非局部随机扩散方程解的HO^¨lder连续性

HO^¨lder Continuous of the Solutions to Nonlocal Stochastic Diffusion Equations

摘要: 本文的目标是获得非局部随机扩散方程解的Hölder连续性。利用Campanato估计和Sobolev嵌入定理，首先证明了非局部随机扩散方程的温和解的Hölder连续性，即解u属于空间C^β(D_T;L^p(Ω))。其次，通过使用尾估计，得到了L^p(Ω;C^β*(D_T)中的温和解的估计。

Abstract: In this paper, we aim to obtain the Hölder continuous of solutions to nonlocal stochastic equations. By using Campanato estimates and Sobolev embedding theorem, we first prove the Hölder con-tinuous of the mild solution of nonlocalstochastic diffusion equations in the sense that the solution u belongs to the space C^β(D_T;L^p(Ω)). Then by using tail estimates, we obtain the estimates of the mild solution in L^p(Ω;C^β*(D_T).

1. 引言

设 $T>0$ ， $D\subset ℝ$ 并且 $D_T:=\left[0,T\right]\times D$ 。设 $\left\{\Omega ,\mathcal{F},{\left\{\mathcal{F}\right\}}_{t\ge 0},\mathbb{P}\right\}$ 是一个给定的过滤概率空间。文章 [1] [2] 获得了 ${\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(ℝ\right)\right)$ 空间(见定义2.1)的奇异随机积分的正则性，其中 $p>1,\theta >0,\delta >0$ 。进一步，借助Sobolev嵌入定理对于任意的 $\theta >1$ ， ${\mathcal{L}}^{p,\theta }\left(D;\delta \right)\mapsto C^{\gamma }\left(D_T;L^P\left(\Omega \right)\right)$ ，获得了在Hölder空间 $C^{\gamma }\left(D_T;L^p\left(\Omega \right)\right)$ ( $\gamma =\frac{3\left(\theta -1\right)}{p}$ )中解的估计。本文的目标是得到 $L^p\left(\Omega ;C^{\beta }\left(D_T\right)\right)$ ( $\beta =\frac{3\left(\theta -1\right)}{p}$ )中解的估计，我们考虑如下方程

$d{u}_t={\Delta }^{\alpha }udt+g\left(t,x\right)d{B}_t^{\hslash },{u|}_{t=0}=0,$ (1)

其中 ${\Delta }^{\alpha }=-{\left(-\Delta \right)}^{\alpha }$ ， $B_t^{\hslash }$ 为具有Hurst参数 $\hslash \in \left(0,\frac{1}{2}\right)$ 的一维分数布朗运动。设方程(1)的mild解记为

$\mathcal{K}g\left(t,x\right):={\displaystyle {\int }_0^t{\displaystyle {\int }_{ℝ}K}}\left(t-r,y\right)g\left(r,x-y\right)dyd{B}^{\hslash }\left(t\right).$

非局部扩散方程的起源可以追溯到1899年分子间作用力的发现，分子间作用力的发现打破了牛顿的传统力学体系。于是，物理学上的突破也引申到数学上，与牛顿力学体系对应的是我们熟知的局部扩散方程。而与分子间作用力对应的方程是非局部扩散方程，作为微分方程的重要分支，非局部扩散方程从20世纪70年代才开始逐步发展并且成熟起来。非局部扩散方程比很多其他方程有更广泛的应用，渗透在生物模型，化学反应模型，物理模型等各个方面。

让我们首先回顾一下关于随机偏微分方程(SPDE)解的正则性的相关结论。与确定性抛物方程一样，随机偏微分方程的正则性研究主要包括两个部分：一个是L_p理论，另一个是Schauder理论(Hölder连续估计)。关于L_p理论的最早结果出现在Krylov的 [3] [4] 中。Kim-Kim [5] 考虑了由Lévy过程驱动的随机偏微分方程的L_p理论。Zhang [6] 获得了一般测度空间上半线性随机偏微分方程的L_p理论并且 [7] 证明了确定性非局部抛物偏微分方程的L_p极大正则性和Cauchy过程驱动的随机微分方程的Krylov估计。

分数布朗运动出现在许多复杂现象模型的应用中。当分数布朗运动的Hurst参数 $\hslash =\frac{1}{2}$ 时，分数布朗运动成为标准布朗运动。分数布朗运动与标准布朗运动的不同之处在于分数布朗运动缺少半鞅性、马尔可夫性和其他经典的随机过程理论 [8] 。分数布朗运动作为一个中心高斯过程，它的特点是其增量的平稳性和中或长记忆的性质。长记忆特性使分数布朗运动成为许多领域建模的候选者 [9] 。对于与分数布朗运动有关的数学模型是近年来的研究热点。如Boufoussi和Hajji [10] 研究了由分数布朗运动驱动的中立型随机泛函微分方程解的存在唯一性和指数渐近行为；Caraball等 [11] 研究了具有分数布朗运动驱动的随机延迟演化方程解的存在性和指数行为；Li和Yan [12] 讨论了在希尔伯特空间上由分数布朗运动驱动的一类中立随机泛函微分方程的遍历性和平稳解；Xu等 [13] 讨论了由分数布朗运动驱动的二阶中立型随机泛函微分方程的全局吸引集和指数衰减。

许多作者对随机偏微分方程的Hölder估计越来越感兴趣。Hsu等 [14] 建立了半线性随机偏微分方程的随机De Giorgi迭代和正则性。Du和Liu [15] 获得了随机偏微分方程解的Schauder估计。结合确定性理论和卷积性质，Debussche等 [16] 建立了抛物型拟线性随机偏微分方程的正则性结果。Kuksin等 [17] 获得了有界域上抛物型随机偏微分方程解的Hölder估计。最近，Tian等 [18] 利用尾估计获得了非局部随机扩散方程温和解的Hölder估计。论文 [19] 考虑了由布朗运动驱动的非局部随机抛物型偏微分方程的正则性。本文旨在将 [19] 的结果推广到由Hurst参数 $\hslash \in \left(0,1/2\right)$ 的分数布朗运动驱动的抛物型非局部随机偏微分方程的正则性。

2. 预备知识

假设对于 $X=\left(t,x\right)\in ℝ\times ℝ$ ， $Y=\left(s,y\right)\in ℝ\times ℝ$ ，有

$\delta \left(X,Y\right):=\max \left\{\left|x-y\right|,{\left|t-s\right|}^{\frac{1}{2}}\right\}.$

假设 $Q_c\left(X\right)$ 是一个以 $X=\left(t,x\right)$ 为圆心， $c>0$ 为半径的球，即

$Q_c\left(X\right):=\left\{Y=\left(s,y\right)\in ℝ\times ℝ:\delta \left(X,Y\right)<c\right\}=\left(t-c^2,t+c^2\right)\times B_c\left(x\right).$

固定任意的 $T\in \left(0,\infty \right)$ ， ${\mathcal{O}}_T:=\left(0,T\right)\times ℝ$ 表示一个光滑的区域。设D是一个有界区域 $D\in ℝ$ ，对于一个点 $X\in D_T$ ， $D\left(X,r\right):=D_T\cap Q_r\left(X\right)$ 并且 $d\left(D\right)=diam\left(D\right)$ 。我们首先介绍Campanato空间的定义。

