Stochastic analysis for measure-valued processes

Abstract

In this paper, we introduce some recent progress on stochastic analysis for measure-valued processes, and propose some problems for further study in this direction. We first recall the notions of derivatives in measures, clarify their relations, and investigate functional inequalities of Dirichlet forms induced by these derivatives and reference probability measures. Then we construct diffusion processes on the Wasserstein space by using image dependent SDEs (stochastic differential equations), investigate the exponential ergodicity of the processes, and establish the Feynman-Kac formula for solutions of PDEs (partial differential equations) on the Wasserstein space. We also introduce Bismut formulas for the $L$ derivative of distribution dependent SDEs.