非局部随机热方程的适定性及其相关问题的研究

In recent era, the theoretical exploration of nonlocal stochastic partial differential equations has become a hot topic in the field of stochastic partial differential equations. According to framework of traditional stochastic partial differential equation, the mathematical model recognized for practical problems can offer the current position and state information. However, it can not give an effective description of the dependence on the state information around and at infinity. In order to better solve such practical problems, this project has been designed as follows. We need to introduce non-local stochastic partial differential equation and establish its basic theory. By means of studing the existence and uniqueness of solutions, Sobolev differentiability and Holder continuity of non-local stochastic heat equation without gradient term and with gradient term under integrable coefficients. Then the results are used to establish the existence and uniqueness of solutions for stochastic differential equation driven by non-Gaussian Levy noise with integrable drift coefficient, and the corresponding stochastic dynamic behavior. Completion of the project will help us to understand the dynamic behavior of solutions of stochastic differential equations and stochastic partial differential equations driven by Levy process. Moreover, subject study enriches the research doors and application of stochastic dynamic system theory.

非局部随机偏微分方程解的理论探索是近来年随机偏微分方程领域研究的一个热点话题。传统的随机偏微分方程模型，虽然其解对当前位置状态能给出很好的刻画，但却不能给出有效的描述对周边及其无穷远处的状态信息的依赖性。为了更好地解决这类实际问题，我们需要引出非局部随机偏微分方程，并建立其基础的理论。本项目通过研究不含梯度项、含有梯度项的非局部随机热方程在系数可积条件下解的存在唯一性、Sobolev可微性、Holder连续性等性质，然后将所得结果用于含有可积漂移系数的非Gaussian Levy噪声驱动的随机微分方程，对其建立解的存在唯一性，并研究相应的随机动力学行为。项目的研究结果将有助于了解Levy过程驱动的随机微分方程和随机偏微分方程解的动力学行为，同时丰富随机动力系统理论研究及其应用。

本项目旨在研究梯度项是非正则系数的非局部随机热方程。梯度项是刻画实际问题的重要因素,例如,工程上通过梯度项刻画的对流换热方式实现许多装置的热交换问题.然而由于数学上研究的困难,使得在工程可应用的理论工具还处在研发阶段,为了解决这类问题,还有许多困难要处理.第一步研究非局部微分算子在梯度扰动下的热核估计.我们考虑了上加性白噪声驱动的非局部偏微分方程,对于可积系数,我们得到了温和解的存在唯一性和Hölder连续性.还研究了分数阶Fokker-Planck方程的分数阶指数,利用Banach空间值Calder-on-Zygmund定理,得到了非局部方程的解的存在唯一性.第二步研究了抛物方程，建立其 Lipschitz和的正则性估计,假设其初始值为零,满足Ladyzhenskaya-Prodi-Serrin 型条件.根据理论结果,我们给出了其对两类方程的应用.首先讨论了随机热方程的正则性,其次讨论了一类奇异漂移系数 SDES 强解的Sobolev可微性.第三步研究了一类具有环境噪声的广义种群动力学方程整体正强解的存在唯一性,而确定性方程的整体存在性不成立.特别地,我们证明了一类随机微分方程在参数满足限制条件时,方程全局解。