定义2.1 [20] 设 $p\ge 1$ ， $\theta \ge 0$ 。如果一个函数u满足

${\left[u\right]}_{{\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)}={\left(\underset{X\in D_T,d\left(D\right)\ge \rho >0}{\sup }\frac{1}{{\left|D\left(X,c\right)\right|}^{\theta }}{{\displaystyle {\int }_{D\left(X,\rho \right)}\left|u\left(Y\right)-u_{X,\rho }\right|}}^{p}dY\right)}^{\frac{1}{p}}<\infty ,u\in L^p\left(D_T\right),$

其中 $\left|D\left(X,\rho \right)\right|$ 表示Borel集 $D\left(X,\rho \right)$ 的Lebesgue测度并且

$u_{X,\rho }:=\frac{1}{D\left(X,\rho \right)}{\displaystyle {\int }_{D\left(X,\rho \right)}u}\left(Y\right)dY,$

则称u属于Campanato空间 ${\mathcal{L}}^{p,\theta }\left(D;\delta \right)$ ，该空间为 $L^p\left(D\right)$ 的子空间。对于 $u\in {\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)$ ，定义

${\Vert u\Vert }_{{\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)}:={\left({\Vert u\Vert }_{L^p{\left(D_T\right)}^p}+{\left[u\right]}_{{\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)}\right)}^{\frac{1}{p}}.$

定义2.2 [19] 设 $0<\gamma \le 1$ ，如果函数u满足

${[u]}_{C^{\gamma }\left({\bar{D}}_T;\delta \right)}:=\underset{X\in D_T,d\left(D\right)\ge \rho >0}{\sup }\frac{\left|u\left(X\right)-u\left(Y\right)\right|}{\delta {\left(X,Y\right)}^{\gamma }}<\infty ,$

则称u属于Hölder空间 $C^{\gamma }\left({\bar{D}}_T;\delta \right)$ 。对于 $u\in C^{\gamma }\left({\bar{D}}_T;\delta \right)$ ，定义

${\Vert u\Vert }_{C^{\gamma }\left({\bar{D}}_T;\delta \right)}:=\underset{D_T}{\sup }\left|u\right|+{\left[u\right]}_{C^{\gamma }\left({\bar{D}}_T;\delta \right)}.$

定义2.3 设 $D_T\subset ℝ$ 是一个域，如果存在 $A>0$ 使得对于任意的 $X\in D_T$ ， $0<\rho \le d\left(D\right)$ ，使得 $\left|D_T\left(X,\rho \right)\right|=\left|D_T\cap Q_{\rho }\left(X\right)\right|\ge A\left|Q_{\rho }\left(X\right)\right|$ ，那么域 $D_T$ 叫做A-类。

下面是一些符号说明。

给定两个集合 $B_1$ 和 $B_2$ ，关系 $B_1=B_2$ 意味着 $B_1\subset B_2$ 并且 $B_2\subset B_1$ 。符号 $f\left(x\right)\approx g\left(x\right)$ 表示有一个独立于x的数字 $0<C<\infty$ ，即一个常数，这样对于每个x，我们都有 $C^{-1}f\left(x\right)\le g\left(x\right)\le Cf\left(x\right)$ 。然后我们有下面关于上面定义的两个空间的比较的关系。

命题2.4 设 $D_T$ 是一个有界区域，则对于任意 $p\ge 1$ 和 $1<\theta <1+\frac{p}{3}$ ，有 ${\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)\cong C^{\gamma }\left({\bar{D}}_T;\delta \right)$ ，其中 $\gamma =\frac{3\left(\theta -1\right)}{p}$ 。

我们想用尾估计来得到随机偏微分方程的解u的有界性结果

$\mathbb{E}{\Vert u\Vert }_{C^{\gamma }\left(\left[0,T\right)\times D\right)}^p\le C,p\ge 1.$

将使用尾估计和Hölder空间与Campanato空间之间的等价，事实上对于任意常数 $M>0$ ，

$\begin{array}{c}\mathbb{E}\left({\left|X\right|}^p\right)={\displaystyle {\int }_{\Omega }{\left|X\right|}^{p}d\mathbb{P}\left(\omega \right)}\\ =p{\displaystyle {\int }_0^{\infty }\mathbb{P}\left\{\left|X\right|>a\right\}{a}^{p-1}da}\\ =p{\displaystyle {\int }_0^M\mathbb{P}\left\{\left|X\right|>a\right\}{a}^{p-1}da}+p{\displaystyle {\int }_M^{\infty }\mathbb{P}\left\{\left|X\right|>a\right\}{a}^{p-1}da}\\ \le M^P+p{\displaystyle {\int }_M^{\infty }\mathbb{P}\left\{\left|X\right|>a\right\}{a}^{p-1}da},\end{array}$

为了得到 $L^p$ 有界性，根据上面的不等式，我们只需要证明第二个积分是有界的。

命题2.5 [21] 设 $u_0\in L^p\left(ℝ,\Omega \right)$ 。考虑柯西问题

${\partial }_{t}u\left(t,x\right)={\Delta }^{\alpha }u\left(t,x\right),t>0,x\in ℝ;u\left(0,x\right)=u_0\left(x\right),$ (2)

那么对于任意的 $0<\beta <1$ ， ${\Vert u\left(t,\cdot \right)\Vert }_{C^{\beta }\left(ℝ\right)}\le C{t}^{-\frac{\beta }{2\alpha }-\frac{1}{2p\alpha }}{\Vert u_0\Vert }_{L^p\left(ℝ\right)}$ ，并且对于任意的 $\tau >0$ ， $\left|u\left(t+\tau ,x\right)-u\left(t,x\right)\right|\le C{t}^{-\beta -\frac{1}{2p\alpha }}{\tau }^{\beta }{\Vert u_0\Vert }_{L^p\left(ℝ\right)}$ 。

我们回顾一些满足 $K_t={\Delta }^{\alpha }K$ 的核函数 $K\left(t,x,y\right)$ 的一些性质(参见 [21] [22] [23] [24] )。

1) 对于任意的 $t>0$ ， ${\Vert K\left(t,\cdot \right)\Vert }_{L^1\left(ℝ\right)}=1$ ；

2) 对于每一个 $t>0$ ， $K\left(t,x,y\right)$ 在区间 $\left(,\infty \right)$ 中属于 $C^{\infty }$ ；

3)对于任意的 $t>0$ ， $x,y\in ℝ$ ， $x\ne y$ ， $K\left(t,x\right)$ 的渐进估计为 $\left|{\nabla }_{x}K\left(t,x,y\right)\right|\approx \left|y-x\right|\min \left\{\frac{t}{{\left|y-x\right|}^5},t^{-\frac{3}{2}}\right\}$ 。

最后，我们介绍关于分数布朗运动的维纳积分的一些基本结果。设 $\left\{\Omega ,\mathcal{F},\mathbb{P},{\left\{\mathcal{F}\right\}}_{t\ge 0}\right\}$ 是一个过滤概率空间。假设 $B^{\hslash }\left(t\right)$ ， $t\in \left[0,T\right]$ 是具有Hurst参数 $\hslash \in \left(0,\frac{1}{2}\right)$ 的一维分数布朗运动。这意味着 $B^{\hslash }$ 是一个中心高斯过程，协方差函数为

