空间非齐次白噪声驱动的分数阶随机热方程的矩估计

数学物理学报 ›› 2022, Vol. 42 ›› Issue (4): 1186-1208.

空间非齐次白噪声驱动的分数阶随机热方程的矩估计

刘俊峰^1,*(),毛磊²(),王志³()

¹ 南京审计大学统计与数据科学学院南京 211815
² 陆军工程大学基础部南京 211100
³ 宁波工程学院理学院浙江宁波 315211

收稿日期:2021-01-29 出版日期:2022-08-26 发布日期:2022-08-08
通讯作者: 刘俊峰 E-mail:jordanjunfeng@163.com;maolei1981@126.com;wangzhi1006@hotmail.com
作者简介:毛磊, E-mail: maolei1981@126.com|王志, E-mail: wangzhi1006@hotmail.com
基金资助:
国家自然科学基金(11701304);教育部人文社会科学基金(18YJCZH101);江苏高校自然科学基金重大项目(18KJA110002);宁波市自然科学基金(2019A610041);王宽诚教育基金会

Moment Bounds for the Fractional Stochastic Heat Equation with Spatially Inhomogeneous White Noise

Junfeng Liu^1,*(),Lei Mao²(),Zhi Wang³()

¹ School of Statistics and Data Science, Nanjing Audit University, Nanjing, 211815
² Department of General Education, Army Engineering University of PLA, Nanjing 211100
³ School of Sciences, Ningbo University of Technology, Zhejiang Ningbo 315211

Received:2021-01-29 Online:2022-08-26 Published:2022-08-08
Contact: Junfeng Liu E-mail:jordanjunfeng@163.com;maolei1981@126.com;wangzhi1006@hotmail.com
Supported by:
the NSFC(11701304);the Humanities and Social Sciences Foundation of the Ministry of Education(18YJCZH101);the Major Research Plan of NSF of the Jiangsu Higher Education Institutions(18KJA110002);the NSF of Ningbo Municipality(2019A610041);the Wong Kuancheng Education Foundation

摘要/Abstract

摘要：

该文主要研究如下伴有Cauchy初值条件的分数阶随机热方程 $\left(\frac{\partial }{\partial t}-{\cal D}_\delta^\alpha\right)u(t, x)=g(u(t, x))\frac{\partial^2}{\partial t\partial x}w_\rho(t, x), \quad 0<t\leq T, \quad x\in{\Bbb R} , $ 其中$\alpha\in(1, 2]$为算子${\cal D}_{\delta}^{\alpha}$的阶数, $\delta (|\delta|\leq2-\alpha)$称为偏度参数, 扩散系数$g(\cdot):{\Bbb R} \mapsto{\Bbb R} $是非随机的可测函数, $\frac{\partial^2}{\partial t\partial x}w_\rho(t, x)$表示空间非齐次白噪声, 在关于非齐次布朗单$w_\rho(t, x)$催化测度$\rho$的适当假设下, 证明了该方程解的存在性、唯一性和Hölder连续性. 同时, 也证明了方程解的矩估计.

关键词: 分数阶随机热方程, 空间非齐次白噪声, Hölder连续性, 矩估计

Abstract:

In this paper, we will study a class of fractional stochastic heat equation of the form $\frac{ \partial}{ \partial t} u(t, x)={\cal D}_\delta^\alpha u(t, x)+g( u(t, x)) \frac{\partial^2}{\partial t\partial x}w_\rho(t, x), \quad 0<t\leq T, \quad x\in{\Bbb R} , $ with $T>0$, where ${\cal D}_\delta^\alpha$ denotes a nonlocal fractional differential operator with $\alpha\in(1, 2]$ and $|\delta|\leq2-\alpha$, and $\frac{\partial^2}{\partial t\partial x}w_\rho(t, x)$ is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet $w_\rho(t, x)$, we prove the existence, uniqueness and Hölder regularity of the solution. Upper and lower moment bounds for the solution are also derived.

Key words: Fractional stochastic heat equation, Spatially inhomogeneous white noise, Hölder regularity, Moment bounds

中图分类号:

O211.63

刘俊峰,毛磊,王志. 空间非齐次白噪声驱动的分数阶随机热方程的矩估计[J]. 数学物理学报, 2022, 42(4): 1186-1208.

Junfeng Liu,Lei Mao,Zhi Wang. Moment Bounds for the Fractional Stochastic Heat Equation with Spatially Inhomogeneous White Noise[J]. Acta mathematica scientia,Series A, 2022, 42(4): 1186-1208.

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