_{\varepsilon}u\right)\partial_{x}J_{\varepsilon}u],\\\
G_{2}(t,u)&=\chi_{R}(\|u\|_{H^{s-3/2}})h(t,u),\\\ u(0,x)&=u_{0}(x)\in
H^{s},\end{aligned}\right.$ (3.2)

where $J_{\varepsilon}$ is the Friedrichs mollifier defined in Section 2.

After mollifying the non-local transport term $(\mathcal{H}u)u_{x}$, we see
that $G_{1,\varepsilon}(\cdot)$ and $G_{2}(t,\cdot)$ are locally Lipschitz
continuous in $H^{s}$. Moreover, the cut-off function
$\chi_{R}(\|\cdot\|_{H^{s-3/2}})$ gives the linear growth condition (cf, Lemma
2.3 and (1.7)), i.e., there are constants $l_{1}=l_{1}(\varepsilon,R)>0$ and
$l_{2}=l_{2}(R)>0$ such that for all $t\geq 0$ and $s>3$,

$\|G_{1,\varepsilon}(u)\|_{H^{s}}\leq l_{1}(1+\|u\|_{H^{s}}),\ \
\|G_{2}(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}\leq
l_{2}(1+\|u\|_{H^{s}}),\ \ t\in[0,T].$ (3.3)

Therefore, for a fixed stochastic basis
$\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ and for $u_{0}\in L^{2}(\Omega;H^{s})$ with $s>3$, the
existence theory of SDEs in Hilbert space (see for example [48, Theorem 4.2.4
with Example 4.1.3] and [33]), (3.2) admits a unique solution
$u_{\varepsilon}\in C([0,\infty),H^{s})$ $\mathbb{P}-a.s.$

We have the following uniform-in-$\varepsilon$ estimate:

######  Proposition 3.1.

Let $\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ be a fixed stochastic basis. Let $s>3$, $R>1$ and
$\varepsilon\in(0,1)$. Assume $h$ satisfies Assumption (A) and $u_{0}\in
L^{2}(\Omega;H^{s})$ is an $H^{s}$-valued $\mathcal{F}_{0}$-measurable random
variable. Let $u_{\varepsilon}\in C([0,\infty);H^{s})$ solve (3.2)
$\mathbb{P}-a.s.$, then for any $T>0$, there are $C=C(R,T,u_{0})>0$
$(i=0,1,2,3,4)$ such that

$\displaystyle\sup_{\varepsilon>0}\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}\leq
C.$ (3.4)

###### Proof.

Using the Itô formula for $\|u_{\varepsilon}\|^{2}_{H^{s}}$, we have that for
any $t>0$,

$\displaystyle{\rm d}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}=\,$ $\displaystyle
2\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left(h(t,u_{\varepsilon}){\rm
d}\mathcal{W},u_{\varepsilon}\right)_{H^{s}}$
$\displaystyle-2\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left(D^{s}J_{\varepsilon}\left[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right],D^{s}u_{\varepsilon}\right)_{L^{2}}{\rm
d}t$
$\displaystyle+\chi^{2}_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\|h(t,u_{\varepsilon})\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}^{2}{\rm
d}t.$

Then, by means of the BDG inequality, (1.7), $H^{s-3/2}\hookrightarrow
W^{1,\infty}$ and (2.7), we find that for some constant $C=C(R)>0$,

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}-\mathbb{E}\|u_{0}\|^{2}_{H^{s}}$
$\displaystyle\lesssim$
$\displaystyle\mathbb{E}\left(\int_{0}^{T}\chi^{2}_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})f^{2}(2\|u_{\varepsilon}\|_{H^{s-3/2}})(1+\|u_{\varepsilon}\|^{2}_{H^{s}})\|u_{\varepsilon}\|^{2}_{H^{s}}{\rm
d}t\right)^{\frac{1}{2}}$
$\displaystyle+2\mathbb{E}\int_{0}^{T}\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left|\left(D^{s}J_{\varepsilon}\left[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right],D^{s}u_{\varepsilon}\right)_{L^{2}}\right|{\rm
d}t$
$\displaystyle+\mathbb{E}\int_{0}^{T}\chi^{2}_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})f^{2}(2\|u_{\varepsilon}\|_{H^{s-3/2}})(1+\|u_{\varepsilon}\|^{2}_{H^{s}}){\rm
d}t$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}\|_{H^{s}}^{2}+C(R)\mathbb{E}\int_{0}^{T}\left(1+\|u_{\varepsilon}\|^{2}_{H^{s}}\right){\rm
d}t$
$\displaystyle+2\mathbb{E}\int_{0}^{T}\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left|\left(D^{s}J_{\varepsilon}\left[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right],D^{s}u_{\varepsilon}\right)_{L^{2}}\right|{\rm
d}t.$

Let $J_{\varepsilon}u_{\varepsilon}=v$. It follows from (2.3), Lemma 2.2,
integration by parts and (2.7) that

$\displaystyle\left|\left(D^{s}J_{\varepsilon}\left[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right],D^{s}u_{\varepsilon}\right)_{L^{2}}\right|\leq$
$\displaystyle\left|\left(\left[D^{s},\mathcal{H}v\right]v_{x},D^{s}v\right)_{L^{2}}\right|+\left|\left((\mathcal{H}v)D^{s}v_{x},D^{s}v\right)_{L^{2}}\right|$
$\displaystyle\leq$ $\displaystyle
C\left(\|\mathcal{H}v\|_{H^{s}}\|v_{x}\|_{L^{\infty}}+\|\mathcal{H}v_{x}\|_{L^{\infty}}\|v\|_{H^{s}}\right)\|v\|_{H^{s}}$
$\displaystyle\leq$ $\displaystyle C\|v\|_{H^{s-3/2}}\|v\|^{2}_{H^{s}},$

which implies

$\displaystyle\mathbb{E}\int_{0}^{T}\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left|\left(D^{s}J_{\varepsilon}\left[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right],D^{s}u_{\varepsilon}\right)_{L^{2}}\right|{\rm
d}t\leq C(R)\mathbb{E}\int_{0}^{T}\|u_{\varepsilon}\|_{H^{s}}^{2}{\rm d}t.$

Therefore we obtain

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}\leq
2\mathbb{E}\|u_{0}\|^{2}_{H^{s}}+C(R)\int_{0}^{T}\left(1+\mathbb{E}\sup_{t^{\prime}\in[0,t]}\|u(t^{\prime})\|_{H^{s}}^{2}\right){\rm
d}t.$

Using Grönwall’s inequality to the above estimate implies that for some
$C=C(R,T,u_{0})>0$,

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}\leq
C(R,T,u_{0}),$

which is (3.4). ∎

###  3.2. Convergence of approximate solutions

Now we are going to show that, there is a subsequence of $u_{\varepsilon}$
converging in $C([0,T],H^{s-3/2})$ almost surely. To this end, for the
solutions $u_{\varepsilon}$ and $u_{\eta}$ to (3.2), we consider the following
problem for $v_{\varepsilon,\eta}=u_{\varepsilon}-u_{\eta}$,

${\rm
d}v_{\varepsilon,\eta}+\left[G_{1,\varepsilon}(u_{\varepsilon})-G_{1,\eta}(u_{\eta})\right]{\rm
d}t=\left[G_{2}(t,u_{\varepsilon})-G_{2}(t,u_{\eta})\right]{\rm
d}\mathcal{W},\ \ v_{\varepsilon,\eta}(0)=0.$ (3.5)

We notice that

$\displaystyle G_{1,\varepsilon}(u_{\varepsilon})-G_{1,\eta}(u_{\eta})$
$\displaystyle=$
$\displaystyle\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})\left[J_{\varepsilon}\left(J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}\right)\right]-\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\left[J_{\eta}\left(J_{\eta}(\mathcal{H}u_{\eta})\partial_{x}J_{\eta}u_{\eta}\right)\right]$
$\displaystyle=$
$\displaystyle\left[\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})-\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\right]J_{\varepsilon}[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}]$
$\displaystyle+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})(J_{\varepsilon}-J_{\eta})[J_{\varepsilon}(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}]$
$\displaystyle+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})J_{\eta}[(J_{\varepsilon}-J_{\eta})(\mathcal{H}u_{\varepsilon})\partial_{x}J_{\varepsilon}u_{\varepsilon}]+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})J_{\eta}[J_{\eta}(\mathcal{H}u_{\varepsilon}-\mathcal{H}u_{\eta})\partial_{x}J_{\varepsilon}u_{\varepsilon}]$
$\displaystyle+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})J_{\eta}[J_{\eta}(\mathcal{H}u_{\eta})\partial_{x}(J_{\varepsilon}-J_{\eta})u_{\varepsilon}]+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})J_{\eta}[J_{\eta}(\mathcal{H}u_{\eta})\partial_{x}J_{\eta}(u_{\varepsilon}-u_{\eta})]$
$\displaystyle\triangleq$ $\displaystyle\sum_{i=1}^{6}q_{i}.$ (3.6)

and

$\displaystyle G_{2}(t,u_{\varepsilon})-G_{2}(t,u_{\eta})$ $\displaystyle=$
$\displaystyle\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})h(t,u_{\varepsilon})-\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})h(t,u_{\eta})$
$\displaystyle=$
$\displaystyle\left[\chi_{R}(\|u_{\varepsilon}\|_{H^{s-3/2}})-\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\right]h(t,u_{\varepsilon})+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})[h(t,u_{\varepsilon})-h(t,u_{\eta})]$
$\displaystyle\triangleq$ $\displaystyle\sum_{i=7}^{8}q_{i}.$ (3.7)

Invoking Itô’s formula in (3.5) and recalling (3.6), (3.7) we find that for
any $t>0$,

$\displaystyle\|v_{\varepsilon,\eta}(t)\|^{2}_{H^{s-3/2}}=$
$\displaystyle\mathcal{Q}_{1}-\int_{0}^{t}\mathcal{Q}_{2}{\rm
d}t^{\prime}+\int_{0}^{t}\mathcal{Q}_{3}{\rm d}t^{\prime},$ (3.8)

where

$\mathcal{Q}_{1}=2\int_{0}^{t}\left(\sum_{i=7}^{8}q_{i}{\rm
d}\mathcal{W},v_{\varepsilon,\eta}\right)_{H^{s-3/2}},\ \
\mathcal{Q}_{2}=2\sum_{i=1}^{6}\left(q_{i},v_{\varepsilon,\eta}\right)_{H^{s-3/2}},\
\
\mathcal{Q}_{3}=\left\|\sum_{i=7}^{8}q_{i}\right\|_{\mathcal{L}_{2}\left(U;H^{s-3/2}\right)}^{2}.$
(3.9)

######  Lemma 3.1.

Let $s>3$. For any $\varepsilon,\eta\in(0,1)$, there is a constant $C>0$ such
that $\mathcal{Q}_{2}$ given by (3.9) satisfies

$\displaystyle|\mathcal{Q}_{2}|\leq$ $\displaystyle
C(1+\|u_{\varepsilon}\|^{2}_{H^{s}}+\|u_{\eta}\|^{2}_{H^{s}})\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}+C(\|u_{\varepsilon}\|^{4}_{H^{s}}+\|u_{\eta}\|^{4}_{H^{s}})\max\\{\varepsilon,\eta\\}.$

###### Proof.

Using the mean value theorem for $\chi_{R}(\cdot)$ and (2.7), we have

$\displaystyle\left\|q_{1}\right\|_{H^{s-3/2}}\lesssim\|v_{\varepsilon,\eta}\|_{H^{s-3/2}}\|u_{\varepsilon}\|^{2}_{H^{s}},$

Using (2.1) and (2.7), we see that

$\displaystyle\left\|q_{i}\right\|_{H^{s-3/2}}\lesssim$
$\displaystyle\max\\{\varepsilon^{1/2},\eta^{1/2}\\}\|u_{\varepsilon}\|^{2}_{H^{s}},\
i=2,3,$ $\displaystyle\left\|q_{4}\right\|_{H^{s-3/2}}\lesssim$
$\displaystyle\|v_{\varepsilon,\eta}\|_{H^{s-3/2}}\|u_{\varepsilon}\|_{H^{s}},$
$\displaystyle\left\|q_{5}\right\|_{H^{s-3/2}}\lesssim$
$\displaystyle\max\\{\varepsilon^{1/2},\eta^{1/2}\\}\|u_{\varepsilon}\|_{H^{s}}\|u_{\eta}\|_{H^{s}}.$