$R_{\hslash }\left(t,s\right)=\mathbb{E}\left(B_t^{\hslash }{B}_s^{\hslash }\right)=\frac{1}{2}\left(t^{2\hslash }+s^{2\hslash }-{\left|t-s\right|}^{2\hslash }\right),$

考虑核

$K_{\hslash }\left(t,s\right)=c_{\hslash }\left[{\left(\frac{t}{s}\right)}^{\hslash -\frac{1}{2}}{\left(t-s\right)}^{\hslash -\frac{1}{2}}-\left(\hslash -\frac{1}{2}\right)s^{\frac{1}{2}-\hslash }{\displaystyle {\int }_s^t{(u-s)}^{\hslash -\frac{1}{2}}}{u}^{\hslash -\frac{3}{2}}du\right],0<\hslash <\frac{1}{2},t>s,$

其中 $c_{\hslash }=\sqrt{\frac{\hslash }{\left(1-2\hslash \right)\beta \left(1-2\hslash ,\hslash +\frac{1}{2}\right)}}$ ， $\beta \left(\cdot ,\cdot \right)$ 表示Beta函数。如果 $t\le s$ ，则 $K_{\hslash }\left(t,s\right)=0$ 。由上式可知

$\left|K_{\hslash }\left(t,s\right)\right|\le 2c_{\hslash }\left({\left(t-s\right)}^{\hslash -\frac{1}{2}}+s^{\hslash -\frac{1}{2}}\right).$ (3)

由 [25] 知，

$\left|\frac{\partial K_{\hslash }}{\partial t}\left(t,s\right)\right|\le c_{\hslash }\left(\frac{1}{2}-\hslash \right){\left(t-s\right)}^{\hslash -\frac{3}{2}}.$ (4)

考虑从 $\mathcal{H}$ 到 $L^2\left(\left[0,T\right]\right)$ 的算子 $K_{\hslash ,T}^{*}$ [13] ，

$\left(K_{\hslash ,T}^{*}\phi \right)\left(s\right)=K_{\hslash }\left(T,s\right)\phi \left(s\right)+{\displaystyle {\int }_s^T\left(\phi \left(r\right)-\phi \left(s\right)\right)}\frac{\partial K_{\hslash }}{\partial r}\left(r,s\right)dr,$ (5)

则 $K_{\hslash ,T}^{*}$ 是一个 $\mathcal{H}$ 和 $L^2\left(\left[0,T\right]\right)$ 之间的等距算子。

考虑 $W=\left\{W_t,t\in \left[0,T\right]\right\}$ ，定义 $W_t=B^{\hslash }\left({\left(K_{\hslash }^{\ast }\right)}^{-1}\left(1_{\left[0,1\right]}\right)\right)$ ，则W是一个维纳过程，并且过程 $B^{\hslash }$ 有积分表示 $B^{\hslash }\left(t\right)={\displaystyle {\int }_0^t{K}_{\hslash }}\left(t,s\right)dW\left(s\right)$ 。此外，对于任何一个 $\phi \in \mathcal{H}$ ，我们都有

${\displaystyle {\int }_0^T\phi }\left(s\right)\text{d}{B}^{\hslash }\left(s\right)={\displaystyle {\int }_0^T\left(K_{\hslash }^{\ast }\phi \right)\left(t\right)dW}\left(t\right).$ (6)

3. Hölder估计

本文的假设如下：

(H1)对于任意的 $t,s\in \left[0,T\right]$ ， $z\in ℝ$ 且 $q>1$ ，有 ${\displaystyle {\int }_{ℝ}{\left|K\left(t,z\right)-K\left(s,z\right)\right|}^q\text{d}z}\le {\left(t-s\right)}^{q{\gamma }_1}$ ，其中 ${\gamma }_1$ 为一个依赖于q的常数。

(H2)对于任意的 $t,s\in \left[0,T\right]$ ， $z\in ℝ$ 且 $p>1$ ，有 ${\displaystyle {\int }_{ℝ}{\left|g\left(t,z\right)-g\left(s,z\right)\right|}^p\text{d}z}\le {\left(t-s\right)}^{p{\gamma }_2}$ ，其中 ${\gamma }_2$ 为一个依赖于p的常数。

定理3.1设D是一个A-类有界域使得 $\bar{D}\subset {\mathcal{O}}_T$ 。假设对于 $p>\frac{1}{\alpha }$ ， $g\in L_{loc}^{\infty }\left({ℝ}_{+};L^p\left(\Omega \times ℝ\right)\right)$ 是一个 ${\mathcal{F}}_t$ 适应过程， $0<\beta <\alpha$ 且 $\left(\alpha -\beta \right)p-1\ge 0$ 。(H1)和(H2)成立。则方程(1)存在mild解u，使得 $u\in {\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(\Omega \right)\right)\cap L^p\left(\Omega ;C^{\beta }\left(D_T\right)\right)$ ，且

${\Vert u\Vert }_{{\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(\Omega \right)\right)}\le C{\Vert g\Vert }_{L^{\infty }\left(\left[0,T\right];L^p\left(\Omega \times ℝ\right)\right)},$ (7)

${\Vert u\Vert }_{C^{\beta }\left(D_T;L^p\left(\Omega \right)\right)}\le C{\Vert g\Vert }_{L^{\infty }\left(\left[0,T\right];L^p\left(\Omega \times ℝ\right)\right)},$ (8)

其中 $\theta =1+\frac{\beta p}{3}$ 。进一步，取 $0<ϵ<\frac{\beta p}{2}$ ， $q>\frac{3}{ϵ}$ ，对于任意的 $1<r<q$ ，

${\Vert u\Vert }_{L^r\left(\Omega ;C^{{\beta }^{\ast }}\left(D_T\right)\right)}\le C{\Vert g\Vert }_{L^{\infty }\left(\left[0,T\right];L^{pq}\left(\Omega \times ℝ\right)\right)},$ (9)

其中 ${\beta }^{\ast }=\beta -\frac{2ϵ}{p}$ 。

证明：在上述假设下方程(1)温和解的存在性是一个经典结果。现在我们证明不等式(7)。由于Campanato空间的定义，只需证明 ${\left[u\right]}_{{\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(\Omega \right)\right)}^p<\infty$ 。

首先，根据定义2.1有，

$\begin{array}{c}{\left[u\right]}_{{\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(\Omega \right)\right)}^p\le \underset{D\left(X,c\right),X\in D_T,0<c\le d\left(D\right)}{\sup }\frac{1}{{\left|D\left(X,c\right)\right|}^{1+\theta }}\mathbb{E}{\displaystyle {\int }_{D\left(X,c\right)}{\displaystyle {\int }_{D\left(X,c\right)}{\left|u\left(t,x\right)-u\left(s,y\right)\right|}^{p}dtdxdsdy}}\\ \le \underset{D\left(X,c\right),X\in D_T,0<c\le d\left(D\right)}{\sup }\frac{1}{{\left|D\left(X,c\right)\right|}^{1+\theta }}\mathbb{E}{\displaystyle {\int }_{D\left(X,c\right)}{\displaystyle {\int }_{D\left(X,c\right)}|{\displaystyle {\int }_0^t{\displaystyle {\int }_{ℝ}K}}\left(t-r,x-z\right)g\left(r,z\right)dzd{B}^{\hslash }\left(r\right)}}\\ {-{\displaystyle {\int }_0^s{\displaystyle {\int }_{ℝ}K}}\left(s-r,y-z\right)g\left(r,z\right)\text{d}z\text{d}{B}^{\hslash }\left(r\right)|}^{p}dtdxdsdy\\ :=\underset{D\left(X,c\right),X\in D_T,0<c\le d\left(D\right)}{\sup }\frac{1}{{\left|D\left(X,c\right)\right|}^{1+\theta }}{\displaystyle {\int }_{D\left(X,c\right)}{\displaystyle {\int }_{D\left(X,c\right)}\mathbb{E}}}{\rm Y}dtdxdsdy.\end{array}$

