The purpose of this paper is to study the Talagrand's $ T_2 $-transportation inequality for the following fourth-order stochastic heat equation driven by a fractional noise:

$ \begin{equation} \begin{cases} \frac{\partial u}{\partial t} = -\Delta^2 u +f(u)+ W^H \ \text{ in }[0, T]\times D, \\ u(0, x) = u_{0}(x), \\ \frac{\partial u}{\partial {\bf n}} = \frac{\partial \Delta u}{\partial {\bf n}} = 0 \text{ on }[0, T]\times\{0, \pi\}, \end{cases} \end{equation} $

(1.1)

where $ \Delta: = \frac{\partial ^2}{\partial x^2}, D = [0, \pi] $, $ f $ is Lipschitz continuous, $ W^H(t, x) = \frac{\partial^2}{\partial t\partial x}B^H(t, x) $ denotes the fractional noise with Hurst parameter $ H\in (1/2, 1) $, which is fractional in time and white in space. See Section 2 for details.

It is well known that Talagrand's transportation inequalities are related closed to the concentration of measure phenomenon, the log-Sobolev and Poincaré inequalities, see for instance monographs [2-4]. We also like to refer the readers to the survey [5] authored by Gozlan and references therein.

Recently, the problem of transportation inequalities and their applications to diffusion processes was widely studied and was still a very active research area both from a theoretical and an applied point of view. Let us first recall the transportation inequality.

Let $ (E, d) $ be a metric space equipped with $ \sigma $-field $ \mathcal B $ such that $ d(\cdot, \cdot) $ is $ \mathcal B\times \mathcal B $ measurable. Given $ p\ge1 $ and two probability measures $ \mu $ and $ \nu $ on $ E $, the Wassersetin distance is defined by

$ {W_{p, d}}(\mu , \nu ): = \mathop {\inf }\limits_{\rm{ \mathsf{ π} }} {\left[ {\int {{d^p}} (x, y)\pi (dx, dy)} \right]^{\frac{1}{p}}}, $

where the infimum is taken over all the probability measures $ \pi $ on $ E\times E $ with marginal distributions $ \mu $ and $ \nu $. The relative entropy of $ \nu $ with respect to $ \mu $ is defined as

$ \begin{equation} \bf H(\nu|\mu): = \left\{ \begin{array}{ll} \int \log \frac{d\nu}{d\mu}d\nu, & \text{if } \nu\ll \mu , \\ +\infty, & \text{otherwise}. \end{array} \right. \end{equation} $

(1.2)

The probability measure $ \mu $ satisfies the $ L^p $-transportation inequality on $ (E, d) $ if there exists a constant $ C>0 $ such that for any probability measure $ \nu $ on $ E $,

$ W_{p, d}(\mu, \nu)\le \sqrt{2C {\bf H}(\nu|\mu)}. $

As usual, we write $ \mu\in {\bf T_{p}}(C) $ for this relation. The properties $ {\bf T_1}(C) $ and $ {\bf T_2}(C) $ are of particular interest. The phenomenon of measure concentration is related to $ {\bf T_1}(C) $ (see [3, 6-8]).

The property $ {\bf T_2}(C) $ is stronger than $ {\bf T_1}(C) $. It was first established by Talagrand [9] for the Gaussian measure, and it was brought into relation with the log-Sobolev inequality, Poincaré inequality, Hamilton-Jacobi's equation, the transportation-information inequality, see [8, 10-13]. The work of Talagrand on the Gaussian measure was generalized by Feyel and Üstünel [14] to the framework of an abstract Wiener space.

With regard to the paths of finite stochastic differential equation (SDE for short), by means of Girsanov transformation and the martingale representation theorem, the $ {\bf T_2}(C) $ w.r.t. the $ L^2 $ and the Cameron-Martin distances were established by Djellout et al. [7]; the $ {\bf T_2}(C) $ w.r.t. the uniform metric was obtained by Wu and Zhang [1]. Bao et al. [15] established the $ {\bf T_2}(C) $ w.r.t. both the uniform and the $ L^2 $ distances on the path space for the segment process associated to a class of neutral function stochastic differential equations. Saussereau [16] studied the $ {\bf T_2}(C) $ for SDE driven by a fractional Brownian motion, and Riedel [17] extent this result to the law of SDE driven by general Gaussian processes by using Lyons' rough paths theory.

