常微、偏微及随机微分方程(英文)

ODE,SDE and PDE

本讲义包含两部分.在第一部分中,我们的目的是证明由属于Sobolev空间的向量场生成的可测映射流的存在性,为此我们讨论常微与运输方程和连续方程的联系.第二部分处理随机微分方程:在系数不满足Lipschitz条件时,我们显式地构造出了强解;在整体Lipschitz条件下,完整地证明了随机同胚流的存在性,并讨论了它与Fokker-Planck方程的关系;最后我们强调了椭圆性在把弱解变成强解中所起的作用.

This note consisting of two parts gives a survey of new developments on the topic concerning ordinary differential equations, stochastic differential equations and partial differential equations. In the first one, our aim is to prove the existence of a flow of measurable maps associated to a vector field belonging to a Sobolev space; to this end, we discuss the link with transport equations and continuity equations. The second part deals with stochastic differential equations： we construct explicitly strong solutions beyond Lipschitz conditions, the existence of a stochastic flow of homeomorphisms under global Lipschitz conditions is fully proved and the relation with Fokker-Planck equations is discussed, and finally we emphasize the role of ellipticity to make weak solutions to strong one.