For $q_{6}$, using (2.3), (2.6) and then integrating by part, we have

$\displaystyle\left(q_{6},v_{\varepsilon,\eta}\right)_{H^{s-3/2}}=$
$\displaystyle\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\int_{\mathbb{R}}D^{s-3/2}[J_{\eta}(\mathcal{H}u_{\eta})\partial_{x}J_{\eta}v_{\varepsilon,\eta}]\cdot
D^{s-3/2}J_{\eta}v_{\varepsilon,\eta}{\rm d}x$ $\displaystyle=$
$\displaystyle\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\int_{\mathbb{R}}[D^{s-3/2},J_{\eta}(\mathcal{H}u_{\eta})]\partial_{x}J_{\eta}v_{\varepsilon,\eta}\cdot
D^{s-3/2}J_{\eta}v_{\varepsilon,\eta}{\rm d}x$
$\displaystyle+\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\int_{\mathbb{R}}J_{\eta}(\mathcal{H}u_{\eta})\partial_{x}D^{s-3/2}J_{\eta}v_{\varepsilon,\eta}\cdot
D^{s-3/2}J_{\eta}v_{\varepsilon,\eta}{\rm d}x$ $\displaystyle=$
$\displaystyle\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\int_{\mathbb{R}}[D^{s-3/2},J_{\eta}(\mathcal{H}u_{\eta})]\partial_{x}J_{\eta}v_{\varepsilon,\eta}\cdot
D^{s-3/2}J_{\eta}v_{\varepsilon,\eta}{\rm d}x$
$\displaystyle-\frac{1}{2}\chi_{R}(\|u_{\eta}\|_{H^{s-3/2}})\int_{\mathbb{R}}J_{\eta}(\mathcal{H}u_{\eta})_{x}\cdot(D^{s-3/2}J_{\eta}v_{\varepsilon,\eta})^{2}{\rm
d}x.$

Using $\chi_{R}(\cdot)\leq 1$, Lemma 2.2, (2.7) and the embedding
$H^{s-3/2}\hookrightarrow W^{1,\infty}$, we have

$\displaystyle\left(q_{6},v_{\varepsilon,\eta}\right)_{H^{s-3/2}}\lesssim$
$\displaystyle\|u_{\eta}\|_{H^{s-3/2}}\|\partial_{x}J_{\eta}v_{\varepsilon,\eta}\|_{L^{\infty}}\|v_{\varepsilon,\eta}\|_{H^{s-3/2}}+\|\partial_{x}J_{\eta}(\mathcal{H}u_{\eta})\|_{L^{\infty}}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}$
$\displaystyle\lesssim$
$\displaystyle\|u_{\eta}\|_{H^{s}}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}.$

Therefore we can put these all together to see that there is a constant $C>0$
such that

$\displaystyle|\mathcal{Q}_{2}|\leq$ $\displaystyle
C(1+\|u_{\varepsilon}\|^{2}_{H^{s}}+\|u_{\eta}\|^{2}_{H^{s}})\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}+C(\|u_{\varepsilon}\|^{4}_{H^{s}}+\|u_{\eta}\|^{4}_{H^{s}})\max\\{\varepsilon,\eta\\},$

which is the desired estimate. ∎

######  Lemma 3.2.

Let $\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ be a fixed stochastic basis. Let $s>3$, $R>1$ and
$\varepsilon\in(0,1)$. Let $u_{\varepsilon}\in C([0,\infty);H^{s})$ solve
(3.2) $\mathbb{P}-a.s.$ For any $T>0$ and $K>1$, we define

$\tau^{T}_{\varepsilon,K}=\inf\left\\{t\geq
0:\|u_{\varepsilon}(t)\|_{H^{s}}\geq K\right\\}\wedge T,$ (3.10)

and

$\tau^{T}_{\varepsilon,\eta,K}=\tau^{T}_{\varepsilon,K}\wedge\tau^{T}_{\eta,K}.$
(3.11)

Then we have

$\lim_{\varepsilon\rightarrow
0}\sup_{\eta\leq\varepsilon}\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-3/2}}=0,\
\ K>1.$ (3.12)

###### Proof.

Recalling (3.8) and (3.9), we have

$\displaystyle\|v_{\varepsilon,\eta}(t)\|^{2}_{H^{s-3/2}}\leq\ $
$\displaystyle|\mathcal{Q}_{1}|+\int_{0}^{t}|\mathcal{Q}_{2}|{\rm
d}t^{\prime}+\int_{0}^{t}|\mathcal{Q}_{3}|{\rm d}t^{\prime}.$ (3.13)

The mean value theorem for $\chi_{R}(\cdot)$ and Assumption (A) yield that

$\displaystyle\|q_{7}\|_{\mathcal{L}_{2}\left(U;H^{s-3/2}\right)}\leq
C\|v_{\varepsilon,\eta}\|_{H^{s-3/2}}f(\|u_{\varepsilon}\|_{H^{s}})\left(1+\|u_{\varepsilon}\|_{H^{s}}\right).$

By (3.11) and Assumption (A), we see that

$\displaystyle\|q_{8}\|_{\mathcal{L}_{2}\left(U;H^{s-3/2}\right)}\leq
C\|v_{\varepsilon,\eta}\|_{H^{s-3/2}}q(K),\ \
t\in[0,\tau^{T}_{\varepsilon,\eta,K}]\ \ \mathbb{P}-a.s.,$

where $q(\cdot)$ is given in Assumption (A). Therefore we find a constant
$C=C(K)>0$ such that

$\displaystyle\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}|\mathcal{Q}_{3}|{\rm
d}t\leq$ $\displaystyle
C(K)\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}{\rm
d}t\leq
C(K)\int_{0}^{T}\mathbb{E}\sup_{t^{\prime}\in[0,\tau^{t}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t^{\prime})\|^{2}_{H^{s-3/2}}{\rm
d}t.$ (3.14)

Then we employ the BDG inequality to (3.8) to find

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t)\|^{2}_{H^{s-3/2}}$
$\displaystyle\leq$ $\displaystyle
C(K)\mathbb{E}\left(\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}\|v_{\varepsilon,\eta}\|^{4}_{H^{s-3/2}}{\rm
d}t\right)^{\frac{1}{2}}+\sum_{i=2}^{3}\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}|\mathcal{Q}_{i}|{\rm
d}t$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}+C(K)\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}{\rm
d}t+\sum_{i=2}^{3}\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}|\mathcal{Q}_{i}|{\rm
d}t$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}+C(K)\int_{0}^{T}\mathbb{E}\sup_{t^{\prime}\in[0,\tau^{t}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t^{\prime})\|^{2}_{H^{s-3/2}}{\rm
d}t+\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}|\mathcal{Q}_{2}|{\rm
d}t,$

On account of Lemma 3.1, we arrive at

$\displaystyle\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}|\mathcal{Q}_{2}|{\rm
d}t\leq$ $\displaystyle
C(K)\mathbb{E}\int_{0}^{\tau^{T}_{\varepsilon,\eta,K}}\|v_{\varepsilon,\eta}\|^{2}_{H^{s-3/2}}{\rm
d}t+C(K)T\max\\{\varepsilon,\eta\\}$ $\displaystyle\leq$ $\displaystyle
C(K)\int_{0}^{T}\mathbb{E}\sup_{t^{\prime}\in[0,\tau^{t}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t^{\prime})\|^{2}_{H^{s-3/2}}{\rm
d}t+C(K)T\max\\{\varepsilon,\eta\\}.$ (3.15)

Now we can put these all together to obtain

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t)\|^{2}_{H^{s-3/2}}\leq
C(K)\int_{0}^{T}\mathbb{E}\sup_{t^{\prime}\in[0,\tau^{t}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t^{\prime})\|^{2}_{H^{s-3/2}}{\rm
d}t+C(K)T\max\\{\varepsilon,\eta\\},$ (3.16)

which means that

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|v_{\varepsilon,\eta}(t)\|^{2}_{H^{s-3/2}}\leq
C(K,T)\max\\{\varepsilon,\eta\\},$ (3.17)

and hence (3.12) holds true. ∎

######  Lemma 3.3.

For any fixed $s>3$ and $T>0$, there is an $\\{\mathcal{F}_{t}\\}_{t\geq 0}$
progressive measurable $H^{s}$-valued process

$u\in L^{2}\left(\Omega;L^{\infty}\left(0,T;H^{s}\right)\right)$ (3.18)

and a countable subsequence of $\\{u_{\varepsilon}\\}$ $($still denoted as
$\\{u_{\varepsilon}\\})$ such that

$u_{\varepsilon}\xrightarrow[]{\varepsilon\rightarrow 0}u\ {\rm in}\
C([0,T];H^{s-3/2})\ \ \mathbb{P}-a.s.$ (3.19)

###### Proof.

We notice that for each $\varepsilon\in(0,1)$, (3.2) has solution
$u_{\varepsilon}$ almost surely. Now we first take $\varepsilon$ to be
discrete such that for all $\varepsilon$, $u_{\varepsilon}$ can be defined on
the same set $\widetilde{\Omega}$ with $\mathbb{P}\\{\widetilde{\Omega}\\}=1$
(Actually, one can pick discrete $\varepsilon=\frac{1}{n}$ from the beginning
in (3.2)). Recall (3.10) and (3.11). For any $\epsilon>0$, by using
Proposition 3.1 and Chebyshev’s inequality, we see that

$\displaystyle\mathbb{P}\left\\{\sup_{t\in[0,T]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-\frac{3}{2}}}>\epsilon\right\\}$
$\displaystyle=\,$
$\displaystyle\mathbb{P}\left\\{\left(\left\\{\tau^{T}_{\varepsilon,\eta,K}<T\right\\}\cup\left\\{\tau^{T}_{\varepsilon,\eta,K}=T\right\\}\right)\cap\left\\{\sup_{t\in[0,T]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-\frac{3}{2}}}>\epsilon\right\\}\right\\}$
$\displaystyle\leq\,$
$\displaystyle\mathbb{P}\left\\{\tau^{T}_{\varepsilon,K}<T\right\\}+\mathbb{P}\left\\{\tau^{T}_{\eta,K}<T\right\\}+\mathbb{P}\left\\{\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-\frac{3}{2}}}>\epsilon\right\\}$
$\displaystyle\leq\,$
$\displaystyle\frac{2C(R,T,u_{0})}{K^{2}}+\mathbb{P}\left\\{\sup_{t\in[0,\tau^{T}_{\varepsilon,\eta,K}]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-\frac{3}{2}}}>\epsilon\right\\}.$

Now (3.12) clearly forces

$\lim_{\varepsilon\rightarrow
0}\sup_{\eta\leq\varepsilon}\mathbb{P}\left\\{\sup_{t\in[0,T]}\|u_{\varepsilon}-u_{\eta}\|_{H^{s-3/2}}>\epsilon\right\\}\leq\frac{2C(R,T,u_{0})}{K^{2}},\
\ K>1.$

Letting $K\rightarrow\infty$, we see that $u_{\varepsilon}$ converges in
probability in $C([0,T];H^{s-3/2})$. Therefore, up to a further subsequence,
(3.19) holds true.

Now we prove (3.18). Indeed, since $H^{s}\hookrightarrow H^{s-3/2}$ is
continuous, there exist continuous maps $\pi_{m}:H^{s-3/2}\to H^{s},\ m\geq 1$
such that

$\|\pi_{m}u\|_{H^{s}}\leq\|u\|_{H^{s}}\ \
\lim_{m\rightarrow\infty}\|\pi_{m}u\|_{H^{s}}=\|u\|_{H^{s}},\ \ u\in
H^{s-3/2},$

where $\|u\|_{H^{s}}\triangleq\infty$ if $u\notin H^{s}.$ For example, one may
take $\pi_{m}$ as the standard mollifier. Then it follows from Proposition 3.1
and Fatou’s lemma that

$\displaystyle\mathbb{E}\sup_{t\in[0,T]}\|u(t)\|^{2}_{H^{s}}\leq$
$\displaystyle\liminf_{m\rightarrow\infty}\mathbb{E}\sup_{t\in[0,T]}\|\pi_{m}u(t)\|^{2}_{H^{s}}$
$\displaystyle\leq$
$\displaystyle\liminf_{m\rightarrow\infty}\liminf_{\varepsilon\rightarrow
0}\mathbb{E}\sup_{t\in[0,T]}\|\pi_{m}u_{\varepsilon}(t)\|^{2}_{H^{s}}$
$\displaystyle\leq$
$\displaystyle\liminf_{m\rightarrow\infty}\liminf_{\varepsilon\rightarrow
0}\mathbb{E}\sup_{t\in[0,T]}\|u_{\varepsilon}(t)\|^{2}_{H^{s}}<C(R,u_{0},T).$

Hence (3.18) holds true. ∎

###  3.3. Global pathwise solution to the cut-off problem

######  Proposition 3.2.

Let $\mathcal{S}=(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathcal{W})$ be a stochastic basis fixed in advance. Let $s>3$, $R>1$ and
$\varepsilon\in(0,1)$. Assume Assumption (A) is satisfied. Let $u_{0}\in
L^{2}(\Omega;H^{s})$ be an $H^{s}$-valued $\mathcal{F}_{0}$ measurable random
variable. Then for any $T>0$, (3.1) has a solution $u\in
L^{2}\left(\Omega;C\left([0,T];H^{s}\right)\right)$. Moreover, there is a
constant $C(R,T,u_{0})>0$ such that

$\mathbb{E}\sup_{t\in[0,T]}\|u\|^{2}_{H^{s}}\leq C(R,T,u_{0}).$

###### Proof.

Since for each $\varepsilon\in(0,1)$, $u_{\varepsilon}$ is
$\\{\mathcal{F}_{t}\\}_{t\geq 0}$ progressive measurable, so is $u$. By Lemma
3.3 and the embedding $H^{s-3/2}\hookrightarrow W^{1,\infty}$, we can send
$\varepsilon\rightarrow 0$ in (3.2) to conclude that $u$ solves (3.1). Now we
only need to prove (1.14). Due to Lemma 3.3, $u\in C([0,T];H^{s-3/2})\cap
L^{\infty}\left(0,T;H^{s}\right)$ almost surely. Since $H^{s}$ is dense in
$H^{s-3/2}$, we see that ([59, page 263, Lemma 1.4]) $u\in
C_{w}\left([0,T];H^{s}\right)$, where $C_{w}\left([0,T];H^{s}\right)$ is the
space of weakly continuous functions with values in $H^{s}$. Therefore to
prove (1.14), we only need to prove the continuity of $[0,T]\ni
t\mapsto\|u(t)\|_{H^{s}}$.