由公式(5)和(6)，有

$\begin{array}{c}\mathbb{E}{\rm Y}\le 2^{p-1}{\left|{\displaystyle {\int }_0^s{\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)}}g\left(r,z\right)dzd{B}^{\hslash }\left(r\right)\right|}^p\\ +2^{p-1}{\left|{\displaystyle {\int }_s^t{\displaystyle {\int }_{ℝ}K}}\left(s-r,y-z\right)g\left(r,z\right)dzd{B}^{\hslash }\left(r\right)\right|}^p\\ \le 2^{p-1}{\left|{\displaystyle {\int }_0^s{K}_{\hslash }^{*}\left[{\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz}\right]}\left(r\right)dW\left(r\right)\right|}^p\\ +2^{p-1}{\left|{\displaystyle {\int }_s^t{K}_{\hslash }^{*}}\left[{\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dz\right]\left(r\right)dW\left(r\right)\right|}^p\end{array}$

$\begin{array}{c}\le 2^{p-1}|{\displaystyle {\int }_0^s[K_{\hslash }\left(s,r\right){\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz\left(r\right)}\stackrel{}{}}\\ +{\displaystyle {\int }_r^s[{\displaystyle {\int }_{ℝ}\left(K\left(t-m,x-z\right)-K\left(s-m,y-z\right)\right)g\left(m,z\right)dz\left(r\right)}}\\ -{\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz\left(r\right)}]{\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\text{d}m]dW\left(r\right)|}^p\\ +2^{p-1}|{\displaystyle {\int }_s^t[K_{\hslash }\left(t,r\right){\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dz\stackrel{}{}}\\ +{{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)dz-{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)g\left(r,z\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm}]dW\left(r\right)|}^p\end{array}$

$\begin{array}{c}\le 2^{2p-2}{\left|{\displaystyle {\int }_0^s{K}_{\hslash }}\left(s,r\right){\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz\left(r\right)dW\left(r\right)}\right|}^p\\ +2^{2p-2}|{\displaystyle {\int }_0^s{\displaystyle {\int }_r^s[{\displaystyle {\int }_{ℝ}\left(K\left(t-m,x-z\right)-K\left(s-m,y-z\right)\right)g\left(m,z\right)dz}}}\\ -{{\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz}]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dmdW\left(r\right)|}^p\\ +2^{2p-2}{\left|{\displaystyle {\int }_s^t{K}_{\hslash }}\left(t,r\right){\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dzdW\left(r\right)\right|}^p\\ +2^{2p-2}{\left|{\displaystyle {\int }_s^t{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)dz-{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)g\left(r,z\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dmdW\left(r\right)}}\right|}^p\end{array}$

$\begin{array}{c}=2^{2p-2}{\left|{\displaystyle {\int }_0^s{K}_{\hslash }}\left(s,r\right){\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g(r,z)dz\left(r\right)}dW\left(r\right)\right|}^p\\ +2^{2p-2}|{\displaystyle {\int }_0^s{\displaystyle {\int }_r^s[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)-K\left(t-r,x-z\right)g\left(r,z\right)dz}}\\ +{{\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)-K\left(s-m,y-z\right)g\left(m,z\right)dz]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dmdW\left(r\right)|}^p\end{array}$

$\begin{array}{c}+2^{2p-2}{\left|{\displaystyle {\int }_s^t{K}_{\hslash }}\left(t,r\right){\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dzdW\left(r\right)\right|}^p\\ +2^{2p-2}{\left|{\displaystyle {\int }_s^t{{\displaystyle \int }}_r^t}\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)dz-{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)g\left(r,z\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dmdW\left(r\right)\right|}^p\end{array}$

$\begin{array}{c}=C_1{\left|{{\displaystyle \int }}_0^s{\left[K_{\hslash }\left(s,r\right){\displaystyle {\int }_{ℝ}\left(K(t-r,x-z)-K(s-r,y-z)\right)g(r,z)dz\left(r\right)}\right]}^{2}dr\right|}^{\frac{p}{2}}\\ +C_1|{{\displaystyle \int }}_0^s[{\displaystyle {\int }_r^s[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)-K\left(t-r,x-z\right)g\left(r,z\right)dz}\\ {+{{\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)-K\left(s-m,y-z\right)g\left(m,z\right)dz]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm]}^{2}dr|}^{\frac{p}{2}}\\ +C_1{\left|{\displaystyle {\int }_s^t{\left[K_{\hslash }\left(t,r\right){\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dz\right]}^{2}dr}\right|}^{\frac{p}{2}}\\ +C_1{\left|{{\displaystyle \int }}_s^t{\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)dz-{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)g\left(r,z\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm}\right]}^{2}dr\right|}^{\frac{p}{2}}\\ :=C_1\left(H_1+H_2+H_3+H_4\right),\end{array}$

其中C₁为一个常数。下面，分别估计H₁，H₂，H₃，H₄。利用不等式(3)和假设(H1)，有

$\begin{array}{c}{H}_1={\left|{\displaystyle {\int }_0^s{\left[K_{\hslash }\left(s,r\right){\displaystyle {\int }_{ℝ}\left(K\left(t-r,x-z\right)-K\left(s-r,y-z\right)\right)g\left(r,z\right)dz\left(r\right)}\right]}^{2}dr}\right|}^{\frac{p}{2}}\\ \le 2^{\frac{p}{2}}{\left|{\displaystyle {\int }_0^s{\left|K_{\hslash }\left(s,r\right)\right|}^2{\left[{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)-K\left(s-r,x-z\right)\right|\cdot \left|g\left(r,z\right)\right|dz\left(r\right)}\right]}^{2}dr}\right|}^{\frac{p}{2}}\\ +2^{\frac{p}{2}}{\left|{\displaystyle {\int }_0^s{\left|K_{\hslash }\left(s,r\right)\right|}^2{\left[{\displaystyle {\int }_{ℝ}\left|K\left(s-r,x-z\right)-K\left(s-r,y-z\right)\right|\cdot \left|g\left(r,z\right)\right|dz\left(r\right)}\right]}^{2}dr}\right|}^{\frac{p}{2}}\\ \le 2^{\frac{p}{2}}{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|{\displaystyle {\int }_0^s{\left|K_{\hslash }\left(s,r\right)\right|}^2{\left[{{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)-K\left(s-r,x-z\right)\right|}}^{q}dz\left(r\right)\right]}^{\frac{2}{q}}dr}\right|}^{\frac{p}{2}}\\ +2^{\frac{p}{2}}{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|{\displaystyle {\int }_0^s{\left|K_{\hslash }\left(s,r\right)\right|}^2{\left[{{\displaystyle {\int }_{ℝ}\left|K\left(s-r,x-z\right)-K\left(s-r,y-z\right)\right|}}^{q}dz\left(r\right)\right]}^{\frac{2}{q}}dr}\right|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^{2p}{\left(C_{\hslash }\right)}^{p}C{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|{\displaystyle {\int }_0^s\left[{\left(s-r\right)}^{2\hslash -1}+r^{2\hslash -1}\right]{\left(t-s\right)}^{2{\gamma }_1}dr}\right|}^{\frac{p}{2}}\\ +2^{2p}{\left(C_{\hslash }\right)}^{p}C{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|x-y\right|}^{\beta p}{\left|{\displaystyle {\int }_0^s{\left(s-r\right)}^{-\frac{\beta }{\alpha }-\frac{1}{p\alpha }}\left[{\left(s-r\right)}^{2\hslash -1}+r^{2\hslash -1}\right]dr}\right|}^{\frac{p}{2}}\\ =2^{2p}{\left(C_{\hslash }\right)}^{p}C{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\hslash }^{-\frac{p}{2}}{s}^{\hslash p}{\left(t-s\right)}^{{\gamma }_{1}p}\\ +2^{2p}{\left(C_{\hslash }\right)}^{p}C{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|x-y\right|}^{\beta p}{\left|\frac{1}{-\frac{\beta }{\alpha }-\frac{1}{p\alpha }+2\hslash }{s}^{-\frac{\beta }{\alpha }-\frac{1}{p\alpha }+2\hslash }+\beta \left(1-\frac{\beta }{\alpha }-\frac{1}{p\alpha },2\hslash \right)s^{-\frac{\beta }{\alpha }-\frac{1}{p\alpha }+2\hslash }\right|}^{\frac{p}{2}}.\end{array}$

