勷勤数学•专家报告

题      目：On Metastability for Stochastic Ordinary Differential Equations

报  告  人：蒋继发 教授  (邀请人：丁维维)

                                           河南师范大学

时      间：7月16日  15：00-16：00

地     点：数科院东楼401

报告人简介:

       蒋继发，河南师范大学教授，被国家人事部授予具有突出贡献的中青年专家，享受国务院政府特殊津贴。曾先后在中国科学技术大学、同济大学、上海师范大学任教，并担任同济大学数学系主任。1993年获安徽省科技进步奖二等奖，2017年获上海市自然科学奖二等奖。安徽省政协第七、八和九届委员会委员。蒋继发教授最早把单调动力系统的研究引入我国，是我国这一方向的学术带头人。其代表性工作发表于Crelle’s Journal、JMPA、SIMA、SIAP、SIADS、JDE等。现从事随机动力系统的研究。自1993年以来持续主持国家自然科学基金面上项目、参加两项重点项目。蒋继发教授在研究生培养方面成绩突出。2004-2006年连续三年被中国科学院评为“优秀研究生导师”。指导的博士毕业生中1名获国家杰出青年科学基金资助、1名入选教育部青年长江学者和拔尖人才、2名获全国优秀博士学位论文奖。

摘      要：

        This talk addresses metastability in stochastic differential equations (SDEs) of the form $$dx_t^{\epsilon}= [-sigma(x_t^{\epsilon})\sigma^T(x_t^{\epsilon})\nabla V(x_t^{\epsilon})+R(x_t^{\epsilon})]dt+\sqrt{2\epsilon}\sigma(x_t^{\epsilon})dW_t \qquad \qquad (1)$$under the assumptions that $\langle R(x),\nabla V(x)\rangle =0$, ${\rm div}R(x)=0$ and $\sigma(x)\sigma^T(x)$ is positive definite pointwise. When the potential $V$ satisfies mild conditions, which hold for polynomial growth at infinity, the system admits a uniform Freidlin-Wentzell large deviations principle and possesses invariant measures.

     Suppose that  the unperturbed system associated with (1) has all maximal equivalent classes $K_1, K_2, ..., K_l$ containing all limit sets.  We rigorously characterize the global phase portraits of both the unperturbed system and its extremal equations. If $K_i$ is Lyapunov stable and the boundary of its attracting basin contains $K_j$, then we prove that the quasipotential $V(K_i,K_j)=V(K_j)-V(K_i)=:\Delta V$ from $K_i$ to $K_j$, which implies that the mean transition time from the stable state $K_i$ to the transition state $K_j$ is exponentially large in $O\left(\frac{\Delta V}{\epsilon}\right)$ with exponentially small probability and gives the most probable transition path as well. 

      After computing the transitive difficult matrix $(V(K_i,K_j))_{l\times l}$ and using the Freidlin-Wentzell procedure, we can determine the concentration of limiting measure for the invariant measures of (1) as $\epsilon \rightarrow 0$. For $l\le 4$, we can classify the metastability of (1). For $l > 4$, we provide a programme to carry out the goal.

        欢迎老师、同学们参加、交流！