Year:    2015

CAM-Net Digest, Vol. 12 (2015), Iss. 15 : p. 4

Abstract

（一）Weak universality of the KPZ equation

Time: August 7, Friday, at 10:00-12:00am 
Location: Center for Mathematical Sciences, Room 1213 
(创新研究院恩明楼12楼13室)

Title: Weak universality of the KPZ equation

Abstract: The KPZ equation is a popular model of one-dimensional interface propagation. From heuristic consideration, it is expected to be "universal" in the sense that any "weakly asymmetric" or "weakly noisy" microscopic model of interface propagation should converge to it if one sends the asymmetry (resp. noise) to zero and simultaneously looks at the interface at a suitable large scale. The only microscopic models for which this has been proven so far all exhibit very particular that allow to perform a microscopic equivalent to the Cole-Hopf transform. The main bottleneck for generalizations to larger classes of models was that until recently it was not even clear what it actually means to solve the equation, other than via the Cole-Hopf transform. In this talk, we will see that there exists a rather large class of continuous models of interface propagation for which convergence to KPZ can be proven rigorously. The main tool for both the proof of convergence and the identification of the limit is the recently developed theory of regularity structures, but with an interesting twist.

报告人：Professor Martin Hairer

Mathematics Department, The University of Warwick
Professor Martin Hairer is one of the world's foremost leaders in the field of stochastic partial differential equations in particular, and in stochastic analysis and stochastic dynamics in general. By bringing new ideas to the subject he made fundamental advances in many important directions such as the study of variants of Hormander's theorem, systematisation of the construction of Lyapunov functions for stochastic systems, development of a general theory of ergodicity for non-Markovian systems, multiscale analysis techniques, theory of homogenisation, theory of path sampling and, most recently, theory of rough paths and the newly introduced theory of regularity structures.

马丁·海尔，奥地利人，现居英国，任职于华威大学。由于马丁在随机偏微分方程理论方面的杰出贡献，尤其是为这些方程建立了一套正则性结构理论，而被授予菲尔兹奖(2014年)。随机偏微分方程传统上对于数学家来说很难处理，海尔开发了一种新的理论框架，让这些方程变得简单许多，不但开启了许多新的纯数学方向，也对科学和工程中的应用有重大意义。

（二）Limit Theorems of Stochastic Processes on Manifolds

Time: August 7, Friday, at 10:00-12:00am 
Location: Center for Mathematical Sciences, Room 1213 
(创新研究院恩明楼12楼13室)

Title: Limit Theorems of Stochastic Processes on Manifolds

Abstract: I will discuss limit of ordinary differential 
equations on manifolds. Such equations are related to stochastic 
homogenisations.

报告人：Professor Xue-Mei Li

Mathematics Institute, The University of Warwick

Professor Xue-Mei Li has been working on geometric analysis of stochastic processes on manifolds, Malliavin calculus, infinite dimensional analysis, L2 Hodge theory, geometric properties of second order differential operators. Her current research interests are: stochastic differential equations with regular and singular coefficients, construction of stochastic flows, hypoelliptic SDEs, limit theorems, homogeneizations, mean field stochastic equations, variational formulation for solutions of Navier-Stokes equations, transport Equations, and the interplay between geometric structures, such as collapsing of manifolds to lower dimensional objects, and limits of diffusion processes.

李雪梅，华裔数学家，2014年菲尔兹奖得主马丁·海尔之妻。她是一位概率论学专家，研究流行上的随机分析，主要为微分几何和概率论的交叉。数学中她有与其他数学家联名命名的公式，好像马丁的工作中用到过此公式。