山东大学数学学院学术沙龙（即通俗数学讲座）2019-2020学年第二期

主讲：Huaizhong Zhao (赵怀忠，Loughborough Univ.，山东大学)

时间：2019年12月18日（周三）16:00-17:00

地点：中心校区知新楼B座924报告厅

题目：随机分析、动力系统及不确定系统的建模

摘要：此报告主要是关于随机系统的动力学性质，其中包括最基本的问题如不变测度、周期测度、以及它们的遍历性。将涉及随机周期理论及其应用的最新进展，作为例子将讲解为什么周期测度及其遍历性是我们解决随机共振等问题中的关键。

我将引入随机周期轨道、周期测度概念，并给出研究存在唯一性的数学工具。我们将用马氏半群的最小生成算子的纯虚谱来刻画周期测度的遍历性。我们证明从谱间隙或不可约性可以得到PS-混合。这些结果可以广泛应用在随机微分方程、随机偏微分方程、随机映射、马氏链等随机系统中。我们进一步将得到周期测度的密度函数的存在性准则以及他们满足的Fokker-Planck方程。

作为例子，我们将考虑Benzi-Parisi-Sutera-Vulpiani关于冰川期周期突变模型。我们证明了此系统的周期测度的存在唯一性，以及噪声存在（非退化）引起的突变的必然性。我们进一步得到突变时间的期望满足的偏微分方程从而得到突变的周期性以及随机共振。

如有时间，我将涉及随机拟周期以及非线性期望下的遍历理论，以及随机周期在研究数据方面的应用。

Title：Stochastic Analysis, Dynamics and Modelling of Uncertainty

Abstract：In this talk, I will mainly talk about dynamics of stochastic systems, invariant and periodic measures and ergodic theory. This includes the recent developments of the random periodic theory and some examples of applications e.g. stochastic resonance.

We introduce the concepts of random periodic paths and periodic measures and give mathematical tools for their existence and uniqueness, their characterization in terms of the pure imaginary spectrum of infinitesimal generators. We also prove that the spectral gap or irreducibility implies PS-mixing. These results apply to many stochastic systems including stochastic differential equations, stochastic partial differential equations, random mappings and Markov chains. We give sufficient conditions for the existence of the density of periodic measures and obtain the Fokker-Planck equation for the density.

As an example, considering Benzi-Parisi-Sutera-Vulpiani’s stochastic resonance model for the transition between the ice age and the interglacial period, we prove a result of the existence and uniqueness of periodic measures which implies the transition exists. We further obtain the PDEs for the expected transition time.

I will also comment on recent results on random quasi-periodicity, dynamics under nonlinear expectations and analysis of data with periodicity.

欢迎本院教师，研究生，和本科生参加！