题目：马尔科夫链的离散Beckner不等式以及Bochner-Bakry-Emery 方法的应用
主讲人：Dr Wen YUE, Austria
时间：2016年 12月21 日 上午11:00
地点：公司主楼1214
主要内容：
这篇文章主要研究了有限状态空间上时间连续的马尔科夫链相对应的离散凸索伯列夫不等式和Beckner不等式。Beckner不等式介于修正对数索伯列夫不等式和庞加莱不等式之间。这些不等式的证明主要基于Bakry-Emery 方法，离散Bochner-类型不等式(首先是Caputo, Dai Pra 和Posta的工作)，以及最近被Fathi和Maas推广到对数墒上的Bochner-类型不等式。这些关于凸熵的理论结果可以被应用于一些常见的马尔科夫链，包括生灭过程，零范围过程，伯努利-拉普拉斯模型，随机位移过程以及一维Fokker-Planck方程的有限容积离散化。
报告人简介：
岳文博士自从2014年三月至今在维也纳工业大学从事博士后的研究工作。在之前的2010年到2014年她在曼彻斯特大学数学学院概率与随机分析专业学习攻读博士学位。她的研究兴趣包括随机过程，随机分析，随机（偏）微分方程，Malliavin 分析，偏微分方程，马尔科夫链和离散粒子系统等等。在博士期间，她主要研究带反射的随机偏微分方程和Malliavin分析并发表相关文章。最近，她开始一些学科内的基于概率论和偏微分方程的交叉合作研究，并且把马尔科夫链和偏微分方程的离散化二者紧密地联系起来，这也正是她的报告中会涉及到的内容。
Title：DISCRETEBECKNER INEQUALITIES VIA THE BOCHNER-BAKRY-EMERY APPROACH FOR MARKOV CHAINS
Abstract：
Discrete convex Sobolev inequalities and Beckner inequalities are derived for time-continuous Markov chains on finite state spaces. Beckner inequalities interpolate between the modified logarithmic Sobolev inequality and the Poincar ́e inequality. Their proof is based on the Bakry-Emery approach and on discrete Bochner-type inequalities established by Caputo, Dai Pra, and Posta and recently extended by Fathi and Maas for logarithmic entropies. The abstract result for convex entropies is applied to several Markov chains, including birth-death processes, zero-range processes, Bernoulli-Laplace models, and random transposition models, and to a finite-volume discretization of a one- dimensional Fokker-Planck equation, applying results by Mielke.
About the speaker：
Dr Wen Yue has been a postdoctoral in Vienna University of Technology, Vienna, Austria from March, 2014 until now. Before she did her PhD study in the group of Probability and Stochastic Analysis, School of Mathematics, University of Manchester, UK, from 2010 to 2014. She has a wide interest in stochastic process, stochastic analysis, stochastic (partial) differential equations, Malliavin calculus, partial differential equations, Markov chains and discrete particle systems and so on. During her PhD period, she studied and published several papers mainly on stochastic(partial) differential equations with reflected walls and Malliavin calculus. In recent two years, she started some intradisciplinary cooperations between probability and PDE, and founded the close relations between Markov chains case and the discretization of PDE's case, which you will see from her talk.