However, we cannot directly apply the Itô formula for $\|u\|^{2}_{H^{s}}$ to
get control of $\mathbb{E}\|u(t)\|_{H^{s}}^{2}$ because we only have $u\in
H^{s}$ and $(\mathcal{H}u)u_{x}\in H^{s-1}$. Indeed, the Itô formula in a
Hilbert space ([12, Theorem 4.32] or [21, Theorem 2.10]) requires
$\left((\mathcal{H}u)u_{x},u\right)_{H^{s}}$ to be well-defined and the Itô
formula under a Gelfand triplet ([41, Theorem I.3.1] or [48, Theorem 4.2.5])
requires the dual product
${}_{H^{s-1}}\langle(\mathcal{H}u)u_{x},u\rangle_{H^{s+1}}$ to be well-
defined. In our case neither of them is satisfied. To this end, we recall the
mollifier $J_{\varepsilon}$ defined in Section 2 and apply the Itô formula to
$\|J_{\varepsilon}u\|^{2}_{H^{s}}$ to obtain

$\displaystyle{\rm d}\|J_{\varepsilon}u(t)\|^{2}_{H^{s}}=$ $\displaystyle
2\chi_{R}(\|u\|_{W^{1,\infty}})\left(J_{\varepsilon}h(t,u){\rm
d}\mathcal{W},J_{\varepsilon}u\right)_{H^{s}}$
$\displaystyle-2\chi_{R}(\|u\|_{W^{1,\infty}})\left(J_{\varepsilon}[(\mathcal{H}u)u_{x}],J_{\varepsilon}u\right)_{H^{s}}{\rm
d}t$
$\displaystyle+\chi^{2}_{R}(\|u\|_{W^{1,\infty}})\|J_{\varepsilon}h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}^{2}{\rm
d}t.$ (3.20)

By (3.18),

$\tau_{N}=\inf\\{t\geq 0:\|u(t)\|_{H^{s}}>N\\}\rightarrow\infty\ \text{as}\
N\rightarrow\infty\ \ \mathbb{P}-a.s.$ (3.21)

Then we only need to prove the continuity up to time $\tau_{N}\wedge T$ for
each $N\geq 1$. We first notice that $J_{\varepsilon}$ satisfies (2.3), (2.6)
and (2.7). Therefore for any $[t_{2},t_{1}]\subset[0,T]$ with $t_{1}-t_{2}<1$,
we use Lemma 2.3, the BDG inequality and Assumption (A) and (3.21) to find

$\displaystyle\mathbb{E}\left[\left(\|J_{\varepsilon}u(t_{1}\wedge\tau_{N})\|^{2}_{H^{s}}-\|J_{\varepsilon}u(t_{2}\wedge\tau_{N})\|^{2}_{H^{s}}\right)^{4}\right]\leq$
$\displaystyle C(N,T)|t_{1}-t_{2}|^{2}.$

Using Fatou’s lemma, we arrive at

$\displaystyle\mathbb{E}\left[\left(\|u(t_{1}\wedge\tau_{N})\|^{2}_{H^{s}}-\|u(t_{2}\wedge\tau_{N})\|^{2}_{H^{s}}\right)^{4}\right]\leq$
$\displaystyle C(N,T)|t_{1}-t_{2}|^{2}.$

This and Kolmogorov’s continuity theorem ensure the continuity of
$t\mapsto\|u(t\wedge\tau_{N})\|_{H^{s}}$, completing the proof. ∎

###  3.4. Concluding the proof of Theorem 1.1

Finally, we are in the position to finish the proof of Theorem 1.1. For the
sake of clarity, we split the proof into three steps.

Step 1: Existence. For $u_{0}(\omega,x)\in L^{2}(\Omega;H^{s})$, we let

$\Omega_{k}=\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\},\ k\in\mathbb{N},\ k\geq 1.$

Since $\mathbb{E}\|u_{0}\|^{2}_{H^{s}}<\infty$, we have

$\displaystyle u_{0}(\omega,x)$ $\displaystyle=\sum_{k\geq
1}u_{0,k}(\omega,x)=\sum_{k\geq
1}u_{0}(\omega,x)\textbf{1}_{\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\}}\ \
\mathbb{P}-a.s.$

On account of Proposition 3.2, we let $u_{k,R}$ be the pathwise global
solution to the cut-off problem (3.1) with initial value $u_{0,k}$ and cut-off
function $\chi_{R}(\cdot)$. Define

$\tau_{k,R}=\inf\left\\{t>0:\sup_{t^{\prime}\in[0,t]}\|u_{k,R}(t^{\prime})\|^{2}_{H^{s}}>\|u_{0,k}\|^{2}_{H^{s}}+2\right\\}.$
(3.22)

Then for any $R>0$ and $k\in\mathbb{N}$, we have
$\mathbb{P}\\{\tau_{k,R}>0\\}=1$. The difficulty here is that we have to take
$R$ to be deterministic. Otherwise Proposition 3.1 will fail. To overcome this
difficulty, we let $R=R_{k}$ be discrete and then denote
$(u_{k},\tau_{k})=(u_{k,R_{k}},\tau_{k,R_{k}})$. It is clear that
$\mathbb{P}\\{\tau_{k}>0,\ \forall\ k\geq 1\\}=1$. Let $E>0$ be the embedding
constant such that $\|\cdot\|_{W^{1,\infty}}\leq E\|\cdot\|_{H^{s}}$ for
$s>3$. Particularly, we take $R^{2}_{k}>E^{2}\|u_{0,k}\|^{2}_{H^{s}}+2E^{2}$,
and then we have

$\mathbb{P}\left\\{\|u_{k}\|^{2}_{W^{1,\infty}}\leq
E^{2}\|u_{k}\|^{2}_{H^{s}}\leq E^{2}\|u_{0,k}\|^{2}_{H^{s}}+2E^{2}<R^{2}_{k},\
\forall\ t\in[0,\tau_{k}],\ \forall\ k\geq 1\right\\}=1,$

which means

$\mathbb{P}\left\\{\chi_{R_{k}}(\|u_{k}\|_{W^{1,\infty}})=1,\ \forall\
t\in[0,\tau_{k}],\ \forall\ k\geq 1\right\\}=1.$

Therefore $(u_{k},\tau_{k})$ is the pathwise solution to (1.2) with initial
value $u_{0,k}$. Notice that

$\displaystyle\textbf{1}_{\Omega_{k}}u_{k}(t\wedge\tau_{k})-\textbf{1}_{\Omega_{k}}u_{0,k}=$
$\displaystyle\int_{0}^{t\wedge\textbf{1}_{\Omega_{k}}\tau_{k}}\textbf{1}_{\Omega_{k}}[(\mathcal{H}u_{k})\partial_{x}u_{k}]\,{\rm
d}t^{\prime}+\int_{0}^{t\wedge\textbf{1}_{\Omega_{k}}\tau_{k}}\textbf{1}_{\Omega_{k}}h(t,u_{k}){\rm
d}\mathcal{W},$
$\textbf{1}_{\Omega_{k}}h(t,u_{k})=h(t,\textbf{1}_{\Omega_{k}}u_{k})-\textbf{1}_{\Omega^{C}_{k}}h(t,0)$

and

$\textbf{1}_{\Omega_{k}}[(\mathcal{H}u_{k})\partial_{x}u_{k}]=(\mathcal{H}\textbf{1}_{\Omega_{k}}u_{k})\partial_{x}\textbf{1}_{\Omega_{k}}u_{k}.$

By Assumption (A), we have
$\|h(t,\mathbf{0})\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}<\infty$. Then we
have

$\displaystyle\textbf{1}_{\Omega_{k}}u_{k}(t\wedge\tau_{k})-\textbf{1}_{\Omega_{k}}u_{0,k}$
$\displaystyle=$
$\displaystyle\textbf{1}_{\Omega_{k}}u_{k}(t\wedge\textbf{1}_{\Omega_{k}}\tau_{k})-u_{0,k}$
$\displaystyle=$
$\displaystyle-\int_{0}^{t\wedge\textbf{1}_{\Omega_{k}}\tau_{k}}(\mathcal{H}\textbf{1}_{\Omega_{k}}u_{k})\partial_{x}(\textbf{1}_{\Omega_{k}}u_{k})\,{\rm
d}t^{\prime}+\int_{0}^{t\wedge\textbf{1}_{\Omega_{k}}\tau_{k}}h(t,\textbf{1}_{\Omega_{k}}u_{k}){\rm
d}\mathcal{W}.$

Therefore $(\textbf{1}_{\Omega_{k}}u_{k},\textbf{1}_{\Omega_{k}}\tau_{k})$ is
a solution to (1.2) with initial data $u_{0,k}$. Since
$\Omega_{k}\bigcap\Omega_{k^{\prime}}=\emptyset$ for $k\neq k^{\prime}$ and
$\bigcup_{k\geq 1}\Omega_{k}$ is a set of full measure, we see that

$\left(u=\sum_{k\geq 1}\textbf{1}_{\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\}}u_{k},\ \
\tau=\sum_{k\geq
1}\textbf{1}_{\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\}}\tau_{k}\right)$

is a pathwise solution to (1.2) corresponding to the initial condition
$u_{0}$. Besides, using (3.22), we have

$\displaystyle\sup_{t\in[0,\tau]}\|u\|_{H^{s}}^{2}=$ $\displaystyle\sum_{k\geq
1}\textbf{1}_{\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\}}\sup_{t\in[0,\tau_{k}]}\|u_{k}\|_{H^{s}}^{2}$
$\displaystyle\leq$ $\displaystyle\sum_{k\geq
1}\textbf{1}_{\\{k-1\leq\|u_{0}\|_{H^{s}}<k\\}}\left(\|u_{0,k}\|^{2}_{H^{s}}+2\right)\leq
2\|u_{0}\|^{2}_{H^{s}}+4.$

Taking expectation gives rise to (1.14).

Step 2: Uniqueness and maximal pathwise solution. With $(u,\tau)$ in hand, we
can extend $(u,\tau)$ to a maximal pathwise solution in the sense of
Definition 1.1 by following the techniques as in [9, 24, 23, 50]. For
uniqueness, we let $(u_{1},\tau_{1})$ and $(u_{2},\tau_{2})$ be two solutions
to (1.2) such that $u_{j}(0)=u_{0}$ almost surely and
$u_{j}(\cdot\wedge\tau_{j})\in
L^{2}\left(\Omega;C\left([0,\infty);H^{s}\right)\right)$ with $s>3$ for
$j=1,2$. Let $\frac{1}{2}<\delta<s-1$ and define

$\tau^{T}_{K}=\inf\left\\{t\geq
0:\|u_{1}(t)\|_{H^{s}}+\|u_{2}(t)\|_{H^{s}}\geq K\right\\}\wedge T,\ \
K\in\mathbb{N},\ T>0.$

Using (1.8) and the definition of $\tau^{T}_{K}$, then the estimate of
$\mathbb{E}\sup_{t\in[0,\tau^{T}_{K}]}\|u_{1}(t)-u_{2}(t)\|^{2}_{H^{\delta}}$
is essential as in the derivation of (3.12) and we have

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau^{T}_{K}]}\|u_{1}(t)-u_{2}(t)\|^{2}_{H^{\delta}}=0.$

If necessary, to guarantee $\tau^{T}_{K}>0$ almost surely, we can first assume
$u_{0}\in L^{\infty}(\Omega;H^{s})$ and then remove this restriction by using
the techniques as in Step 1. Hence we obtain uniqueness and the details are
omitted here for brevity.