根据假设(H1)，(H2)和公式(4)，有

$\begin{array}{c}{H}_2=|{\displaystyle {\int }_0^s[{\displaystyle {\int }_r^s[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)-K\left(t-r,x-z\right)g\left(r,z\right)dz}}\\ +{{{\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)-K\left(s-m,y-z\right)g\left(m,z\right)dz]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm]}^{2}dr|}^{\frac{p}{2}}\\ \le 2^{\frac{p}{2}}|{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)-K\left(t-r,x-z\right)g\left(r,z\right)dz\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr\\ +{{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)-K\left(s-m,y-z\right)g\left(m,z\right)dz\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^p|{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left(K\left(t-m,x-z\right)-K\left(t-r,x-z\right)\right)g\left(m,z\right)dz}\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr\\ +{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)\left(g\left(m,z\right)-g\left(r,z\right)\right)dz\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr\\ +{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left(K\left(s-r,y-z\right)-K\left(s-m,y-z\right)\right)g\left(r,z\right)dz}\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr\\ +{{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}K}\left(s-m,y-z\right)\left(g\left(r,z\right)-g\left(m,z\right)\right)dz\right]}\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm\right]}}^{2}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^p|{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)-K\left(t-m,x-z\right)\right|\cdot \left|g\left(m,z\right)\right|dz}\right]}\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm\right]}}^{2}dr\\ +{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)\right|\cdot \left|g\left(m,z\right)-g\left(r,z\right)\right|dz}\right]}\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm\right]}}^{2}dr\\ +{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left|K\left(s-r,y-z\right)-K\left(s-m,y-z\right)\right|\cdot \left|g(r,z)\right|dz}\right]\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm}\right]}}^{2}dr\\ +{{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}\left|K\left(s-m,y-z\right)\right|\cdot \left|g\left(m,z\right)-g\left(r,z\right)\right|dz}\right]}\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm\right]}}^{2}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^p{\left(C_{\hslash }\right)}^p|{{\displaystyle {\int }_0^s\left[{{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}{\left|K\left(t-r,x-z\right)-K\left(t-m,x-z\right)\right|}^q\left|g\left(m,z\right)\right|dz}\right]}}^{\frac{1}{q}}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm\right]}}^{2}dr{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^2\\ +{{\displaystyle {\int }_0^s\left[{{\displaystyle {\int }_r^s\left[{\displaystyle {\int }_{ℝ}{\left|K\left(t-r,x-z\right)\right|}^{q}dz}\right]}}^{\frac{1}{q}}{\left[{\displaystyle {\int }_{ℝ}{\left|g\left(m,z\right)-g\left(r,z\right)\right|}^{p}dz}\right]}^{\frac{1}{p}}dm{\left(m-r\right)}^{\hslash -\frac{3}{2}}\right]}}^{2}dr\\ +{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s{\left[{\displaystyle {\int }_{ℝ}{\left|K\left(s-r,y-z\right)-K\left(s-m,y-z\right)\right|}^q}\left|g\left(r,z\right)\right|dz\right]}^{\frac{1}{q}}}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm\right]}}^{2}dr{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^2\\ +{{{\displaystyle {\int }_0^s\left[{\displaystyle {\int }_r^s{\left[{\displaystyle {\int }_{ℝ}{\left|K\left(s-m,y-z\right)\right|}^{q}dz}\right]}^{\frac{1}{q}}{\left[{\displaystyle {\int }_{ℝ}{\left|g\left(m,z\right)-g\left(r,z\right)\right|}^{p}dz}\right]}^{\frac{1}{p}}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm}\right]}}^{2}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}=2^p{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p|\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}+{\displaystyle {\int }_0^s\frac{1}{\hslash +{\gamma }_2-\frac{1}{2}}}{\left(t-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }}{\left(s-r\right)}^{\hslash +{\gamma }_2-\frac{1}{2}}dr\\ {+\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}+{\displaystyle {\int }_0^s\beta }\left(\frac{3q\alpha -\left(p-1\right)}{2q\alpha },\hslash +{\gamma }_2-\frac{1}{2}\right){\left(s-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2-\frac{1}{2}}dr|}^{\frac{p}{2}}\\ \le 2^p{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p|\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}+{\displaystyle {\int }_0^t\frac{1}{\hslash +{\gamma }_2-\frac{1}{2}}}{\left(t-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2-\frac{1}{2}}dr\\ {+\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}+{\displaystyle {\int }_0^s\beta }\left(\frac{3q\alpha -\left(p-1\right)}{2q\alpha },\hslash +{\gamma }_2-\frac{1}{2}\right){\left(s-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2-\frac{1}{2}}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^p{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p|\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}\\ +\frac{1}{\hslash +{\gamma }_2-\frac{1}{2}}\frac{1}{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}{t}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}+\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{s}^{2\hslash +2{\gamma }_1}\\ +{\beta \left(\frac{3q\alpha -\left(p-1\right)}{2q\alpha },\hslash +{\gamma }_2-\frac{1}{2}\right)\frac{1}{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}{s}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}|}^{\frac{p}{2}},\end{array}$