For the infinite stochastic partial differential equation (SPDE for short), Wu and Zhang [18] studied the $ {\bf T_2}(C) $ w.r.t. $ L^2 $-metric by Galerkin's approximation. By Girsanov's transformation, Boufoussi and Hajji [19] obtained the $ {\bf T_2}(C) $ w.r.t. $ L^2 $-metric for the stochastic heat equation driven by space-time white noise and driven by fractional noise.

Recently, the fourth-order stochastic heat equation driven by a fractional noise was actively studied, for example, Jiang et al. [20] proved the existence, uniqueness and regularity of the solution, and established the large deviation principle of the equation with a small perturbation; Yang and Jiang [21] obtained the moderate deviation principle; Liu et al. [22] proved a weak approximation for SPDE (1.1).

In this paper, we shall study the $ {\bf T_2}(C) $ w.r.t. the uniform metric for the fourth-order stochastic heat equation driven by a fractional noise. This gives a positive answer to Boufoussi and Hajji's question in Remark 10 of [19].

The rest of this paper is organized as follows. In Section 2, we first give the definition of the fractional integral and the solution to eq. (1.1), and then state the main results of this paper. In Section 3, we prove the main result.

In this section, we first recall the fractional integral and give the known results about the solution of eq. (1.1), taking from Jiang et al. [20], and then state the main result of this paper.

Let $ W^H(t, x) = \frac{\partial ^2}{\partial t\partial x}B^H(t, x) $ be a fractional noise, where $ B^H = \{B^H(t, x), t\in[0, T], x\in D\} $ is a central Gaussian process defined on a complete probability space $ (\Omega, \mathcal F, \mathbb P) $, with the covariance function

$ {{{\rm{Cov}}}} (B^H(t, A), B^H(s, B)) = \frac12\left(t^{2H}+s^{2H}-|t-s|^{2H}\right)\lambda(A\cap B) $

with $ s, t\in [0, T] $ and $ A, B\in \mathcal B(D) $, $ \lambda $ is the Lebesgue measure. Denote by $ \Xi $ the set of step functions on $ [0, T]\times D $. Let $ \Theta $ be the Hilbert space defined as the closure of $ \Xi $ w.r.t. the scalar product determined by

$ \begin{equation} \langle 1_{[0, t]\times A}, 1_{[0, s]\times B} \rangle_{\Theta} = \mathbb{E} \left[B^H(t, A), B^H(s, B) \right]. \end{equation} $

(2.1)

Note that the covariance kernel of the fractional Brownian motion

$ R_H(t, s) = \frac12\left(t^{2H}+s^{2H}-|t-s|^{2H}\right) $

can be written as

$ R_H(t, s) = \int_0^{t\wedge s}K_H(t, r) K_H(s, r)dr, $

where

$ K_H(t, s) = c_H(t-s)^{H-\frac12}+c_H\left(\frac12-H \right) \int_s^t(u-s)^{H-\frac32}\left(1-\left(\frac{s}{u}\right)^{\frac12-H}\right)du $

with $ c_H = \sqrt{\frac{2H\Gamma\left(3/2-H\right)}{\Gamma\left(H+1/2\right)\Gamma(2-2H)}} $. Let

$ (K_H^*\phi)(s, x) = K_H(T, s)\phi(s, x)+ \int_s^T\left(\phi(r, x)-\phi(s, x)\frac{\partial K_H(r, s)}{\partial r} \right)dr. $

In particular,

$ \begin{equation} (K_H^* 1_{[0, t]\times A})(s, x) = K_H(t, s)1_{[0, t]\times A}(s, x). \end{equation} $

(2.2)

Then for any pair of step functions $ \varphi $ and $ \psi $ in $ \Xi $, we have

$ \begin{equation} \langle K_H^*\varphi, K_H^* \psi \rangle_{L^2([0, T]\times D)} = \langle \varphi, \psi \rangle_{\Theta}. \end{equation} $

(2.3)

Since the operator $ K_H^* $ provides an isometry between the Hilbert space $ \Theta $ and $ L^2([0, T]\times D) $, for any $ t\in [0, T] $ and $ A\in \mathcal B(D) $, there exists a space-time white noise $ W(t, A) $, which is defined by

$ \begin{equation} W(t, A) = B^H((K_H^*)^{-1}1_{[0, t]\times A}) \end{equation} $

(2.4)

satisfying that

$ \begin{equation} B^H(t, A) = \int_0^t \int_A K_H(t, s)W(ds, dy). \end{equation} $

(2.5)

See Nualart [23] for more properties about the fractional noise.