Step 3: Blow-up criterion. We first define

$\displaystyle\tau_{1,m}=\inf\left\\{t\geq 0:\|u(t)\|_{H^{s}}\geq m\right\\},\
\ \ \tau_{2,n}=\inf\left\\{t\geq
0:\|u_{x}(t)\|_{L^{\infty}}+\|\mathcal{H}u_{x}(t)\|_{L^{\infty}}\geq
n\right\\},$

and then let $\displaystyle\tau_{1}=\lim_{m\rightarrow\infty}\tau_{1,m}$ and
$\displaystyle\tau_{2}=\lim_{n\rightarrow\infty}\tau_{2,n}$. We notice that
for fixed $m,n>0$, even if $\mathbb{P}\\{\tau_{1,m}=0\\}$ or
$\mathbb{P}\\{\tau_{2,n}=0\\}$ is larger than $0$, for a.e. $\omega\in\Omega$,
there is $m>0$ or $n>0$ such that $\tau_{1,m},\tau_{2,n}>0$. By continuity of
$\|u(t)\|_{H^{s}}$ and the uniqueness of $u$, it is easy to check that
$\tau_{1}=\tau_{2}$ is actually the maximal existence time $\tau^{*}$ of $u$
in the sense of Definition 1.1. Therefore to prove (1.15), we only need to
verify that

$\tau_{1}=\tau_{2}\ \ \mathbb{P}-a.s.$ (3.23)

The approach here is motivated by [9, 3]. Since $H^{s}\hookrightarrow
W^{1,\infty}$ and $\mathcal{H}$ is continuous in $H^{s}$ (cf. (2.7)), there
exists a constant $M>0$ such that,

$\displaystyle\sup_{t\in[0,\tau_{1,m}]}\left(\|u_{x}(t)\|_{L^{\infty}}+\|\mathcal{H}u_{x}\|_{L^{\infty}}\right)\leq
M\sup_{t\in[0,\tau_{1,m}]}\|u(t)\|_{H^{s}}\leq([M]+1)m,$

where $[M]$ denotes the integer part of $M$. Therefore we have
$\tau_{1,m}\leq\tau_{2,([M]+1)m}\leq\tau_{2}$ $\mathbb{P}-a.s.,$ which means
that $\tau_{1}\leq\tau_{2}$ $\mathbb{P}-a.s.$ Now we only need to prove
$\tau_{2}\leq\tau_{1}$ $\mathbb{P}-a.s.$ We do the following claim:

Claim:

$\displaystyle\mathbb{P}\left\\{\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|u(t)\|_{H^{s}}<\infty\right\\}=1,\ \ \forall\
n_{1},n_{2}\in\mathbb{N}.$ (3.24)

As is explained before, we cannot directly apply the Itô formula for
$\|u\|^{2}_{H^{s}}$ to get control of $\mathbb{E}\|u(t)\|_{H^{s}}^{2}$.
Similar to (3.20), by applying $J_{\varepsilon}$ to (1.2) and using the Itô
formula for $\|J_{\varepsilon}u\|^{2}_{H^{s}}$, we have that for any $t>0$,

$\displaystyle{\rm d}\|J_{\varepsilon}u(t)\|^{2}_{H^{s}}=$
$\displaystyle\left(J_{\varepsilon}h(t,u){\rm
d}\mathcal{W},J_{\varepsilon}u\right)_{H^{s}}-2\left(D^{s}J_{\varepsilon}\left[(\mathcal{H}u)u_{x}\right],D^{s}J_{\varepsilon}u\right)_{L^{2}}\,{\rm
d}t+\|J_{\varepsilon}h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}^{2}\,{\rm
d}t.$ (3.25)

By the BDG inequality, we have

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|J_{\varepsilon}u(t)\|^{2}_{H^{s}}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\|J_{\varepsilon}u_{0}\|^{2}_{H^{s}}+C\mathbb{E}\left(\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}\|J_{\varepsilon}h(t,u)\|^{2}_{\mathcal{L}_{2}(\mathcal{U};H^{s})}\|J_{\varepsilon}u\|^{2}_{H^{s}}{\rm
d}t\right)^{\frac{1}{2}}$
$\displaystyle+2\mathbb{E}\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}|\left(D^{s}J_{\varepsilon}\left[(\mathcal{H}u)u_{x}\right],D^{s}J_{\varepsilon}u\right)_{L^{2}}|{\rm
d}t+\mathbb{E}\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}\|J_{\varepsilon}h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}^{2}{\rm
d}t.$

Then (1.7) and (2.7) lead to

$\displaystyle C\mathbb{E}\left(\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}\|J_{\varepsilon}h(t,u)\|^{2}_{\mathcal{L}_{2}(\mathcal{U};H^{s})}\|J_{\varepsilon}u\|^{2}_{H^{s}}{\rm
d}t\right)^{\frac{1}{2}}$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|J_{\varepsilon}u\|_{H^{s}}^{2}+Cf^{2}(2n_{1})\int_{0}^{n_{2}}\left(1+\mathbb{E}\|u\|_{H^{s}}^{2}\right){\rm
d}t.$

By Lemma 2.3, we find

$\displaystyle 2\mathbb{E}\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}|\left(D^{s}J_{\varepsilon}\left[(\mathcal{H}u)u_{x}\right],D^{s}J_{\varepsilon}u\right)_{L^{2}}|{\rm
d}t\leq Cn_{1}\int_{0}^{n_{2}}\left(1+\mathbb{E}\|u\|_{H^{s}}^{2}\right){\rm
d}t.$ (3.26)

It follows from (1.7) that for some constant $C>0$,

$\displaystyle\mathbb{E}\int_{0}^{\tau_{2,n_{1}}\wedge
n_{2}}\|J_{\varepsilon}h(t,u)\|_{\mathcal{L}_{2}(\mathcal{U};H^{s})}^{2}\,{\rm
d}t\leq
Cf^{2}(2n_{1})\int_{0}^{n_{2}}\left(1+\mathbb{E}\|u\|_{H^{s}}^{2}\right){\rm
d}t,$

Therefore we combine the above estimates, use (2.7), and then send
$\varepsilon\rightarrow 0$ in the resulting inequality to obtain

$\displaystyle\mathbb{E}\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|u(t)\|^{2}_{H^{s}}\leq
C\mathbb{E}\|u_{0}\|^{2}_{H^{s}}+C\int_{0}^{n_{2}}\left(1+\mathbb{E}\sup_{t^{\prime}\in[0,t\wedge\tau_{2,n_{1}}]}\|u(t^{\prime})\|_{H^{s}}^{2}\right){\rm
d}t.$ (3.27)

Then Grönwall’s inequality shows that for each $n_{1},n_{2}\in\mathbb{N}$,
there is a constant $C=C(n_{1},n_{2},u_{0})>0$ such that

$\mathbb{E}\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|u(t)\|^{2}_{H^{s}}<C(n_{1},n_{2},u_{0}),$

which gives (3.24) and concludes the claim.

Hence (3.24) implies that for all $n_{1},n_{2}\in\mathbb{N}$,
$\mathbb{P}\left\\{\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|u(t)\|_{H^{s}}<\infty\right\\}=1$. On the other hand, it is easy to
see that for all $n_{1},n_{2}\in\mathbb{N}$,

$\displaystyle\left\\{\sup_{t\in[0,\tau_{2,n_{1}}\wedge
n_{2}]}\|u(t)\|_{H^{s}}<\infty\right\\}\subset\bigcup_{m\in\mathbb{N}}\left\\{\tau_{2,n_{1}}\wedge
n_{2}\leq\tau_{1,m}\right\\}\subset\left\\{\tau_{2,n_{1}}\wedge
n_{2}\leq\tau_{1}\right\\}.$

Consequently,

$\displaystyle\mathbb{P}\left\\{\tau_{2}\leq\tau_{1}\right\\}=\mathbb{P}\left\\{\bigcap_{n_{1}\in\mathbb{N}}\left\\{\tau_{2,n_{1}}\leq\tau_{1}\right\\}\right\\}=\mathbb{P}\left\\{\bigcap_{n_{1},n_{2}\in\mathbb{N}}\left\\{\tau_{2,n_{1}}\wedge
n_{2}\leq\tau_{1}\right\\}\right\\}=1.$ (3.28)

Combining the above three steps, we complete the proof of Theorem 1.1.

##  4\. Proof of Theorem 1.2

To begin with, we can follow the steps as in the proof of Theorem 1.1 to
obtain that, if $u_{0}$ is an $H^{s}$-valued $\mathcal{F}_{0}$-measurable
random variable satisfying $\mathbb{E}\|u_{0}\|^{2}_{H^{s}}<\infty$ with
$s>3$, then (1.3) has a unique pathwise solution $u\in H^{s}$ with maximal
existence time $\tau^{*}$. Now the target is to show that
$\mathbb{P}\\{\tau^{*}=\infty\\}=1$. To this end, we define

$\displaystyle\widehat{\tau}_{k}=\inf\left\\{t\geq 0:\|u(t)\|_{H^{s-1}}\geq
k\right\\},\ \ k\geq 1\ \ \text{and}\ \
\widehat{\tau^{*}}=\lim_{k\rightarrow\infty}\widehat{\tau}_{k}.$

Recalling Remark 1.1, we have

$\widehat{\tau^{*}}=\tau^{*}\ \ \mathbb{P}-a.s.$ (4.1)

Therefore we only need to show $\mathbb{P}\\{\widehat{\tau^{*}}=\infty\\}=1$.
Applying the Itô formula to $\|u(t)\|^{2}_{H^{s-1}}$ gives

$\displaystyle{\rm d}\|u\|^{2}_{H^{s-1}}=\,$ $\displaystyle
2\left(\alpha(t,u),u\right)_{H^{s-1}}\,{\rm
d}W-2\left((\mathcal{H}u)u_{x},u\right)_{H^{s-1}}\,{\rm
d}t+\|\alpha(t,u)\|^{2}_{H^{s-1}}\,{\rm d}t.$ (4.2)

Let $\mathcal{G}\in\mathfrak{G}$. On account of the Itô formula, we derive

$\displaystyle{\rm d}\mathcal{G}(\|u\|^{2}_{H^{s-1}})=\,$ $\displaystyle
2\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left(\alpha(t,u),u\right)_{H^{s-1}}\,{\rm
d}W$
$\displaystyle+\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left\\{-2\left((\mathcal{H}u)u_{x},u\right)_{H^{s-1}}+\|\alpha(t,u)\|^{2}_{H^{s-1}}\right\\}\,{\rm
d}t$
$\displaystyle+2\mathcal{G}^{\prime\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|^{2}\,{\rm
d}t.$

Recall that in Assumption (B),

$\mathcal{M}(t)=2Q(\|u_{x}(t)\|_{L^{\infty}}+\|\mathcal{H}u_{x}(t)\|_{L^{\infty}})\|u(t)\|^{2}_{H^{s-1}}+\|\alpha(t,u)\|^{2}_{H^{s-1}}.$

Hence taking expectation, using inequality (2.7), Lemma 2.3 and Assumption (B)
we find that for any $t>0$,

$\displaystyle\mathbb{E}\mathcal{G}(\|u(t)\|^{2}_{H^{s-1}})$
$\displaystyle=\,$
$\displaystyle\mathbb{E}\mathcal{G}(\|u_{0}\|^{2}_{H^{s-1}})+\mathbb{E}\int_{0}^{t}\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left\\{-2\left((\mathcal{H}u)u_{x},u\right)_{H^{s-1}}+\|\alpha(t^{\prime},u)\|^{2}_{H^{s-1}}\right\\}\,{\rm
d}t^{\prime}$
$\displaystyle+\mathbb{E}\int_{0}^{t}2\mathcal{G}^{\prime\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t^{\prime},u),u\right)_{H^{s-1}}\right|^{2}\,{\rm
d}t^{\prime}$ $\displaystyle\leq\,$
$\displaystyle\mathbb{E}\left\\{\mathcal{G}(\|u_{0}\|^{2}_{H^{s-1}})+\mathbb{E}\int_{0}^{t}\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\mathcal{M}(t^{\prime})+2\mathcal{G}^{\prime\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t^{\prime},u),u\right)_{H^{s-1}}\right|^{2}\right\\}\,{\rm
d}t^{\prime}$ $\displaystyle\leq\,$
$\displaystyle\mathbb{E}\mathcal{G}(\|u_{0}\|^{2}_{H^{s-1}})+K_{1}t-\mathbb{E}\int_{0}^{t}K_{2}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t^{\prime},u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t^{\prime},$ (4.3)

which shows that there exists a constant $C=C(u_{0},K_{1},K_{2},t)>0$ such
that

$\displaystyle\mathbb{E}\int_{0}^{t}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t^{\prime},u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t^{\prime}\leq C(u_{0},K_{1},K_{2},t).$ (4.4)

Moreover, for any $T>0$, it follows from Assumption (B) and the BDG inequality
that

$\displaystyle\mathbb{E}\sup_{t\in[0,{T}]}\mathcal{G}(\|u\|^{2}_{H^{s-1}})$
$\displaystyle\leq\,$
$\displaystyle\mathbb{E}\mathcal{G}(\|u_{0}\|^{2}_{H^{s-1}})+C\mathbb{E}\left(\int_{0}^{T}\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}\,{\rm
d}t\right)^{\frac{1}{2}}$
$\displaystyle+K_{1}T+K_{2}\mathbb{E}\int_{0}^{T}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t$ $\displaystyle\leq\,$
$\displaystyle\mathbb{E}\mathcal{G}(\|u_{0}\|^{2}_{H^{s}})+\frac{1}{2}\mathbb{E}\sup_{t\in[0,{T}]}\left(1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})\right)+C\mathbb{E}\int_{0}^{T}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t$
$\displaystyle+K_{1}T+K_{2}\mathbb{E}\int_{0}^{T}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t.$

Thus, using (4.4) we obtain

$\displaystyle\mathbb{E}\sup_{t\in[0,{T}]}\mathcal{G}(\|u\|^{2}_{H^{s-1}})$
$\displaystyle\leq\,C(u_{0},K_{1},T)+C(K_{2})\mathbb{E}\int_{0}^{T}\frac{\left\\{\mathcal{G}^{\prime}(\|u\|^{2}_{H^{s-1}})\left|\left(\alpha(t,u),u\right)_{H^{s-1}}\right|\right\\}^{2}}{1+\mathcal{G}(\|u\|^{2}_{H^{s-1}})}\,{\rm
d}t$ $\displaystyle\leq\,C(u_{0},K_{1},K_{2},T).$

We can infer from the above estimate that

$\mathbb{P}\\{\widehat{\tau^{*}}<T\\}\leq\mathbb{P}\\{\widehat{\tau}_{k}<T\\}\leq\mathbb{P}\left\\{\sup_{t\in[0,T]}\mathcal{G}(\|u\|^{2}_{H^{s-1}})\geq\mathcal{G}(k^{2})\right\\}\leq\frac{C(u_{0},K_{1},K_{2},T)}{\mathcal{G}(k^{2})}.$

Therefore, since $\lim_{x\rightarrow\infty}\mathcal{G}(x)=\infty$, one can
send $k\rightarrow\infty$ to identify that
$\mathbb{P}\\{\widehat{\tau^{*}}<T\\}=0$. Since $T>0$ is arbitrary, we have
that $\mathbb{P}\\{\widehat{\tau^{*}}=\infty\\}=1$ which shows the desired
assertion.