其中 $q=\frac{p}{p-1}$ 。因为 $\alpha p>1$ ，所以 $q\alpha -\left(p-1\right)>0$ 。由公式(3)有

$\begin{array}{c}{H}_3={\left|{\displaystyle {\int }_s^t{\left[K_{\hslash }\left(t,r\right){\displaystyle {\int }_{ℝ}K}\left(s-r,y-z\right)g\left(r,z\right)dz\right]}^2\text{d}r}\right|}^{\frac{p}{2}}\\ \le {\left|{\displaystyle {\int }_s^t{\left|K_{\hslash }\left(t,r\right)\right|}^2{\left[{\displaystyle {\int }_{ℝ}\left|K\left(s-r,y-z\right)\right|\cdot \left|g\left(r,z\right)\right|dz}\right]}^{2}dr}\right|}^{\frac{p}{2}}\\ \le {\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|{\displaystyle {\int }_s^t{\left|K_{\hslash }\left(t,r\right)\right|}^2{\left[{\displaystyle {\int }_{ℝ}{\left|K\left(s-r,y-z\right)\right|}^{q}dz}\right]}^{\frac{2}{q}}dr}\right|}^{\frac{p}{2}}\\ \le 2^{2p}{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|{\displaystyle {\int }_s^t\left({\left(t-r\right)}^{2\hslash -1}+r^{2\hslash -1}\right){\left(t-r\right)}^{\frac{2q\alpha -\left(q-1\right)}{q\alpha }}dr}\right|}^{\frac{p}{2}}\end{array}$

$\le 2^{2p}{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|\frac{1}{\frac{2q\alpha -\left(q-1\right)}{q\alpha }+2\hslash }{\left(t-s\right)}^{\frac{2q\alpha -\left(q-1\right)}{q\alpha }+2\hslash }+\beta \left(2\hslash ,\frac{3q\alpha -\left(q-1\right)}{q\alpha }\right)t^{\frac{2q\alpha -\left(q-1\right)}{q\alpha }+2\hslash }\right|}^{\frac{p}{2}}.$

利用假设(H1)和公式(4)，有

$\begin{array}{c}{H}_4={\left|{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}K}\left(t-m,x-z\right)g\left(m,z\right)-K\left(t-r,x-z\right)g\left(r,z\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm}\right]}^2}dr\right|}^{\frac{p}{2}}\\ \le 2^p|{{\displaystyle {\int }_s^t\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}\left(K\left(t-m,x-z\right)-K\left(t-r,x-z\right)\right)g\left(m,z\right)dz}\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm}\right]}}^{2}dr\\ +{{{\displaystyle {\int }_s^t\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}K}\left(t-r,x-z\right)\left(g\left(m,z\right)-g\left(r,z\right)\right)dz\right]\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)dm}\right]}}^{2}dr|}^{\frac{p}{2}}\\ \le 2^p|{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)-K\left(t-m,x-z\right)\right|\cdot \left|g\left(m,z\right)\right|dz}\right]\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm}\right]}^2}dr\\ +{{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t\left[{\displaystyle {\int }_{ℝ}\left|K\left(t-r,x-z\right)\right|\cdot \left|g\left(m,z\right)-g\left(r,z\right)\right|dz}\right]\left|\frac{\partial K_{\hslash }}{\partial m}\left(m,r\right)\right|dm}\right]}^2}dr|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}\le 2^p{\left(C_{\hslash }\right)}^p|{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t{\left[{\displaystyle {\int }_{ℝ}{\left|K\left(t-r,x-z\right)-K\left(t-m,x-z\right)\right|}^q\left|g\left(m,z\right)\right|dz}\right]}^{\frac{1}{q}}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm}\right]}^2}dr{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^2\\ +{{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t{\left[{\displaystyle {\int }_{ℝ}{\left|K\left(t-r,x-z\right)\right|}^{q}dz}\right]}^{\frac{1}{q}}{\left[{\displaystyle {\int }_{ℝ}{\left|g\left(m,z\right)-g\left(r,z\right)\right|}^{p}dz}\right]}^{\frac{1}{p}}{(m-r)}^{\hslash -\frac{3}{2}}dm}\right]}^2}dr|}^{\frac{p}{2}}\\ \le 2^p{\left(C_{\hslash }\right)}^p|{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t{\left(m-r\right)}^{{\gamma }_1}}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm\right]}^{2}dr}{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^2\\ +{{\displaystyle {\int }_s^t{\left[{\displaystyle {\int }_r^t{\left(t-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }}}{\left(m-r\right)}^{{\gamma }_2}{\left(m-r\right)}^{\hslash -\frac{3}{2}}dm\right]}^{2}dr}{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^2|}^{\frac{p}{2}}\end{array}$

$\begin{array}{c}=2^p{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p|\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{\left(t-s\right)}^{2\hslash +2{\gamma }_1}{+{\displaystyle {\int }_s^t\frac{1}{\hslash +{\gamma }_2-\frac{1}{2}}}{\left(t-r\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2-\frac{1}{2}}dr|}^{\frac{p}{2}}\\ \le 2^p{\left(C_{\hslash }\right)}^p{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p|\frac{1}{{\left(\hslash +{\gamma }_1-\frac{1}{2}\right)}^2\left(2\hslash +2{\gamma }_1\right)}{\left(t-s\right)}^{2\hslash +2{\gamma }_1}\\ +{\frac{1}{\hslash +{\gamma }_2-\frac{1}{2}}\frac{1}{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}{\left(t-s\right)}^{\frac{q\alpha -\left(p-1\right)}{2q\alpha }+\hslash +{\gamma }_2+\frac{1}{2}}|}^{\frac{p}{2}}.\end{array}$

假设 $D\left(X,c\right)=D_T\cap Q_c$ 且 $Q_c=Q_c\left(t_0,x_0\right)$ 。根据假设 $\left(t,x\right)\in Q_c\left(t_0,x_0\right)$ ， $\left(s,y\right)\in Q_c\left(t_0,x_0\right)$ ，有

$0\le t-s\le 2c^2,\left|x-y\right|\le \left|x-x_0\right|+\left|y-y_0\right|\le 2c.$

由A-类有界区域的定义，有

${\left[u\right]}_{{\mathcal{L}}^{p,\theta }\left(\left(D_T;\delta \right);L^p\left(\Omega \right)\right)}^p\le C_2{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p,$

其中 $C_2$ 为一个常数， $\theta =1+\frac{\beta p}{3}$ ，即不等式(7)成立。因此利用命题2.1可以得到不等式(8)。

下面证明不等式(9)。为了使用尾估计，我们首先考虑以下估计。设 $\left(t_0,x_0\right)\in D_T\subset {\mathcal{O}}_T$ 且

$Q_c\left(t_0,x_0\right)=\left(t_0-c^2,t_0+c^2\right)\times B_c\left(x_0\right),$

则有 ${\bar{D}}_T\subset Q_{d\left(D\right)}\left(t_0,x_0\right)$ 。设 $\left(t_1,x_1\right),\left(t_2,x_2\right)\in D_T$ ， $Q_i:=D_T\cap Q_{c_i}\left(t_i,x_i\right),i=1,2$ 并且

$F\left(t_i,x_i,c_i\right)=\frac{1}{{\left|Q_i\right|}^{1+\theta }}{\displaystyle {\int }_{Q_i}{\displaystyle {\int }_{Q_i}{\left|u\left(t,x\right)-u\left(s,y\right)\right|}^p}}dtdxdsdy=\frac{1}{{\left|Q_i\right|}^{1+\theta }}{\displaystyle {\int }_{Q_i}{\displaystyle {\int }_{Q_i}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^p}}dtdxdsdy.$