Definition 2.1   A random field $ u: = \{u(t, x), t\in[0, T], x\in D\} $ is said to be a solution of eq. (1.1), if for any $ \phi\in \mathcal C _0^{\infty}([0, T]\times D) $, $ u $ satisfies that for any $ t\in [0, T] $,

$ \begin{align} \langle u(t), \phi(t)\rangle = &\langle u_0, \phi(0)\rangle+ \int_0^t\langle u(s), \left(\frac{\partial }{\partial s}+\Delta^2\right)\phi(s) \rangle ds\\ &+ \int_0^t \langle f(u(s)), \phi(s) \rangle ds+ \int_0^t\langle W^H(s), \phi(s)ds. \end{align} $

(2.6)

Let $ G_t(x, y) $ be the Green function of $ \frac{\partial }{\partial t}+\Delta^2 $ with Neuman boundary conditions. Then by Walsh's theory [24], (2.6) is equivalent to the following form

$ \begin{align} u(t, x) = & \int_D G_t(x, y)u_0(y)dy+ \int_0^t \int_D G_{t-s}(s, y)f(u(s, y))dyds\\ &+ \int_0^t \int_D G_{t-s}(x, y) B^H(ds, dy) \end{align} $

(2.7)

for $ t\in[0, T] $ and $ x\in D $, where the last term is defined by

$ \int_0^t \int_D G_{t-s}(x, y) B^H(ds, dy) = \int_0^t \int_D K_H^*(t, s)G_{t-s}(x, y) W (ds, dy). $

Assume that the coefficient $ f: \mathbb{R}\rightarrow \mathbb{R} $ is Lipschitz continuous, i.e., there exists a constant $ L>0 $ such that

$ \begin{equation} |f(z_1)-f(z_2)|\le L|z_1-z_2|, \ \ \ z_1, z_2\in \mathbb{R}. \end{equation} $

(2.8)

The existence and uniqueness of the solution of eq. (1.1) was established in Jiang et al. [20].

Theorem 2.2 [20]   Let $ H\in \left(1/2, 1 \right) $. Under Lipschitzian condition (2.8), for any $ \mathcal F_0 $-adapted function $ u_0 $ such that $ \mathbb{E}\left[ \int_D|u_0(x)|^2dx\right]<\infty $, eq. (1.1) has a unique adapted mild solution $ \{u(t, x);(t, x)\in[0, T]\times D \} $, and moreover

$ \mathop {\sup }\limits_{(t, x)] \in [0, T] \times D} \mathbb{E}\left[|u(t, x)|^2\right]<\infty. $

Remark 2.2 The initial condition $ u_0 $ in Jiang et al. [20] is

$ \begin{equation} \mathop {\sup }\limits_{x \in [0, 1]} \mathbb{E}\left[|u_0(x)|^2\right]<\infty. \end{equation} $

(2.9)

In view of eq.(3.5) in the proof Theorem 3.1 of [20], it follows that condition (2.9) can be replaced by $ \mathbb{E}\left[ \int_D|u_0(x)|^2dx\right]<\infty $.

Let $ \mathcal C: = \mathcal C([0, T]\times {D}; \mathbb{R}) $ be the space of all continuous functions on $ [0, T]\times {D} $, which is endowed with

$ d_{\infty}(u, v) = \mathop {\sup }\limits_{(t, x) \in [0, T] \times D}|u(t, x)-v(t, x)|, \ \ \ \forall u, v\in \mathcal C. $

Let $ \mathbb{P}_u $ be the law of $ \{u(t, x); (t, x)\in[0, T]\times D\} $ on $ \mathcal C $. The main result of this paper is the following transportation inequality for $ \mathbb{P}_u $.

Theorem 2.3 Let $ H\in \left(3/4, 1 \right) $. Under Lipschitzian condition (2.8), for any $ \mathcal F_0 $-adapted function $ u_0 $ such that $ \mathbb{E}\left[ \int_D|u_0(x)|^2dx\right]<\infty $, there exists some constant $ C>0 $ such that the probability measure $ \mathbb{P}_u $ satisfies $ T_2(C) $ on the space $ \mathcal C([0, T]\times {D}) $ endowed with the metric $ d_{\infty} $.

For each $ t\in [0, T] $, denote by $ \mathcal F_t^{B^H} $ the $ \sigma $-field generated by the random variables $ \{B^H(s, A), s\in[0, t], A\in \mathcal B({D})\} $ and the sets of probability zero. We denote by $ \mathcal P $ the $ \sigma $-field of progressively measurable subsets of $ [0, T]\times \Omega $.