######  Remark 4.1.

We remark that the using of Lyapunov is motivated by the non-explosion text
[36], see also [7, 49, 51]. In the above proof, (2.8) is used to obtain (4.3),
but we remark that (2.9) is also used implicitly. Indeed, as in the proof of
(3.23), (2.9) is used to obtain (3.26), and here (4.1) also requires (2.9)
because it is a consequence of (3.23). In this work we estimate $H^{s-1}$ norm
(i.e., $\mathbb{E}\sup_{t\in[0,{T}]}\mathcal{G}(\|u\|^{2}_{H^{s-1}})$) and use
the fact (4.1) to prove global existence. Let us stress that this is different
from the recent work [51], where the authors estimate $H^{s}$ norm of the
solution to show global existence.

##  5\. Proof of Theorem 1.3

Due to the linear nature of the noise we use the Girsanov type transformation

$v=\frac{1}{\beta(\omega,t)}u,\ \ \beta(\omega,t)={\rm
e}^{\int_{0}^{t}b(t^{\prime}){\rm
d}W_{t^{\prime}}-\int_{0}^{t}\frac{b^{2}(t^{\prime})}{2}{\rm d}t^{\prime}}.$
(5.1)

In the following lemma we show that the defined process $v$ is the solution to
a random PDE enjoying desired regularity properties.

######  Lemma 5.1.

Let $s>3$, $b(t)$ satisfies assumption (C) and fix
$\mathcal{S}=\left(\Omega,\mathcal{F},\mathbb{P},\\{\mathcal{F}_{t}\\}_{t\geq
0},W\right)$. Assume that $u_{0}(\omega,x)$ is an $H^{s}$-valued
$\mathcal{F}_{0}$-measurable random variable with
$\mathbb{E}\|u_{0}\|^{2}_{H^{s}}<\infty$ and $(u,\tau^{*})$ is the
corresponding unique maximal pathwise solution to (1.4). Then for
$t\in[0,\tau^{*})$, the process $v$ as in (5.1) is a solution
$\mathbb{P}-a.s.$ to

$\left\\{\begin{aligned}
&v_{t}+\beta(\mathcal{H}v)v_{x}=0,\quad\beta(\omega,t)={\rm
e}^{\int_{0}^{t}b(t^{\prime}){\rm
d}W_{t^{\prime}}-\int_{0}^{t}\frac{b^{2}(t^{\prime})}{2}{\rm d}t^{\prime}},\\\
&v(\omega,0,x)=u_{0}(\omega,x),\quad x\in\mathbb{R}\end{aligned}\right.$ (5.2)

with $v\in C\left([0,\tau^{*});H^{s}\right)\bigcap
C^{1}([0,\tau^{*});H^{s-1})$ $\mathbb{P}-a.s.$

###### Proof.

Applying Theorem 1.1 for the particular case $h(t,u)=b(t)u$ and noticing that
$b(t)$ satisfies Assumption (C) (and therefore $h(t,u)=b(t)u$ satisfies
Assumption (A)), we infer that equation (1.4) has a unique maximal pathwise
solution $(u,\tau^{*})$. Itô’s formula yields

${\rm d}\frac{1}{\beta}=-b(t)\frac{1}{\beta}{\rm
d}W+b^{2}(t)\frac{1}{\beta}{\rm d}t,$

and hence straightforward computation shows that

${\rm d}v=-\beta(\mathcal{H}v)v_{x}{\rm d}t,$ (5.3)

yielding the first equation in (5.2). At time $t=0$,
$v(\omega,0,x)=u_{0}(\omega,x)$ since $\beta(\omega,0)=1$ almost surely, so
$v$ satisfies (5.2) almost surely. Furthermore, Theorem 1.1 shows that $u\in
C\left([0,\tau^{*});H^{s}\right)$ $\mathbb{P}-a.s.$, hence $v\in
C\left([0,\tau^{*});H^{s}\right)$ and $v\in
C^{1}\left([0,\tau^{*});H^{s-1}\right)$ $\mathbb{P}-a.s.$. ∎

Invoking Lemma 5.1 we have that a.e. $\omega\in\Omega$ the process
$v(\omega,t,x)$ solves (5.2) on $[0,\tau^{*})$ and $v\in
C\left([0,\tau^{*});H^{s}\right)\bigcap C^{1}([0,\tau^{*});H^{s-1})$ for
$s>3$. In particular, by the Sobolev embedding, $v\in
C\left([0,\tau^{*});C^{1}\right)$, therefore for a.e. $\omega\in\Omega$, the
particle trajectory mapping related to the process $v$ given by

$\left\\{\begin{aligned} &\frac{{\rm d}\phi(\omega,t,x)}{{\rm
d}t}=\beta(\omega,t)\mathcal{H}v(\omega,t,\phi(\omega,t,x)),\quad
t\in[0,\tau^{*}),\\\ &\phi(\omega,0,x)=x,\quad
x\in\mathbb{R},\end{aligned}\right.$ (5.4)

has a unique solution $\phi(\omega,t,x)\in
C^{1}([0,\tau^{*})\times\mathbb{R})$. Now for a.e. $\omega\in\Omega$, we let
$x_{0}=x_{0}(\omega)\in\mathbb{R}$ be the point that $u_{0}$ attains it global
maximum, i.e.,

$v(\omega,0,x_{0}(\omega))=u_{0}(\omega,x_{0}(\omega))=\displaystyle\max_{x\in\mathbb{R}}u_{0}(\omega,x),\mbox{
for a.e. }\omega\in\Omega.$

Then we focus on the particle trajectory mapping from $x_{0}$ in (5.4), i.e.,

$\left\\{\begin{aligned} &\frac{{\rm d}\phi(\omega,t,x_{0})}{{\rm
d}t}=\beta(\omega,t)\mathcal{H}v(\omega,t,\phi(\omega,t,x_{0})),\quad
t\in[0,\tau^{*}),\\\ &\phi(\omega,0,x_{0})=x_{0}.\end{aligned}\right.$ (5.5)

On the other hand, by the transport nature of equation (5.2) the value of $v$
is constant along characteristics and since $v(\omega,0,x_{0})$ attains a
global maximum, we have that

$\partial_{x}v(\omega,t,\phi(\omega,t,x_{0}))=0,\ \ t\in[0,\tau^{*})\ \
\mathbb{P}-a.s.$ (5.6)

Computing the quantity $\Lambda v(\omega,t,\phi(\omega,t,x_{0}))$, i.e., the
evolution of the $\Lambda$ operator of $v$ along the trajectory, we see that
for a.e. $\omega\in\Omega$ and $t\in[0,\tau^{*})$, there holds

$\displaystyle\frac{{\rm d}\Lambda v(\omega,t,\phi(\omega,t,x_{0}))}{{\rm
d}t}$ $\displaystyle=\,$ $\displaystyle\Lambda
v_{t}(\omega,t,\phi(\omega,t,x_{0}))+\frac{{\rm d}\Lambda
v(\omega,t,\phi(\omega,t,x_{0}))}{{\rm d}\phi(\omega,t,x_{0}))}\frac{{\rm
d}\phi(\omega,t,x_{0})}{{\rm d}t}$ $\displaystyle=\,$
$\displaystyle-\beta(\omega,t)\left\\{\Lambda[(\mathcal{H}v)v_{x}](\omega,t,\phi(\omega,t,x_{0}))-(\mathcal{H}v)(\omega,t,\phi(\omega,t,x_{0}))\Lambda
v_{x}(\omega,t,\phi(\omega,t,x_{0}))\right\\},$ (5.7)

where chain rule is used in the former equality and the fact that $v$ solves
(5.2) and the particle trajectory equation (5.5) in the latter. Let
$\tilde{v}=\mathcal{H}v$. Then we have $v_{x}=\Lambda\tilde{v}$ and $\Lambda
v_{x}=-\tilde{v}_{xx}$ and then we can rewrite equation (5.7) as

$\displaystyle\frac{{\rm d}\Lambda v(\omega,t,\phi(\omega,t,x_{0}))}{{\rm
d}t}$ $\displaystyle=$
$\displaystyle-\beta(\omega,t)\left[\Lambda(\tilde{v}\Lambda\tilde{v})(\omega,t,\phi(\omega,t,x_{0}))+\tilde{v}(\omega,t,\phi(\omega,t,x_{0}))\tilde{v}_{xx}(\omega,t,\phi(\omega,t,x_{0}))\right].$
(5.8)

Now we denote $z_{0}\triangleq\phi(\omega,t,x_{0})$ and omit the dependence of
$\omega$ and $t$ in (5.8) for simplicity if there is no ambiguity.

###  5.1. An identity for the fractional Laplacian $\Lambda$

######  Lemma 5.2.

For a.e. $\omega\in\Omega$ and $t\in[0,\tau^{*})$, there holds the following
equation

$\displaystyle\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=-\frac{1}{2}(\Lambda
v(z_{0}))^{2}-\frac{1}{\pi}\left\lVert\frac{\mathcal{H}v(z_{0})-\mathcal{H}v(\cdot)}{z_{0}-\cdot}\right\rVert^{2}_{\dot{H}^{\frac{1}{2}}}.$
(5.9)

###### Proof.

We remark that (5.9) has been obtained in [53, Proposition 3.5] in the
deterministic case. However, notice that one cannot assume without loss of
generality (as in the deterministic case) that $\tilde{v}(z_{0})=0$ which
simplifies the proof of (5.9) (see Remark 1.4) and hence we present also the
complete proof here.

Recalling (5.6) and $v_{x}=\Lambda\tilde{v}$, we have
$\Lambda\tilde{v}(z_{0})=0$ for $t\in[0,\tau^{*})$ almost surely. Then,
invoking the integral representation (2.5) for $\alpha=1$, we arrive at

$\displaystyle\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})=$
$\displaystyle\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{\tilde{v}(z_{0})\Lambda\tilde{v}(z_{0})-\tilde{v}(y)\Lambda\tilde{v}(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y$ $\displaystyle=$
$\displaystyle\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{-\tilde{v}(y)\Lambda\tilde{v}(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y,\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$ (5.10)

On the other hand, we have that

$\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=-\tilde{v}(z_{0})\Lambda(\Lambda\tilde{v})(z_{0})=\tilde{v}(z_{0})\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{\Lambda\tilde{v}(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y,\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.,$

where we have used in the first equality the fact that
$\partial_{xx}=-\Lambda^{2}$ and the semigroup property of the fractional
Laplace operator $\Lambda^{2}=\Lambda\Lambda$. Therefore, we have that

$\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{\left(\tilde{v}(z_{0})-\tilde{v}(y)\right)\Lambda\tilde{v}(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y,\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$ (5.11)

Recalling the notation $\tilde{v}(z_{0})=\tilde{v}(\omega,t,z_{0})$, then for
a.e. $\omega\in\Omega$ and $t\in[0,\tau^{*})$ we define

$\overline{v}(\omega,t,y)\triangleq\tilde{v}(\omega,t,z_{0})-\tilde{v}(\omega,t,y).$
(5.12)

Since $\overline{v}(\omega,t,z_{0})=0$, factorizing the root implies that
there exists a process $\eta(\omega,t,y)$ such that

$\overline{v}(\omega,t,y)=(z_{0}-y)\eta(\omega,t,y),\ \ t\in[0,\tau^{*})\ \
\mathbb{P}-a.s.$ (5.13)

Therefore we can observe that

$\eta(\omega,t,z_{0})=-\overline{v}_{x}(\omega,t,z_{0}),\
\mathcal{H}\eta(\omega,t,z_{0})=\Lambda\overline{v}(\omega,t,z_{0}),\ \
t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$ (5.14)

Again, we drop $\omega$ and $t$ if there is no ambiguity. Then (5.11) reduces
to

$\displaystyle\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=$
$\displaystyle-\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{\overline{v}(y)\Lambda\overline{v}(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y$ $\displaystyle=$
$\displaystyle-\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{(z_{0}-y)\eta(y)[\Lambda((z_{0}-\cdot)\eta(\cdot))](y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y,\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Using the linearity of the fractional Laplacian operator and Lemma 2.4, we
have

