非对称随机微分方程的泛函不等式  

Functional inequalities for non-symmetric stochastic differential equations

作　　者：王凤雨[1] Fengyu Wang

机构地区：[1]北京师范大学数学科学学院

出　　处：《中国科学：数学》2020年第1期167-178,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11771326和11431014)资助项目

摘　　要：通常非对称Markov半群比相应的对称半群有更好的分析性质.例如, Wang (2017)给出一类超压缩(因此,在L2和相对熵下指数遍历)的非对称Markov半群,其对称半群甚至不遍历.本文讨论反方向的问题:在什么条件下,非对称Markov半群和相应的对称半群享有同等的性质.分别对于由Brown运动和Lévy跳过程驱动的随机微分方程,本文得到了非对称半群和对称半群在一些重要性质方面地位对等的充分必要条件,这些性质包括指数收敛性、一致可积性、H-超压缩性和S-超有界性.A non-symmetric Markov semigroup usually has better properties than the corresponding symmetric one. For example, Wang(2017) provides a class of non-symmetric Markov semigroups which are hypercontractive(and thus converge exponentially in both L2 and entropy), but the symmetric ones are even not ergodic. In this paper, we consider the inverse problem: search for reasonable conditions to ensure that a non-symmetric Markov semigroup and its symmetrization share the properties of exponential convergence, uniform integrability,hypercontractivity, and super boundedness. Since in the symmetric case these properties are precisely characterized by functional inequalities of the Dirichlet form, the key point of the study is to prove these inequalities for non-symmetric Markov processes. Stochastic differential equations driven by Brown motion or Lévy jump process are investigated.

关 键 词：非对称半群 泛函不等式 指数收敛性 一致可积性 超有界性 

分 类 号：O211.63[理学—概率论与数理统计]