By formula (2.5), we know that $ \mathcal F_t^{B^H}\subset \mathcal F_t $, where $ \mathcal F_t $ is the filtration of the white-noise considered in (2.5).

Proof Clearly, it is enough to prove the result for any probability measure $ \mathbb{Q} $ on $ \mathcal C $ such that $ \mathbb{Q}\ll \mathbb{P}_u $ and $ H( \mathbb{Q}| \mathbb{P}_u)<\infty $. We divide the proof into two steps.

Step 1 We shall closely follow the arguments in [7]. Define

$ \tilde{\mathbb{Q}}: = \frac{d \mathbb{Q}}{d \mathbb{P}_u}(u(\cdot, \cdot)) \mathbb{P}. $

The process $ \{u(t, x), (t, x)\in[0, T]\times {D}\} $ is adapted with respect to the white-noise filtration $ \mathcal F_t = \sigma\{W(s, A), s\le t, A\in \mathcal B(D)\} $. Notice that $ \frac{d \mathbb{Q}}{d \mathbb{P}_u}(u(\cdot, \cdot)) $ is an $ \mathcal F_T $-measurable random variable. Since $ \mathbb{Q} $ is a probability measure on $ \mathcal C $ and the law of $ u $ under $ \mathbb{P} $ is $ \mathbb{P}_u $, then

$ \int_{\Omega}\frac{d \mathbb{Q}}{d \mathbb{P}_u}(u)d \mathbb{P} = \int_{{\mathcal C}}\frac{d \mathbb{Q}}{d \mathbb{P}_u}(\omega)d \mathbb{P}_u(\omega) = \mathbb{Q}({\mathcal C}) = 1. $

Then $ \frac{d \mathbb{Q}}{d \mathbb{P}_u}(u(\cdot, \cdot)) $ is integrable and the process

$ M_t: = \mathbb{E}_{ \mathbb{P}}\left[ \frac{d \mathbb{Q}}{d \mathbb{P}_u}(u) \mid\mathcal F_t\right], \ \ \ 0\le t\le T $

is an $ \mathcal F_t $-martingale. Let $ \tau = \inf\{t\in [0, T];M_t = 0\}\wedge T $ with the convention that $ \inf\emptyset = +\infty $. Then $ \tau $ is an $ (\mathcal F_t) $-stopping time and $ \tilde{\mathbb{Q}}(\tau = T) = 1. $ Thus, we can write

$ M_t = 1_{[t<\tau]}\exp\left(L_t-\frac12\langle L, L\rangle_t\right), $

where

$ L_t = \int_0^t\frac{dM_s}{M_s}, \ \ \ \forall t<\tau. $

Following the argument as that in Djellout et al. [7], we have

$ \begin{equation} {\bf H}( \mathbb{Q}| \mathbb{P}_u) = {\bf H}(\widetilde{\mathbb{Q}}| \mathbb{P}) = \mathbb{E}_{ \mathbb{P}}\left[M_T\log M_T\right] = \mathbb{E}_{\widetilde{\mathbb{Q}}}[\log M_T]. \end{equation} $

(3.1)

On the other hand, consider

$ \tau_n: = \inf\{t\in[0, \tau];\langle L, L\rangle_t = n \}\wedge \tau, \ \ \ n\ge1. $

Then $ M $ is $ \mathbb{P}- $a.s. continuous and since $ \tau_n\uparrow \tau $ a.s., we have, by (3.1),

$ \begin{equation} {\bf H}( \mathbb{Q}| \mathbb{P}_u) = {\bf H}(\widetilde Q| \mathbb{P}) = \lim\limits_{n\rightarrow \infty} \mathbb{E}_{\widetilde{\mathbb{Q}}}[ L_{T\wedge \tau_n}-\frac12\langle L, L\rangle_{T\wedge \tau_n}]. \end{equation} $

(3.2)

By Girsanov's theorem, $ \{L_{t\wedge \tau_n}-\frac12\langle L, L\rangle_{t\wedge \tau_n}\}_{t\ge0} $ is a $ \widetilde{\mathbb{Q}} $-square integrable martingale with respect to the filtration $ (\mathcal F_t) $. Then, by the martingale representation theorem w.r.t. the Brownian sheet (see [25, Theorem 1.3]), there exists an adapted process $ (h_n(t, x))_{t\le T, x\in {D}} $ such that $ \int_0^t \int_D h_n^2(s, x)dsdx<\infty, \mathbb{P}- $a.s., and