$[\Lambda((z_{0}-\cdot)\eta(\cdot))](y)=z_{0}\Lambda\eta(y)-y\Lambda\eta(y)+\mathcal{H}\eta(y)=(z_{0}-y)\Lambda\eta(y)+\mathcal{H}\eta(y)$

and hence

$\displaystyle\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})$
$\displaystyle+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{-(z_{0}-y)\eta(y)\left((z_{0}-y)\Lambda\eta(y)+\mathcal{H}\eta(y)\right)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y$
$\displaystyle=\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{-(z_{0}-y)^{2}\eta(y)\Lambda\eta(y)\,{\rm
d}y}{\lvert
z_{0}-y\rvert^{2}}-\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{(z_{0}-y)\eta(y)\mathcal{H}\eta(y)}{\lvert
z_{0}-y\rvert^{2}}\,{\rm d}y$
$\displaystyle=-\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\eta(y)\Lambda\eta(y)\,{\rm
d}y+\frac{1}{\pi}\mbox{p.v.}\int_{\mathbb{R}}\frac{\eta(y)\mathcal{H}\eta(y)}{y-z_{0}}\,{\rm
d}y$
$\displaystyle=-\frac{1}{\pi}\left\lVert\Lambda^{1/2}\eta\right\rVert^{2}_{L^{2}}+[\mathcal{H}(\eta\mathcal{H}\eta)](z_{0}),\
\ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Therefore, applying the identity (2.10) we can rewrite the above equation as

$\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})=-\frac{1}{2}(\eta(z_{0}))^{2}+\frac{1}{2}(\mathcal{H}\eta(z_{0}))^{2}-\frac{1}{\pi}\left\lVert\Lambda^{1/2}\eta\right\rVert^{2}_{L^{2}},\
\ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Using (5.12), (5.13), (5.14) and noticing $\Lambda\tilde{v}(z_{0})=0$, we
arrive at

$\displaystyle\Lambda(\tilde{v}\Lambda\tilde{v})(z_{0})+\tilde{v}(z_{0})\tilde{v}_{xx}(z_{0})$
$\displaystyle=-\frac{1}{2}(\overline{v}_{x}(z_{0}))^{2}+\frac{1}{2}(\Lambda\overline{v}(z_{0}))^{2}-\frac{1}{\pi}\left\lVert\frac{\overline{v}(\cdot)}{z_{0}-\cdot}\right\rVert^{2}_{\dot{H}^{\frac{1}{2}}}$
$\displaystyle=-\frac{1}{2}\left(\tilde{v}_{x}(z_{0})\right)^{2}-\frac{1}{\pi}\left\lVert\frac{\tilde{v}(z_{0})-\tilde{v}(\cdot)}{z_{0}-\cdot}\right\rVert^{2}_{\dot{H}^{\frac{1}{2}}},\
\ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Then (5.9) is a direct consequence of the above equation and the fact
$\tilde{v}=\mathcal{H}v$. ∎

###  5.2. Proof of Theorem 1.3

Now we are in the position to prove Theorem 1.3. To begin with, we can infer
from Lemma 5.2 and (5.8) that

$\displaystyle\frac{{\rm d}\Lambda v(z_{0})}{{\rm
d}t}=\beta\left[\frac{1}{2}(\Lambda
v(z_{0}))^{2}+\frac{1}{\pi}\left\lVert\frac{\mathcal{H}v(z_{0})-\mathcal{H}v(\cdot)}{z_{0}-\cdot}\right\rVert^{2}_{\dot{H}^{\frac{1}{2}}}\right]\geq\beta\frac{1}{2}(\Lambda
v(z_{0}))^{2},\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Let $F(\omega,t)\triangleq\Lambda v(\omega,t,z_{0})$. Then the above estimate
becomes

$\frac{{\rm d}F(\omega,t)}{{\rm
d}t}\geq\frac{1}{2}\beta(\omega,t)F^{2}(\omega,t),\ \ t\in[0,\tau^{*})\ \
\mathbb{P}-a.s.$ (5.15)

Let $0<K<1$ and define $\Omega^{*}\triangleq\\{\omega:\beta(t)\geq
Ke^{-\frac{b^{*}}{2}t}\mbox{ for all }t\\}.$ If $F(\omega,0)>\frac{b^{*}}{K}$
almost surely, then $\tau^{*}<\infty$ for a.e. $\omega\in\Omega^{*}$. Indeed,
integrating (5.15) leads to

$-\frac{1}{F(\omega,t)}+\frac{1}{F(\omega,0)}\geq\frac{1}{2}\int_{0}^{t}\beta(\omega,t^{\prime})\,{\rm
d}t^{\prime},\ \ t\in[0,\tau^{*})\ \ \mathbb{P}-a.s.$

Since $F(\omega,0)>\frac{b^{*}}{K}$ almost surely, (5.15) means that $F$ is
increasing almost surely. Therefore we restrict the above inequality to
$\omega\in\Omega^{*}$ and we arrive at

$\frac{1}{F(\omega,0)}\geq\frac{1}{2}K\int_{0}^{\tau^{*}}e^{-\frac{b^{*}}{2}t^{\prime}}dt^{\prime}=\frac{K}{b^{*}}\left(1-e^{-\frac{b^{*}}{2}\tau^{*}}\right)$

and hence

$\frac{1}{F(\omega,0)}-\frac{K}{b^{*}}\left(1-e^{-\frac{b^{*}}{2}\tau^{*}}\right)\geq
0.$

By the assumption $F(\omega,0)>\frac{b^{*}}{K}$ almost surely, we arrive at

$\frac{K}{b^{*}}e^{-\frac{b^{*}}{2}\tau^{*}}\geq\frac{K}{b^{*}}-\frac{1}{F(0)}>0,\quad\mbox{
a.e }\omega\in\Omega^{*}.$

Thus $\tau^{*}<\infty$ a.e. on $\Omega^{*}$ as desired. Recalling that
$\beta(\omega,t)={\rm e}^{\int_{0}^{t}b(t^{\prime}){\rm
d}W_{t^{\prime}}-\int_{0}^{t}\frac{b^{2}(t^{\prime})}{2}{\rm d}t^{\prime}}$,
we have shown that

$\mathbb{P}\\{\tau^{*}<\infty\\}\geq\mathbb{P}\\{\beta(t)\geq
Ke^{-\frac{b^{*}}{2}t}\mbox{ for all }t\\}$

which implies that

$\mathbb{P}\\{\tau^{*}<\infty\\}\geq\mathbb{P}\\{{\rm
e}^{\int_{0}^{t}b(t^{\prime})\rm{d}W_{t^{\prime}}}>K\mbox{ for all }t\\}$

since $b^{2}(t)<b^{*}$ for all $t>0$. The proof is now complete.

##  6\. Proof of Theorem 1.4

In this section, we provide the proof of Theorem 1.4. As is mentioned in
Remark 1.5, since we cannot get an explicit expression of the solution to
(1.2), we start with constructing some approximate solutions from which (1.22)
can be established. Similarly as before we divide the proof into several
subsections.

###  6.1. Approximate solutions and actual solutions

Following [44], we construct the approximate solution as follows. First, we
fix two functions $\phi$, $\tilde{\phi}\in C_{c}^{\infty}$ such that

$\phi(x)=\left\\{\begin{aligned} &1,\ \text{if}\ |x|<1,\\\ &0,\ \text{if}\
|x|\geq 2,\end{aligned}\right.\quad\text{and}\quad\tilde{\phi}(x)=1\
\text{if}\ x\in\text{supp}\ \phi.$ (6.1)

Next, we construct the following sequence of approximate solutions

$\displaystyle u_{m,n}=u_{h}+u_{l},\ \ m\in\\{-1,1\\},$ (6.2)

where

  * •

$u_{h}=u_{h,m,n}$ is the high-frequency part defined by

$\displaystyle
u_{h}=u_{h,m,n}(t,x)=n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt),\ \ n\in\mathbb{N}.$ (6.3)

  * •

$u_{l}=u_{l,m,n}$ is the low-frequency part defined as the solution to the
following problem:

$\left\\{\begin{aligned}
&\partial_{t}u_{l}+(\mathcal{H}u_{l})\partial_{x}u_{l}=0,\quad
x\in\mathbb{R},\ t>0,\\\
&u_{l}(0,x)=-\mathcal{H}\left(mn^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right),\quad
x\in\mathbb{R}.\end{aligned}\right.$ (6.4)

In (6.3)-(6.4), $\delta>0$ is a parameter that will be determined later in the
proof.

Let us consider the problem (1.2) with the initial data $u_{m,n}(0,x)$, i.e.,

$\left\\{\begin{aligned} &{\rm d}u+(\mathcal{H}u)u_{x}{\rm d}t=h(t,u){\rm
d}\mathcal{W},\quad x\in\mathbb{R},\ t>0,\\\
&u(0,x)=-\mathcal{H}\left(mn^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right)+n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx),\quad
x\in\mathbb{R}.\end{aligned}\right.$ (6.5)

Since Assumption (D) implies Assumption (A), Theorem 1.1 immediately yields
that for each fixed $n\in\mathbb{N}$, (6.5) has a unique pathwise solution
$(u^{m,n},\tau^{m,n})$ such that $u^{m,n}\in
C\left([0,\tau^{m,n}];H^{s}\right)$ $\mathbb{P}-a.s.$ with $s>3$.

###  6.2. Estimates on the errors

Substituting (6.2) into (1.2), we define the error $\mathcal{E}(\omega,t,x)$
as

$\displaystyle\mathcal{E}(\omega,t,x)=\,u_{m,n}(t,x)-u_{m,n}(0,x)+\int_{0}^{t}(\mathcal{H}u_{m,n})\partial_{x}u_{m,n}\,{\rm
d}t^{\prime}-\int_{0}^{t}h(t^{\prime},u_{m,n})\,{\rm d}\mathcal{W}\ \
\mathbb{P}-a.s.$

By using (6.2) and (6.4), we reformulate $\mathcal{E}(\omega,t,x)$ as

$\displaystyle\mathcal{E}$ $\displaystyle(\omega,t,x)$ $\displaystyle=\,$
$\displaystyle
u_{l}(t,x)-u_{l}(0,x)+\int_{0}^{t}(\mathcal{H}u_{l})\partial_{x}u_{l}\,{\rm
d}t^{\prime}+u_{h}(t,x)-u_{h}(0,x)$
$\displaystyle+\int_{0}^{t}(\mathcal{H}u_{l})\partial_{x}u_{h}+(\mathcal{H}u_{h})(\partial_{x}u_{l}+\partial_{x}u_{h})\,{\rm
d}t^{\prime}-\int_{0}^{t}h(t^{\prime},u_{m,n})\,{\rm d}\mathcal{W}$
$\displaystyle=\,$ $\displaystyle
u_{h}(t,x)-u_{h}(0,x)+\int_{0}^{t}(\mathcal{H}u_{l})\partial_{x}u_{h}+(\mathcal{H}u_{h})(\partial_{x}u_{l}+\partial_{x}u_{h})\,{\rm
d}t^{\prime}-\int_{0}^{t}h(t^{\prime},u_{m,n})\,{\rm d}\mathcal{W}\ \
\mathbb{P}-a.s.$ (6.6)

The following lemma shows the decay estimate for the low-frequency part of
$u_{m,n}$.

######  Lemma 6.1.

Let $|m|=1$, $s>3$, $\delta\in(0,2)$ and $n\gg 1$. Then there exists a
$T_{l}>0$ such that for all $n\gg 1$, the initial value problem (6.4) has a
unique smooth solution $u_{l}=u_{l,m,n}\in C([0,T_{l}];H^{s})$ such that
$T_{l}$ does not depend on $n$. Besides, for all fixed $r>0$, there exists a
constant $C=C_{r,\tilde{\phi},T_{l}}>0$ such that $u_{l}$ satisfies

$\|u_{l}(t)\|_{H^{r}}\leq C|m|n^{\frac{\delta}{2}-1},\ \ t\in[0,T_{l}].$ (6.7)

###### Proof.

For $|m|=1$ and any fixed $n\geq 1$, since $u_{l}(0,x)\in H^{\infty}$, by
applying Theorem 1.1 with $h=0$ and deterministic initial data, we see that
for any $s>3$, (6.4) has a unique (deterministic) solution $u_{l}=u_{l,m,n}\in
C\left([0,T_{l}];H^{s}\right)$. We will show that there exists a lower bound
of the existence time, i.e., there is a $T_{l}>0$ such that for all $n\gg 1$,
$u_{l}=u_{l,m,n}$ exists on $[0,T_{l}]$ and satisfies (6.7). The proof of
Lemma 6.1 consists of three main steps.

