具有测度值系数的椭圆算子及扩散过程  

Diffusion processes associated with elliptic operators with measure-valued coefficients

作　　者：张土生 魏茸 杨赛赛 Tusheng Zhang;Rong Wei;Saisai Yang

机构地区：[1]School of Mathematics,University of Manchester,Manchester M139PL,UK [2]中国科学技术大学数学科学学院,合肥230026

出　　处：《中国科学：数学》2021年第11期1955-1974,共20页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11971456,11671372和11721101);中国博士后科学基金(批准号:BH0010000027)资助项目。

摘　　要：本文介绍一类系数是测度值的椭圆算子及其对应的扩散过程,这里考虑的测度值系数属于Kato类K_(d,1).首先,介绍漂移项为测度值μ的Brown运动及反射Brown运动的弱解的存在唯一性以及弱解的转移概率密度函数估计.然后,给出关于无穷小算子L=12∆+μ·∇的二阶椭圆方程的Dirichlet边界值问题和Neumann边界值问题的弱解的存在唯一性及概率表示.最后,介绍漂移项为测度值μ的Brown运动的小时间渐近性和大偏差原理.In this paper,we give a survey on diffusion processes associated with elliptic operators with measurevalued coefficients.We first introduce the existence and uniqueness of Brownian motion and reflected Brownian with measure-valued drifts as well as the Gaussian type estimates of the transition functions.We then study the Dirichlet and Neumann boundary value problems for the associated generators of the diffusion processes and obtain the probabilistic representations for the solutions.we also present the small time asymptotics and large deviations for the Brownian motions with drifts.

关 键 词：椭圆算子 扩散过程 Kato类 Dirichlet边界值问题 Neumann边界值问题 小时间渐近性 大偏差 

分 类 号：O177[理学—数学]