$ L_n(t) = \int_0^t \int_D h_n(s, x)W(ds, dx), \ \ \mathbb{P}-\hbox{a.s.}, t\in [0, \tau]. $

Recall that $ \widetilde {\mathbb{Q}}[\lim\limits_{n\rightarrow \infty}\tau_n = \tau = T] = 1 $. Let

$ h\left( {t, x} \right): = \left\{ \begin{array}{l} {h_n}\left( {t, x} \right), \;\;\;{\rm{if}}\;t \le {\tau _n}, \\ 0, \;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;t > \tau , \end{array} \right.$

(3.3)

and

$ \begin{equation} L(t) = \int_0^t \int_D h(s, x)W(ds, dx), \ \ \tilde {\mathbb{Q}}-\text {a.s.}, t\in[0, \tau]. \end{equation} $

(3.4)

Combining (3.2) and (3.4), we obtain that

$ {\bf H}( \mathbb{Q}| \mathbb{P}_u) = {\bf H}(\tilde {\mathbb{Q}}| \mathbb{P}) = \frac12 \mathbb{E}_{\tilde {\mathbb{Q}}}\left[ \int_0^T \int_D h^2(s, x)dsdx\right]. $

Step 2 By Girsanov's theorem for the Brownian sheet (see [26, Proposition 1.6]), under $ \widetilde {\mathbb{Q}} $,

$ \widetilde W(dt, dx): = -h(t, x)dtdx+W(dt, dx) $

is a space-time white noise, where $ W $ is the space-time noise considered in (2.5).

Consider the $ \widetilde {\mathbb{Q}} $-fractional noise $ \widetilde W^H $ associated to $ \widetilde W $ defined by

$ \widetilde W^H(t, x) = \int_0^t \int_0^x K_H(t, s)d\widetilde W(s, y) = \int_0^t \int_0^x K_H(t, s)dW(s, y)-(K_Hh)(t, x) $

with

$ (K_H h)(t, x): = \int_0^t \int_0^x K_H(t, s)h(s, y)dsdy. $

Since for any $ H>1/2 $, the kernel $ K_H(t, s) $ is

$ K_H(t, s) = c_H s^{\frac12-H} \int_s^t (u-s)^{H-\frac32}u^{H-\frac12}du, $

by Fubini's theorem, we have

$ \begin{align*} (K_H h)(t, x) = &c_H \int_0^t \int_0^x\left(s^{\frac12-H} \int_s^t (u-s)^{H-\frac32}u^{H-\frac12}du\right)h(s, y)ds dy\\ = :& \int_0^t \int_0^x g(s, y)dsdy, \end{align*} $

where $ g(s, y) = c_H s^{H-\frac12} \int_0^s r^{\frac12-H}(s-r)^{H-\frac32}h(r, y)dr $. Consequently, under $ \widetilde{ \mathbb{Q}} $, the process $ u(t, x) $ satisfies that for all $ t\ge0, x\in {D} $,

$ \begin{align} u(t, x) = & \int_D G_t(x, y)u_0(y)dy+ \int_0^t \int_D G_{t-s}(s, y)f(u(s, y))dsdy\\ &+ \int_0^t \int_D G_{t-s}(x, y) \tilde W^H(ds, dy) + \int_0^t \int_D G_{t-s}(x, y)g(s, y)ds dy. \end{align} $

(3.5)

Now, consider $ \{v(t, x), (t, x)\in[0, T]\times {D}\} $ the solution of equation (1.1). By the uniqueness of the solution in Theorem 2.2, we know that the law of $ \{v(t, x), (t, x)\in[0, T]\times {D}\} $ is $ \mathbb{P}_u $. Thus, $ \{(u(t, x), v(t, x)), (t, x)\in[0, T]\times {D}\} $ under $ \tilde{ \mathbb{Q}} $, is a coupling of $ ( \mathbb{Q}, \mathbb{P}_u) $, which implies that

$ \begin{equation} \left(W_{2, d_{\infty}}( \mathbb{Q}, \mathbb{P}_u)\right)^2\le \mathbb{E}_{\tilde{ \mathbb{Q}}}(d_{\infty}(u, v)^2) = \mathbb{E}_{\tilde{ \mathbb{Q}}}\left[\sup\limits_{t\in[0, T], x\in {D}}|u(t, x)-v(t, x)|^2 \right]. \end{equation} $

(3.6)

Next, we estimate the distance between $ u $ and $ v $ with respect to $ d_{\infty} $-metric.