显然

$\begin{array}{l}F\left(t_1,x_1,c_1\right)-F\left(t_2,x_2,c_2\right)=\left[F\left(t_1,x_1,c_1\right)-F\left(t_2,x_1,c_1\right)\right]\\ +\left[F\left(t_2,x_1,c_1\right)-F\left(t_2,x_2,c_1\right)\right]\\ +\left[F\left(t_2,x_2,c_1\right)-F\left(t_2,x_2,c_2\right)\right]\\ :=I_1+I_2+I_3.\end{array}$ (10)

对于 $I_1$ ，

$\begin{array}{l}{I}_1=F\left(t_1,x_1,c_1\right)-F\left(t_2,x_1,c_1\right)\\ =\frac{1}{{\left|Q_1\right|}^{1+\theta }}{\displaystyle {\int }_{Q_1}{\displaystyle {\int }_{Q_1}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^p}}dtdxdsdy\\ =\frac{1}{{\left|Q_1\right|}^{1+\theta }}{\displaystyle {\int }_{Q_1}{\displaystyle {\int }_{Q_1}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^p}}dtdxdsdy-\frac{1}{{\left|Q_{12}\right|}^{1+\theta }}{\displaystyle {\int }_{Q_{12}}{\displaystyle {\int }_{Q_{12}}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^p}}dtdxdsdy\\ =\frac{1}{{\left|Q_1\right|}^{1+\theta }}\{{\int }_{Q_1\backslash Q_{12}}{\int }_{Q_1\backslash Q_{12}}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^{p}dtdxdsdy-{\int }_{Q_{12}\backslash Q_1}{\int }_{Q_{12}\backslash Q_1}|{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^{p}dtdxdsdy\}\\ +\left[\frac{1}{{\left|Q_1\right|}^{1+\theta }}-\frac{1}{{\left|Q_{12}\right|}^{1+\theta }}\right]{\displaystyle {\int }_{Q_{12}}{\displaystyle {\int }_{Q_{12}}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^p}}dtdxdsdy\\ :=I_{11}+I_{12}\text{,}\end{array}$

其中 $Q_{12}=D_T\cap Q_{c_1}\left(t_2,x_1\right)$ 。为简单起见，我们假设 $\left|Q_1\right|\ge \left|Q_{12}\right|$ 。相反的，我们可以改变 $Q_1$ 和 $Q_{12}$ 的位置。因此 $I_{12}\le 0$ 。现在考虑 $I_{11}$ 这一项。在给出 $I_{11}$ 的估计之前，我们首先回顾一下我们的目标。为了应用尾部估计，我们希望得到 $I_{11}$ 的估计，即存在 $\delta >0$ ，使得 $\mathbb{E}{I}_{11}\le C{\left(t_1-t_2\right)}^{\delta }$ 。显然

$\left|Q_1\backslash Q_{12}\right|\le C\left(t_1-t_2\right)c_1\text{,}\left|Q_1\right|\approx C{c}_1^3.$

假设 $t>s$ ，则

类似不等式(7)的证明有

$\mathbb{E}{\rm Y}\le C{c}_1^{\beta p}{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p.$

因为 $\left(t,x\right)\in Q_1$ ， $\left(s,y\right)\in Q_1$ ，所以 $0\le t-s\le 2c_1^2$ ， $\left|x-y\right|\le \left|x-x_1\right|+\left|y-x_1\right|\le 2c_1$ ，利用上述不等式和A-类有界区域的性质有

$\mathbb{E}{\int }_{Q_1\backslash Q_{12}}{\int }_{Q_1\backslash Q_{12}}\mathbb{E}{\rm Y}dtdxdsdy\le C\left(p,T\right)c_1^{\beta p}{\left|Q_1\backslash Q_{12}\right|}^2{\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p.$

因为 $D_T$ 是一个A-类有界区域，所以 $2c_1\le \left|D\right|$ ，

$A\left|Q_{c_1}\left(t_1,x_1\right)\right|\le \left|Q_1\right|\le \left|Q_{c_1}\left(t_1,x_1\right)\right|,$

$A\left|Q_{c_1}\left(t_1,x_1\right)\backslash Q_{c_1}\left(t_2,x_1\right)\right|\le \left|Q_1\backslash Q_{12}\right|\le \left|Q_{c_1}\left(t_1,x_1\right)\backslash Q_{c_1}\left(t_2,x_1\right)\right|.$

注意到

$\left|Q_{c_1}\left(t_1,x_1\right)\right|\approx C{c}_1^3,\left|Q_{c_1}\left(t_1,x_1\right)\backslash Q_{c_1}\left(t_2,x_1\right)\right|\le C{c}_1\left[c_1^2\wedge \left(t_1-t_2\right)\right],$

其中C是一个不依赖于 $c_1$ 的正常数。显然 $Q_1\backslash Q_{12}\subset Q_1$ ，取 $0<\delta <\frac{\beta p}{2}$ 有

$\mathbb{E}{\int }_{Q_1\backslash Q_{12}}{\int }_{Q_1\backslash Q_{12}}{\left|\mathcal{K}g\left(t,x\right)-\mathcal{K}g\left(s,y\right)\right|}^{p}dtdxdsdy\le C\left(C_0,D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|Q_1\right|}^{2+\frac{\beta p-2\delta }{3}}{\left|t_1-t_2\right|}^{\delta }.$

类似地，我们可以得到

$\mathbb{E}{\int }_{Q_{12}\backslash Q_1}{\int }_{Q_{12\backslash Q_1}}{\left|u\left(t,x\right)-u\left(s,y\right)\right|}^{p}dtdxdsdy\le C\left(D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|Q_1\right|}^{2+\frac{\beta p-2\delta }{3}}{\left|t_1-t_2\right|}^{\delta }.$

由 $I_{12}\le 0$ 有

$\mathbb{E}{I}_1\le C\left(D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|t_1-t_2\right|}^{\delta },$ (11)

其中 $\theta =1+\frac{\beta p-2\delta }{3}$ 。

接下来我们估计 $I_2$ 。利用事实

$\left|\left[D\cap Q_{c_1}\left(t_2,x_1\right)\right]\backslash \left[D\cap Q_{c_1}\left(t_2,x_2\right)\right]\right|\le C{c}_1^2\left|x_1-x_2\right|,$

类似于 $I_1$ 的估计，可以取 $0<\delta <\frac{\beta p}{2}$ 使得

$\mathbb{E}{I}_2=\mathbb{E}\left[F\left(t_2,x_1,c_1\right)-F\left(t_2,x_2,c_1\right)\right]\le C\left(D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|x_1-x_2\right|}^{\delta },$ (12)

其中 $\theta =1+\frac{\beta p-\delta }{3}$ 。

下面估计 $I_3$ 。利用事实

$\left|\left[D\cap Q_{c_1}\left(t_2,x_2\right)\right]\backslash \left[D\cap Q_{c_2}\left(t_2,x_2\right)\right]\right|\le C{c}_1^2\left(c_1-c_2\right),c_1\ge c_2,$

类似于 $I_1$ 的估计有

$\mathbb{E}{I}_3=\mathbb{E}\left[F\left(t_2,x_2,c_1\right)-F\left(t_2,x_2,c_2\right)\right]\le C\left(D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p{\left|c_1-c_2\right|}^{\delta },$ (13)

