报告题目：Irreducibility of SPDEs driven by pure jump noise

报 告 人：翟建梁 副教授 中国科学技术大学

邀请人：李童庆、薄立军

报告时间：2022年12月1日（周四）10:00-11:30

腾讯会议ID：433-426-520，密码：2022

报告人简介：翟建梁，中国科学技术大学副教授，2010年于中国科学院数学与系统科学研究院获得博士学位，先后在北京大学、英国曼彻斯特大学、伦敦国王学院进行博士后研究和Research Fellow工作。主持国家自然科学基金青年基金和面上基金各一项，参加国家自然科学基金重点项目两项。主要研究方向是随机偏微分方程，主要学术贡献：纯跳Levy过程驱动的随机偏微分方程的大偏差原理、中偏差原理、不可约性等；随机环境下流体中趋化模型研究；平稳测度支撑的渐近行为的研究。已在中国科学：数学，J. Euro. Math. Soc., J. Math. Pures Appl.，J. Funct. Anal.等国际权威的概率杂志上发表论文30余篇。

报告摘要：The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. In the literature, there are very few results on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise. The existing methods on this topic are basically along the same lines as that for the Gaussian case. They heavily rely on the fact that the driving noises are additive type and more or less in the class of stable processes. The use of such methods to deal with the case of other types of additive pure jump noises appears to be unclear, let alone the case of multiplicative noises.

In this paper, we develop a new, effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild, and in some sense they are necessary and sufficient. This leads to not only significantly improving all of the results in the literature, but also to new irreducibility results of a much larger class of equations driven by pure jump noise with much weaker requirements than those treatable by the known methods. As a result, we are able to apply the main results to SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrodinger equations, Euler equations etc. We emphasize that under our setting the driving noises could be compound Poisson processes, even allowed to be infinite dimensional. It is somehow surprising.

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