Abstract

In this talk, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of polynomial growth and possible degenerate driving noises. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains and extend Freidlin and Wentzell's asymptotics theorem to stochastic systems with unbounded coefficients. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and  van der Pol Duffing equation whose noises are naturally degenerate. We also construct two examples to match the global phase portraits of Freidlin and Wentzell's unperturbed systems and to explicitly  compute their transition difficulty matrices. Other applications include stochastic May-Leonard system and random systems with infinitely many  equivalent classes.

个人简介

张土生，中国科学技术大学教授，国际知名的概率论专家，曾任英国曼彻斯特大学终身教授和概率统计系主任， 入选长江学者，国家特聘专家。现担任《Stochastic Processes and Their Applications》、《Journal of Theoretical Probability》、《Applied Mathematics and Optimization》等国际著名刊物的编委。张教授的研究成果包括两本专著（由Spinger出版）和100多篇发表在国际一流杂志的论文。他在随机微分方程、随机偏微分方程、马氏唯一性、Dirichlet 型和大偏差的研究等方面有突出贡献。