其中 $\theta =1+\frac{\beta p-\delta }{3}$ 。因此根据公式(10)，(11)，(12)和(13)，有

$\mathbb{E}{\left|F\left(t_1,x_1,c_1\right)-F\left(t_2,x_2,c_2\right)\right|}^q\le C\left(D,T\right){\Vert g\Vert }_{D^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^{pq}{\left(\left|t_1-t_2\right|+\left|x_1-x_2\right|+\left|c_1-c_2\right|\right)}^{\delta q},$

其中 $\theta =1+\frac{\beta p-2\delta }{3},\left(t_i,x_i\right)\in D_T$ 且 $0<c_i\le \left|D\right|,i=1,2$ 。

设 $D_T={\left[0,1\right]}^2$ ， $c\in \left[0,2\right]$ 。引入一系列集合，

${\mathcal{S}}_n=\left\{z\in {\mathbb{Z}}^3\mid z{2}^{-n}\in {\left(0,1\right)}^2\times \left(0,2\right)\right\},n\in \mathbb{N}.$

对于任意的 $e=\left(e_1,e_2,e_3\right)\in {\mathbb{Z}}^3$ 使得 ${\left|e\right|}_{\infty }={\max }_{1<i<3}\left|e_j\right|=1$ ，并且对于任意的 $z,z+e\in {\mathcal{S}}_n$ ，定义 $v_z^{n,e}=\left|F\left(\left(z+e\right)2^{-n}\right)-F\left(z{2}^{-n}\right)\right|$ 。从上面的讨论，可以得到

$\mathbb{E}{\left|v_z^{n,e}\right|}^q\le C\left(\beta ,C_0,D,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^{pq}{2}^{-n\delta q}:=\hat{C}{2}^{-n\delta q}.$

对于任意的 $\tau >0$ ， $K>0$ ，

${\mathcal{A}}_{z,\tau }^{n,e}=\left\{\omega \in \Omega |v_z^{n,e}\ge K{\tau }^n,z,z+e\in {\mathcal{S}}_n\right\},$

从而

$\mathbb{P}\left({\mathcal{A}}_{z,\tau }^{n,e}\right)\le \frac{\mathbb{E}{\left|v_z^{n,e}\right|}^q}{K^q{\tau }^{qn}}\le \frac{\hat{C}{2}^{-n\cancel{q}}}{K^q{\tau }^{qn}}.$

对于每一个n，事件的总数 ${\mathcal{A}}_{z,\tau }^{n,e},z,z+e\in {\mathcal{S}}_n$ 不大于 $2^3{3}^3$ 。因此并集为

${\mathcal{A}}_{\tau }^n={\displaystyle {\cup }_{z,z+e\in {\mathcal{S}}_n}\left({\displaystyle {\cup }_{{\Vert e\Vert }_{\infty }=1}{\mathcal{A}}_{z,\tau }^{n,e}}\right)},$

并集的概率为

$\mathbb{P}\left({\mathcal{A}}_{\tau }^n\right)\le \frac{\hat{C}{2}^{-n\delta q}}{K^q{\tau }^{qn}}{2}^{3n}\le \hat{C}{K}^{-q}{\left(\frac{2^3}{{\left(2^{\delta }\tau \right)}^q}\right)}^n.$

设 $\tau =2^{-v\delta }$ ，其中 $v>0$ 且 $\left(1-v\right)\delta q\ge 3$ 。事件 $\mathcal{A}={\displaystyle {\cup }_{n\ge 1}{\mathcal{A}}_{\tau }^n}$ 发生的概率为

$\mathbb{P}\left(\mathcal{A}\right)\le C\hat{C}{K}^{-q}.$ (14)

对于任意一点 $\xi =\left(t,x,c\right)\in {\left(0,1\right)}^2\times \left(0,2\right)$ 有 $\xi ={\displaystyle \underset{i=0}{\overset{\infty }{\sum }}{e}_i}{2}^{-i}\left({\Vert e_i\Vert }_{\infty }\le 1\right)$ ， ${\xi }_k={\displaystyle \underset{i=0}{\overset{k}{\sum }}{e}_i}{2}^{-i}$ 且 ${\xi }_0=0$ 。对于任意的 $\omega \notin \mathcal{A}$ ， $\left|F\left({\xi }_{k+1}\right)-F\left({\xi }_k\right)\right|<K{\tau }^{k+1}$ ，这意味着

$\left|F\left(t,x,c\right)\right|\le {\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\left|F\left({\xi }_{k+1}\right)-F\left({\xi }_k\right)\right|}<K{\displaystyle \underset{k=1}{\overset{\infty }{\sum }}{\tau }^k}\le K{\left(2^{\nu \delta }-1\right)}^{-1}.$ (15)

设 $v_1=\underset{\left(t,x,c\right)\in {\left(0,1\right)}^2\times \left(0,2\right)}{\sup }\left|F\left(t,x,c\right)\right|$ ，由于F是连续的，则 $v_1=\underset{\left(t,x,c\right)\in {\left(0,1\right)}^2\times \left(0,2\right)}{\sup }\left|F\left(t,x,c\right)\right|$ 。对于 $0<r<q$ ，

$\mathbb{E}{v}_1^r=r{\displaystyle {\int }_0^{\infty }{a}^{r-1}}\mathbb{P}\left(v_1\ge a\right)da=r{\displaystyle {\int }_0^{\gamma K}{a}^{r-1}}\mathbb{P}\left(v_1\ge a\right)da+r{\displaystyle {\int }_{\gamma K}^{\infty }{a}^{r-1}}\mathbb{P}\left(v_1\ge a\right)da.$ (16)

选择 $\gamma \ge {\left(2^{v\delta }-1\right)}^{-1}$ ，利用(14)，(15)和(16)有

$\mathbb{E}{v}_1^r\le {\left(\gamma K\right)}^r+C{\hat{C}}^{q}r{\displaystyle {\int }_{\gamma K}^{\infty }{a}^{r-1-q}}\text{d}a\le {\left(\gamma K\right)}^r+C\hat{C}r{\left(cK\right)}^{r-q},$

从而

$\mathbb{E}{v}_1^r\le C\left(D,d,T\right){\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left({ℝ}^d\right)\right)\right)}^{pr},$

如果选择 $K={\Vert g\Vert }_{L^p\left(\Omega ;L^{\infty }\left(\left[0,T\right];L^p\left(ℝ\right)\right)\right)}^p$ 。通过使用嵌入不等式 $L^p\left(\Omega ;{\mathcal{L}}^{p,\theta }\left(D_T;\delta \right)\right)\cong L^p\left(\Omega ;C^{\gamma }\left({\bar{D}}_T;\delta \right)\right)$ ，得到不等式 。

4. 总结

本文利获得了非局部随机扩散方程解的Hölder连续性。首先利用Campanato估计证明了非局部随机扩散方程的温和解的Hölder连续性，即解u属于空间 $C^{\beta }\left(D_T;L^p\left(\Omega \right)\right)$ ，其中 $\beta$ 为确定的Hölder连续指数。其次，通过使用尾估计和Sobolev嵌入定理得到了 $L^p\left(\Omega ;C^{{\beta }^{\ast }}\left(D_T\right)\right)$ 中温和解的估计。更重要的是，我们在 $\beta$ 和 ${\beta }^{\ast }$ 两个指数之间给出了一个明确的公式。

基金项目

天津师范大学研究生科研创新项目资助(2022KYCX114Y)。

NOTES

^*通讯作者。