Irreducibility of SPDEs driven by pure jump noise

主讲人：翟建梁

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.

主讲人简介：翟建梁，中国科学技术大学副教授。2010年获得中国科学院数学与系统科学研究院概率论与数理统计博士学位，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。主要研究方向是Levy过程驱动的随机偏微分方程。已发表论文20余篇, 包括“J. Math. Pures Appl.”、“J. Funct. Anal.”、“Bernoulli”、“J. Differential Equations”、“C.R.Math.Acad.Sci.Paris”等国际重要杂志。主要学术贡献：Levy过程驱动的随机偏微分方程的鞅解存在性和马氏选择、时间正则性、大偏差原理、中偏差原理等；平稳测度支撑的渐近行为的研究。主持国家自然科学基金青年基金一项，主持国家自然科学基金面上项目一项，参加国家自然科学基金重点项目一项。

邀请人：吴付科

时间：2023年3月19日（星期日）14:00-15:30