2024-3-1
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2024年2月28日下午,欧洲科学院院士Michael Röckner应国家数学与交叉科学中心邀请做客综合论坛,并做了题为“NONLINEAR FOKKER-PLANCK-KOLMOGOROV EQUATIONS AND NONLINEAR MARKOV PROCESSES”的学术报告,多名师生参加了本次报告会。
自上个世纪中叶以来,随机分析领域的许多工作都致力于(抛物)线性偏微分方程(PDEs),更准确地说,线性Fokker-Planck-Kolmogorov方程(FPKEs)和随机微分方程(SDEs),或更一般的马尔可夫过程之间的关系。它最突出的例子是经典的热方程和布朗运动给出的马尔可夫过程。本次报告是关于非线性模型的,也就是说,非线性FPKEs与McKean- Vlasov SDEs (Nemytskii型)之间的关系,或者更一般地说,McKean意义上的非线性马尔可夫过程。这个问题是由McKean在他1966年发表的一篇影响深远的论文中提出的,这次报告是关于这个领域的最新发展。报告将包括非线性FPKEs的分布解的存在唯一性结果,以及McKean-Vlasov SDEs的弱解的存在唯一性结果。此外,报告还给出了相应路径分布形成非线性马尔可夫过程的判据。这些结果可以应用到包括多孔介质方程(包括用非局部算子代替拉普拉斯算子并可能被输运项扰动的多孔介质方程)及其相关的非线性马尔可夫过程。同时也讨论了涡量形式的二维Navier -Stokes方程及其相关的非线性马尔可夫过程。
Michael Röckner,欧洲科学院院士,德国洪堡奖学金评选委员会委员,曾担任德国数学会理事长,德国比勒菲尔德大学交叉学科研究中心执行主任。他是概率论和随机分析领域的国际顶尖专家,多项工作在Ann. of Prob., Comm. Pure Appl. Math., J. Eur. Math. Soc. 等国际概率顶尖杂志发表,完成了近400篇论文,Google引用达1万9千余次。担任或曾担任10余个国际学术杂志的编委。