Step 1: Estimate $\|u_{l}(0,x)\|_{H^{r}}$. Let
$g(x)=mn^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)$. For $n\gg 1$, by
using (2.7), we have that

$\displaystyle\|u_{l}(0,x)\|^{2}_{H^{r}}=\,\left\|-\mathcal{H}\left(mn^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right)\right\|^{2}_{H^{r}}$
$\displaystyle\leq\left\|mn^{-1}\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)\right\|^{2}_{H^{r}}$
$\displaystyle=\int_{\mathbb{R}}(1+|\xi|^{2})^{r}|\widehat{g}(\xi)|^{2}\,{\rm
d}\xi$
$\displaystyle=m^{2}n^{2\delta-2}\int_{\mathbb{R}}(1+|\xi|^{2})^{r}\left|\widehat{\tilde{\phi}}(n^{\delta}\xi)\right|^{2}\,{\rm
d}\xi$
$\displaystyle=m^{2}n^{\delta-2}\int_{\mathbb{R}}\left(1+\left|\frac{z}{n^{\delta}}\right|^{2}\right)^{r}\left|\widehat{\tilde{\phi}}(z)\right|^{2}\,{\rm
d}z$ $\displaystyle\leq
m^{2}n^{\delta-2}\int_{\mathbb{R}}\left(1+\left|z\right|^{2}\right)^{r}\left|\widehat{\tilde{\phi}}(z)\right|^{2}\,{\rm
d}z\leq Cm^{2}n^{\delta-2},$

for some constant $C=C_{r,\tilde{\phi}}>0$. Therefore we find that

$\displaystyle\|u_{l}(0,x)\|_{H^{r}}\leq C|m|n^{\frac{\delta}{2}-1}.$

Step 2: Proof of (6.7) for $r>3/2$. In this case, we apply Lemma 2.2, (2.7),
$H^{r}\hookrightarrow W^{1,\infty}$ and integration by parts to find

$\displaystyle\frac{1}{2}$ $\displaystyle\frac{\,{\rm d}}{\,{\rm
d}t}\|u_{l}\|^{2}_{H^{r}}$ $\displaystyle=\,$
$\displaystyle-\int_{\mathbb{R}}D^{r}u_{l}D^{r}\left((\mathcal{H}u_{l})\partial_{x}u_{l}\right)\,{\rm
d}x$ $\displaystyle\leq\,$
$\displaystyle\left|\left(D^{r}u_{l},D^{r}\left((\mathcal{H}u_{l})\partial_{x}u_{l}\right)\right)_{L^{2}}\right|$
$\displaystyle\leq\,$
$\displaystyle\left|\left([D^{r},\mathcal{H}u_{l}]\partial_{x}u_{l},D^{r}u_{l}\right)_{L^{2}}\right|+\left|\left((\mathcal{H}u_{l})D^{r}\partial_{x}u_{l},D^{r}u_{l}\right)_{L^{2}}\right|$
$\displaystyle\lesssim\,$
$\displaystyle\left(\|D^{r}(\mathcal{H}u_{l})\|_{L^{2}}\|\partial_{x}u_{l}\|_{L^{\infty}}+\|\partial_{x}\left(\mathcal{H}u_{l}\right)\|_{L^{\infty}}\|D^{r-1}\partial_{x}u_{l}\|_{L^{2}}\right)\|u_{l}\|_{H^{r}}+\left|\frac{1}{2}\int_{\mathbb{R}}\left(\mathcal{H}u_{l}\right)\partial_{x}(D^{r}u_{l})^{2}\,{\rm
d}x\right|$ $\displaystyle\lesssim\,$
$\displaystyle\|\partial_{x}u_{l}\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}+\|\partial_{x}\left(\mathcal{H}u_{l}\right)\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\|u_{l}\|_{W^{1,\infty}}\|u_{l}\|_{H^{r}}^{2}+\|\mathcal{H}u_{l}\|_{W^{1,\infty}}\|u_{l}\|_{H^{r}}^{2}$
$\displaystyle\leq\,$ $\displaystyle C\|u_{l}\|^{3}_{H^{r}},\ \ C=C_{r}>0.$

Solving the above inequality gives

$\|u_{l}\|_{H^{r}}\leq\frac{\|u_{l}(0)\|_{H^{r}}}{1-Ct\|u_{l}(0)\|_{H^{r}}},\
\ 0\leq t<\frac{1}{C\|u_{l}(0)\|_{H^{r}}}.$

Remember that $u_{l}=u_{l,m,n}$. Then we define the time interval
$[0,T_{l,m,n}]$ such that

$\|u_{l}\|_{H^{r}}\leq 2\|u_{l}(0)\|_{H^{r}},\ \ t\in[0,T_{l,m,n}],\ \
T_{l,m,n}=\frac{1}{2C\|u_{l}(0)\|_{H^{r}}}.$ (6.8)

By Step 1, we have that for $|m|=1$,
$T_{l,m,n}\gtrsim\frac{1}{2Cn^{\frac{\delta}{2}-1}}\rightarrow\infty,\
\text{as}\ n\rightarrow\infty.$ Therefore we can find a common time interval
$[0,T_{l}]$ such that

$\|u_{l}\|_{H^{r}}\leq 2\|u_{l}(0)\|_{H^{r}}\leq C|m|n^{\frac{\delta}{2}-1},\
\ t\in[0,T_{l}],\ \ C=C_{r,\tilde{\phi}}>0,$ (6.9)

which is (6.7).

Step 3: Proof of (6.7) for $0<r\leq 3/2$. Applying Lemma 2.2, (2.7), we have

$\displaystyle\frac{1}{2}\frac{\,{\rm d}}{\,{\rm d}t}\|u_{l}\|^{2}_{H^{r}}$
$\displaystyle=\,$
$\displaystyle-\int_{\mathbb{R}}D^{r}u_{l}D^{r}\left((\mathcal{H}u_{l})\partial_{x}u_{l}\right)\,{\rm
d}x$ $\displaystyle\leq\,$
$\displaystyle\left|\left(D^{r}u_{l},D^{r}\left((\mathcal{H}u_{l})\partial_{x}u_{l}\right)\right)_{L^{2}}\right|$
$\displaystyle\leq\,$
$\displaystyle\left|\left([D^{r},\mathcal{H}u_{l}]\partial_{x}u_{l},D^{r}u_{l}\right)_{L^{2}}\right|+\left|\left((\mathcal{H}u_{l})D^{r}\partial_{x}u_{l},D^{r}u_{l}\right)_{L^{2}}\right|$
$\displaystyle\lesssim\,$
$\displaystyle\left(\|D^{r}(\mathcal{H}u_{l})\|_{L^{2}}\|\partial_{x}u_{l}\|_{L^{\infty}}+\|\partial_{x}\left(\mathcal{H}u_{l}\right)\|_{L^{\infty}}\|D^{r-1}\partial_{x}u_{l}\|_{L^{2}}\right)\|u_{l}\|_{H^{r}}+\left|\frac{1}{2}\int_{\mathbb{R}}\left(\mathcal{H}u_{l}\right)\partial_{x}(D^{r}u_{l})^{2}\,{\rm
d}x\right|$ $\displaystyle\lesssim\,$
$\displaystyle\|\partial_{x}u_{l}\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}+\|\partial_{x}\left(\mathcal{H}u_{l}\right)\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}.$

It follows from the embedding $H^{r+\frac{3}{2}}\hookrightarrow W^{1,\infty}$
that

$\displaystyle\frac{1}{2}\frac{\,{\rm d}}{\,{\rm
d}t}\|u_{l}\|^{2}_{H^{r}}\lesssim\,$
$\displaystyle\|\partial_{x}u_{l}\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}+\|\partial_{x}\left(\mathcal{H}u_{l}\right)\|_{L^{\infty}}\|u_{l}\|_{H^{r}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\|u_{l}\|_{W^{1,\infty}}\|u_{l}\|^{2}_{H^{r}}+\|\mathcal{H}u_{l}\|_{W^{1,\infty}}\left\|u_{l}\right\|_{H^{r}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\|u_{l}\|_{H^{r+\frac{3}{2}}}\|u_{l}\|_{H^{r}}^{2}+\|\mathcal{H}u_{l}\|_{H^{r+\frac{3}{2}}}\|u_{l}\|_{H^{r}}^{2}$
$\displaystyle\lesssim\,$
$\displaystyle\|u_{l}\|_{H^{r+\frac{3}{2}}}\|u_{l}\|_{H^{r}}^{2}.$

Using the conclusion of Step 2 for $r+\frac{3}{2}>\frac{3}{2}$, we have

$\displaystyle\frac{\,{\rm d}}{\,{\rm
d}t}\|u_{l}\|_{H^{r}}\lesssim\|u_{l}\|_{H^{r}}\|u_{l}(0)\|_{H^{r+\frac{3}{2}}},\
\ t\in[0,T_{l}],$

and hence

$\displaystyle\|u_{l}(t)\|_{H^{r}}\lesssim\|u_{l}(0)\|_{H^{r}}+\int_{0}^{t}\|u_{l}\|_{H^{r}}\|u_{l}(0)\|_{H^{r+\frac{3}{2}}}\,{\rm
d}t,\ \ t\in[0,T_{l}].$

Applying Grönwall’s inequality to the above inequality, we have

$\displaystyle\|u_{l}\|_{H^{r}}\lesssim\|u_{l}(0)\|_{H^{r}}\exp\left\\{\|u_{l}(0)\|_{H^{r+\frac{3}{2}}}T_{l}\right\\},\
\ t\in[0,T_{l}].$

Since $\delta\in(0,2)$, we can infer from Step 1 that
$\exp\left\\{\|u_{l}(0)\|_{H^{r+\frac{3}{2}}}T_{l}\right\\}<C(r,\tilde{\phi},T_{l})$
for some constant $C(r,\tilde{\phi},T_{l})>0$. Therefore we see that there
exists a constant $C=C_{r,\tilde{\phi},T_{l}}>0$ such that

$\displaystyle\|u_{l}\|_{H^{r}}\leq C|m|n^{\frac{\delta}{2}-1},\ \
t\in[0,T_{l}],$

concluding the desired estimate (6.7). ∎

The above result implies that the $H^{s}$-norm of $u_{l}$, the low-frequency
part of the approximate solution defined by (6.2), is decaying. For the high-
frequency part $u_{h}$, due to Lemma 2.5, its $H^{s}$-norm is bounded. To sum
up, let $T_{l}$ be given in Lemma 6.1, for any $s>0$, there is a constant
$M=M_{s,\tilde{\phi},\phi,T_{l}}>0$ such that

$\|u_{m,n}(t)\|_{H^{s}}\lesssim M,\ \ t\in[0,T_{l}].$ (6.10)

Moreover, although not strictly necessary, we can infer from (6.9) that $M$
can be independent of $T_{l}$ when $s>3/2$.

####  6.2.1. Estimating the error $\mathcal{E}$

Recall (6.2). By using (6.1), we have that $\phi=\tilde{\phi}\phi$. Then by
(6.3) and $u_{l}(0,x)$ in (6.4), we see that for $m\in\\{-1,1\\}$,

$\displaystyle u_{h}(t,x)-$ $\displaystyle u_{h}(0,x)$ $\displaystyle=\,$
$\displaystyle
n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt)-n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx)$
$\displaystyle=\,$ $\displaystyle
m^{-1}m\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt)-m^{-1}m\tilde{\phi}\left(\frac{x}{n^{\delta}}\right)n^{-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx)$
$\displaystyle=\,$ $\displaystyle
m^{-1}\left(\mathcal{H}u_{l}(0,x)\right)n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt)-m^{-1}\left(\mathcal{H}u_{l}(0,x)\right)n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx)$
$\displaystyle=\,$
$\displaystyle\int_{0}^{t}\left(\mathcal{H}u_{l}(0,x)\right)n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt^{\prime})\,{\rm d}t^{\prime}.$

Furthermore,

$\displaystyle\int_{0}^{t}(\mathcal{H}u_{l})\partial_{x}u_{h}\,{\rm
d}t^{\prime}=\,$
$\displaystyle-\int_{0}^{t}(\mathcal{H}u_{l})(t^{\prime})n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt^{\prime})\,{\rm d}t^{\prime}$
$\displaystyle+\int_{0}^{t}(\mathcal{H}u_{l})(t^{\prime})n^{-\frac{3\delta}{2}-s}\partial_{x}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt^{\prime})\,{\rm d}t^{\prime}.$ (6.11)

Thus, (6.2) becomes

$\displaystyle\mathcal{E}(\omega,t,x)=\,$
$\displaystyle\int_{0}^{t}[(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t^{\prime})]n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt^{\prime})\,{\rm d}t^{\prime}$
$\displaystyle+\int_{0}^{t}(\mathcal{H}u_{l})(t^{\prime})n^{-\frac{3\delta}{2}-s}\partial_{x}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt^{\prime})\,{\rm d}t^{\prime}$
$\displaystyle+\int_{0}^{t}(\mathcal{H}u_{h})(\partial_{x}u_{l}+\partial_{x}u_{h})\,{\rm
d}t^{\prime}-\int_{0}^{t}h(t^{\prime},u_{m,n})\,{\rm d}\mathcal{W}$
$\displaystyle=\,$ $\displaystyle\int_{0}^{t}E\,{\rm
d}t^{\prime}-\int_{0}^{t}h(t^{\prime},u_{m,n})\,{\rm d}\mathcal{W}\ \
\mathbb{P}-a.s.,$ (6.12)

where

$\displaystyle E=\,E(\omega,t,x)=\,$
$\displaystyle[(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)]n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt)$
$\displaystyle+(\mathcal{H}u_{l})(t)n^{-\frac{3\delta}{2}-s}\partial_{x}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt)+(\mathcal{H}u_{h})(\partial_{x}u_{l}+\partial_{x}u_{h}).$ (6.13)

Now we shall estimate the $H^{\sigma_{0}}$-norm of the error $\mathcal{E}$,
where $\sigma_{0}$ is given in Assumption (D). Actually, we will show that the
$H^{\sigma_{0}}$-norm of $\mathcal{E}$ is decaying.

######  Lemma 6.2.

Let $n\gg 1$, $s>3$, $\frac{3}{4}<\delta<1$. Let $T_{l}$ be given in Lemma
6.1, and $\sigma_{0}$ be given in Assumption (D). Let

$r_{s}=-s-1+\sigma_{0}+\delta.$ (6.14)

Then $r_{s}<0$ and the error $\mathcal{E}$ given by (6.12) satisfies

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}\leq
Cn^{2r_{s}},$

where $C=C(\sigma_{0},\tilde{\phi},\phi,T_{l})>0$ is a constant independent of
$n$.

