随机动力系统的逼近和跑出问题

项目名称： 随机动力系统的逼近和跑出问题

项目编号： No.11501549

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 申俊

作者单位： 中国科学院数学与系统科学研究院

项目金额： 17万元

中文摘要： 本项目主要研究随机动力系统的逼近和跑出问题。由于随机问题的不确定性，研究带有乘性白噪音的随机微分方程，我们不仅需要大量随机分析的理论，而且应用起来非常不方便。与前人讨论了随机微分方程的Wong-Zakai逼近和数值解不同，本项目考虑一种受布朗运动驱动的无界随机受迫。我们研究这种噪音驱动下的新系统，讨论它的轨道、光滑不变流形、不变叶层和吸引子等与原系统的关系，即给出是逐点收敛或者一致收敛的条件。这样新系统就可以作为原系统的一个逼近系统。但与原系统不同，它可以作为一个带有随机参数的微分方程，允许我们逐点研究其定性性质。因此与原系统相比，它分析起来更为方便。至于跑出问题，我们将考虑两方问题，其一是带有随机白噪音扰动的系统，我们研究其轨道跑出确定系统不变流形邻域的概率；其二就是对于逼近系统，我们可以研究他们的大偏差估计和跑出平衡点、吸引子或者不变流形的概率与原系统的关系。

中文关键词： 随机动力系统；不变流形；不变叶层；逼近；跑出问题

英文摘要： This project is mainly concerned with the approximation of random dynamical systems and exit problem. Due to the uncertainties of the random problem, for the study of stochastic differential equation with a multiplicative white noise, we need not only to know a lot of random analysis theory , but it is very inconvenient to apply. Unlike predecessors discussed the Wong-Zakai approximation and numerical solution for stochastic differential equation, in this project we will consider an unbounded random forcing driven by a Brownian motion. We will study a new system driven by the noise and discuss the relation on orbits、smooth invariant manifolds、invariant foliations and attractors between the new system and the old one, that is, we will give the conditions they are pointwise or uniformly convergence to those of original system. Then the new system can be taken as an approximated one. However, unlike the original system, it may be taken as a differential system with random parameter and admits us to study pointwisely. Thus compare with the original system, it is more convenient to analyze. As for the exit problem, we will consider two issues. One is the noise-induced deviations of the sample paths from the neighborhood of invariant manifolds for deterministic dynamical systems, the other is the relationship of the large deviation estimate and the probability of exiting from equilibrium points、attractors or invariant manifolds between the approximated system and the original one.

英文关键词： Random dynamical system;Invariant manifold;Invariant foliation;Approximation;Exit problem