For any $ (t, x)\in[0, T]\times {D} $,

$ \begin{align} u(t, x)-v(t, x) = & \int_0^t \int_DG_{t-s}(x, y)\left[f(s, u(s, y))-f(s, v(s, y))\right]ds dy\\ &+ \int_0^t \int_D G_{t-s}(x, y)g(s, y)ds dy\\ = : &I_1(t, x)+I_2(t, x). \end{align} $

(3.7)

By (2.8) and Hölder's inequality, we have

$ \begin{equation} \begin{aligned} \label{I1} & \mathbb{E}\left[\sup\limits_{(s, x)\in[0, T]\times {D}}|I_1(t, x)| ^2\right]\notag\\ \le\ & c(L)\left(\sup\limits_{(s, x)\in [0, T]\times D} \int_0^t \int_D G_s^2(x, y)dsdy\right) \times \mathbb{E}\left[ \int_0^t\sup\limits_{(r, x)\in[0, s]\times D}|u(r, x)-v(r, x)|^2dr\right]. \end{aligned} \end{equation} $

We use the approach in Boufoussi and Hajji [19] to estimate the second term $ I_2 $:

$ \begin{align} I_2(t, x) = & \int_0^t \int_DG_{t-s}(x, y)g(s, y)dyds\\ = & c_H \int_D \int_0^t u^{\frac12-H}h(u, y)\left( \int_u^t s^{H-\frac12}(s-u)^{H-\frac32}G_{t-s}(x, y)ds\right)dudy. \end{align} $

(3.8)

By Lemma 1.2 of [27], we know that

$ |G_t(x, y)|\le\frac{c_1}{t^{\frac14}}\exp\left\{-c_2\frac{|x-y|^{\frac{4}{3}}}{t^{\frac13}}\right\}. $

Thus, we have

$ \begin{align} \left| \int_u^t s^{H-\frac12}(s-u)^{H-\frac32}G_{t-s}(x, y)ds\right| \le &\left| \int_u^t s^{H-\frac12}(s-u)^{H-\frac32}\frac{C_T}{(t-s)^{\frac14}}ds\right|\\ \le & C_T T^{H-\frac12}\beta\left(H-\frac12, \frac34\right)(t-u)^{H-\frac54}. \end{align} $

(3.9)

Putting (3.8) and (3.9) together, we have

$ \begin{align} &\sup\limits_{(t, x)\in [0, T]\times D}|I_2(t, x)|^2\\ \le &c_H^2C_T^2 T^{2H-1}\beta^2\left(H-\frac12, \frac34\right)\beta\left(2-2H, 2H-\frac32\right)\times \int_0^T \int_D h^2(s, y) dsdy. \end{align} $

(3.10)

Putting (3.7), (3.8) and (3.10) together,

$ \begin{align} & \mathbb{E}_{\widetilde{ \mathbb{Q}}}\left[\sup\limits_{(s, x)\in [0, t]\times D}|u(s, x)-v(s, x)|^2\right]\\ \le & c(L) \int_0^t \mathbb{E}_{\widetilde{ \mathbb{Q}}} \left[\sup\limits_{(r, x)\in [0, s]\times {D}}|u(r, x)-v(r, x)|^2\right]ds+ \gamma \mathbb{E}_{\widetilde{ \mathbb{Q}}}\left[ \int_0^t \int_D h^2(s, y) dsdy\right], \end{align} $

(3.11)

here $ \gamma = c_H^2C_T^2 T^{2H-1}\beta^2\left(H-\frac12, \frac34\right)\beta\left(2-2H, 2H-\frac32\right) $.

Applying Gronwall's inequality to $ g(t): = \mathbb{E}_{\widetilde{ \mathbb{Q}}}\left[\sup\limits_{(s, x)\in[0, t]\times{D}}|u(s, x)-v(s, x)|^2 \right] $, by (5.11), we have

$ \mathbb{E}_{\widetilde{ \mathbb{Q}}}\left[\sup\limits_{(t, x)\in[0, T]\times D}|u(t, x)-v(t, x)|^2\right] \le \gamma\exp^{c(L)T} \mathbb{E}_{\widetilde{ \mathbb{Q}}}\left[ \int_0^T \int_D h^2(s, y)dsdy\right], $

which is the desire result.

The proof is completed.