###### Proof.

It is obvious by construction that $r_{s}$ given by (6.14) is negative.
Combining (6.13), the embedding $H^{\sigma_{0}}\hookrightarrow L^{\infty}$ and
Lemmas 6.1 and 2.5, we find that for $t\in[0,T_{l}]$,

$\displaystyle\|E\|_{H^{\sigma_{0}}}\leq\,$
$\displaystyle\left\|\left[(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right]n^{1-\frac{\delta}{2}-s}\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt)\right\|_{H^{\sigma_{0}}}$
$\displaystyle+\left\|(\mathcal{H}u_{l})(t)n^{-\frac{3\delta}{2}-s}\partial_{x}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt)\right\|_{H^{\sigma_{0}}}$
$\displaystyle+\left\|(\mathcal{H}u_{h})\partial_{x}u_{l}\right\|_{H^{\sigma_{0}}}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{h}\right\|_{H^{\sigma_{0}}}$
$\displaystyle\lesssim\,$ $\displaystyle
n^{1-\frac{\delta}{2}-s}\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}\left\|\phi\left(\frac{x}{n^{\delta}}\right)\sin(nx-
mt)\right\|_{H^{\sigma_{0}}}$
$\displaystyle+n^{-\frac{3\delta}{2}-s}\left\|(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}\left\|\partial_{x}\phi\left(\frac{x}{n^{\delta}}\right)\cos(nx-
mt^{\prime})\right\|_{H^{\sigma_{0}}}$
$\displaystyle+\left\|(\mathcal{H}u_{h})\partial_{x}u_{l}\right\|_{H^{\sigma_{0}}}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{h}\right\|_{H^{\sigma_{0}}}$
$\displaystyle\lesssim\,$ $\displaystyle
n^{1-s+\sigma_{0}}\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}+n^{-s-1+\sigma_{0}-\frac{1}{2}\delta}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{l}\right\|_{H^{\sigma_{0}}}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{h}\right\|_{H^{\sigma_{0}}}$
$\displaystyle\lesssim\,$ $\displaystyle
n^{1-s+\sigma_{0}}\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}+n^{r_{s}}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{l}\right\|_{H^{\sigma_{0}}}+\left\|(\mathcal{H}u_{h})\partial_{x}u_{h}\right\|_{H^{\sigma_{0}}}.$
(6.15)

For
$\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}$,
it follows from the fundamental theorem of calculus and
$H^{\sigma_{0}}\hookrightarrow L^{\infty}$ that for $t\in[0,T_{l}]$,

$\displaystyle\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}\lesssim\,$
$\displaystyle\|u_{l}(0)-u_{l}(t)\|_{H^{\sigma_{0}}}$ $\displaystyle=\,$
$\displaystyle\left\|\int_{0}^{t}\partial_{t}u_{l}(t^{\prime})\,{\rm
d}t^{\prime}\right\|_{H^{\sigma_{0}}}$ $\displaystyle\lesssim\,$
$\displaystyle\int_{0}^{t}\|(\mathcal{H}u_{l})\partial_{x}u_{l}\|_{H^{\sigma_{0}}}\,{\rm
d}t^{\prime}$ $\displaystyle\lesssim\,$
$\displaystyle\int_{0}^{t}\|u_{l}\|_{H^{\sigma_{0}+1}}^{2}\,{\rm
d}t^{\prime}\lesssim\,n^{\delta-2}T_{l},$

where we used (6.4) with $t\in[0,T_{l}]$, Lemma 6.1 and the embedding
$H^{\sigma_{0}+1}\hookrightarrow W^{1,\infty}$. Therefore,

$\displaystyle
n^{1-s+\sigma_{0}}\left\|(\mathcal{H}u_{l})(0)-(\mathcal{H}u_{l})(t)\right\|_{H^{\sigma_{0}}}\lesssim
n^{1-s+\sigma_{0}+\delta-2}T_{l}=n^{r_{s}}T_{l},\ \ t\in[0,T_{l}].$ (6.16)

Next, applying Lemma 6.1 and 2.5, we have for $t\in[0,T_{l}]$,

$\displaystyle\left\|(\mathcal{H}u_{h})\partial_{x}u_{l}\right\|_{H^{\sigma_{0}}}\lesssim\,$
$\displaystyle\|u_{h}\|_{H^{\sigma_{0}}}\|u_{l}\|_{H^{\sigma_{0}+1}}$
$\displaystyle\lesssim\,$ $\displaystyle
n^{-s+\sigma_{0}}n^{\frac{\delta}{2}-1}=\,n^{-s+\sigma_{0}+\frac{\delta}{2}-1}\lesssim
n^{r_{s}},$ (6.17)
$\displaystyle\left\|(\mathcal{H}u_{h})\partial_{x}u_{h}\right\|_{H^{\sigma_{0}}}\lesssim\,$
$\displaystyle\|u_{h}\|_{H^{\sigma_{0}}}\|u_{h}\|_{H^{\sigma_{0}+1}}$
$\displaystyle\lesssim\,$ $\displaystyle
n^{-s+\sigma_{0}}n^{-s+\sigma_{0}+1}=\,n^{-2s+2\sigma_{0}+1}\lesssim
n^{r_{s}}.$ (6.18)

Here in (6.18) we used the assumption $\sigma_{0}\in(3/2,7/4)$ to guarantee
$n^{-2s+2\sigma_{0}+1}\lesssim n^{r_{s}}$. Inserting (6.16), (6.17) and (6.18)
into (6.15), we finally obtain

$\displaystyle\|E\|_{H^{\sigma_{0}}}\lesssim n^{r_{s}},\ \ t\in[0,T_{l}].$
(6.19)

With (6.19) at hand, we are in the position to estimate
$\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}$.
Invoking Itô formula in (6.12) leads to

$\displaystyle\|\mathcal{E}(t,x)\|_{H^{\sigma_{0}}}^{2}\leq\ $
$\displaystyle\left|-2\int_{0}^{t}(h(t^{\prime},u_{m,n})\,{\rm
d}\mathcal{W},\mathcal{E})_{H^{\sigma_{0}}}\right|+2\int_{0}^{t}\left|(E,\mathcal{E})_{H^{\sigma_{0}}}\right|\,{\rm
d}t^{\prime}+\int_{0}^{t}\|h(t^{\prime},u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t^{\prime}.$

Taking supremum with respect to $t\in[0,T_{l}]$ and using the BDG inequality,
we can find some $\overline{C}>0$ such that

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}$
$\displaystyle\leq\,$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}+\overline{C}\mathbb{E}\int_{0}^{T_{l}}\|h(t,u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t+\overline{C}\int_{0}^{T_{l}}\left[\mathbb{E}\|E\|_{H^{\sigma_{0}}}^{2}+\mathbb{E}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}\right]\,{\rm
d}t.$

By virtue of (6.19), we arrive at

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}\lesssim
T_{l}n^{2r_{s}}+\mathbb{E}\int_{0}^{T_{l}}\|h(t,u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t+\int_{0}^{T_{l}}\mathbb{E}\sup_{t^{\prime}\in[0,t]}\|\mathcal{E}(t^{\prime})\|_{H^{\sigma_{0}}}^{2}\,{\rm
d}t.$

Now, we estimate
$\|h(t,u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}$. For any
fixed $s>3$, on account of Assumption (D), Lemmas 2.5 and 6.1, we can pick
$\kappa>2(1+\frac{s-\sigma_{0}-1}{2-\delta})$ such that

$\displaystyle\|h(t,u_{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\lesssim\left({\rm
e}^{\frac{-1}{\|u_{m,n}\|_{H^{\sigma_{0}}}}}\right)^{2}\lesssim\left(n^{-s+\sigma_{0}}+n^{\frac{\delta}{2}-1}\right)^{2\kappa}\lesssim
n^{2r_{s}},$

which gives

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}\lesssim
T_{l}n^{2r_{s}}+\int_{0}^{T_{l}}\mathbb{E}\sup_{t^{\prime}\in[0,t]}\|\mathcal{E}(t^{\prime})\|_{H^{\sigma_{0}}}^{2}\,{\rm
d}t.$

Obviously, for each $n\geq 1$,
$\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}$ is
finite and $T_{l}>0$ is fixed. Then by the Grönwall inequality, we have

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}]}\|\mathcal{E}(t)\|_{H^{\sigma_{0}}}^{2}\leq
Cn^{2r_{s}}.$

The proof is completed. ∎

####  6.2.2. Estimating $u_{m,n}-u^{m,n}$

Recall the approximate solutions $u_{m,n}$ given by (6.2). Then we have the
following estimates on the difference between the actual solutions and the
approximate solutions.

######  Lemma 6.3.

Let $s>3$, $\frac{3}{4}<\delta<1$, $\sigma_{0}$ be given in Assumption (D) and
$r_{s}<0$ be given in (6.14). For any $R>1$, we define

$\displaystyle\tau^{m,n}_{R}:=\inf\\{t>0:\|u^{m,n}\|_{H^{s}}>R\\}.$ (6.20)

Then for $n\gg 1$ and $T_{l}>0$ given in Lemma 6.1,

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u_{m,n}-u^{m,n}\|_{H^{\sigma_{0}}}^{2}\leq
Cn^{2r_{s}},$ (6.21)
$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|u_{m,n}-u^{m,n}\|_{H^{2s-\sigma_{0}}}^{2}\leq
Cn^{2s-2\sigma_{0}},$ (6.22)

where $C=C(s,\sigma_{0},\tilde{\phi},\phi,T_{l},R)>0$ is a constant
independent of $n$.

###### Proof.

Let $v=v_{m,n}=u_{m,n}-u^{m,n}$. Then $v$ satisfies $v(0)=0$ and

$\displaystyle
v(t)+\int_{0}^{t}\left(\left(\mathcal{H}v\right)\partial_{x}u_{m,n}+\left(\mathcal{H}u^{m,n}\right)\partial_{x}v\right)\,{\rm
d}t^{\prime}=-\int_{0}^{t}h(t^{\prime},u^{m,n})\,{\rm
d}\mathcal{W}+\int_{0}^{t}E\,{\rm d}t^{\prime}.$

where (6.12) is used. For $T_{l}>0$, we use the Itô formula to find

$\displaystyle\|v(t)\|_{H^{\sigma_{0}}}^{2}=\,$
$\displaystyle-2\int_{0}^{t}\left(h(t^{\prime},u^{m,n})\,{\rm
d}\mathcal{W},v\right)_{H^{\sigma_{0}}}+2\int_{0}^{t}(E,v)_{H^{\sigma_{0}}}\,{\rm
d}t^{\prime}-2\int_{0}^{t}\left(\left(\mathcal{H}v\right)\partial_{x}u_{m,n},v\right)_{H^{\sigma_{0}}}\,{\rm
d}t^{\prime}$
$\displaystyle-2\int_{0}^{t}\left(\left(\mathcal{H}u^{m,n}\right)\partial_{x}v,v\right)_{H^{\sigma_{0}}}\,{\rm
d}t^{\prime}+\int_{0}^{t}\|h(t^{\prime},u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t^{\prime}.$

Taking supremum with respect to $t\in[0,T_{l}\wedge\tau^{m,n}_{R}]$, and then
using the BDG inequality yield that for some $\overline{C}>0$,

$\displaystyle\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|v(t)\|_{H^{\sigma_{0}}}^{2}$
$\displaystyle\lesssim\,$ $\displaystyle
C\mathbb{E}\left(\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\|v\|_{H^{\sigma_{0}}}^{2}\|h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t\right)^{1/2}+2\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|(E,v)_{H^{\sigma_{0}}}\right|\,{\rm
d}t$
$\displaystyle+2\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|\left(\left(\mathcal{H}v\right)\partial_{x}u_{m,n},v\right)_{H^{\sigma_{0}}}\right|\,{\rm
d}t+2\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|\left(\left(\mathcal{H}u^{m,n}\right)\partial_{x}v,v\right)_{H^{\sigma_{0}}}\right|\,{\rm
d}t$
$\displaystyle+\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\|h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t$ $\displaystyle\leq\,$
$\displaystyle\frac{1}{2}\mathbb{E}\sup_{t\in[0,T_{l}\wedge\tau^{m,n}_{R}]}\|v(t)\|_{H^{\sigma_{0}}}^{2}+\overline{C}\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|(E,v)_{H^{\sigma_{0}}}\right|\,{\rm
d}t$
$\displaystyle+\overline{C}\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|\left(\left(\mathcal{H}v\right)\partial_{x}u_{m,n},v\right)_{H^{\sigma_{0}}}\right|\,{\rm
d}t+\overline{C}\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\left|\left(\left(\mathcal{H}u^{m,n}\right)\partial_{x}v,v\right)_{H^{\sigma_{0}}}\right|\,{\rm
d}t$
$\displaystyle+\overline{C}\mathbb{E}\int_{0}^{T_{l}\wedge\tau^{m,n}_{R}}\|h(t,u^{m,n})\|_{\mathcal{L}_{2}(\mathcal{U};H^{\sigma_{0}})}^{2}\,{\rm
d}t.$

Recall that (6.10) gives $\|u_{m,n}\|_{H^{s}}\lesssim
M_{s,\tilde{\phi},\phi,T_{l}}$ on $t\in[0,T_{l}\wedge\tau^{m,n}_{R}]$. Hence
we can infer from Assumption (D) that for some $\overline{C}>0$