Image Dependent Conditional McKean-Vlasov SDEs for Measure-Valued Diffusion Processes

TL;DR

This paper introduces a new class of measure-dependent stochastic differential equations influenced by the solution's image, establishing well-posedness, a Feynman-Kac formula, and ergodicity results for measure-valued diffusions.

Contribution

It develops a novel framework for mean field SDEs with common noise depending on the solution's image, with weaker conditions for well-posedness and new PDE connections.

Findings

Strong well-posedness under weaker monotone conditions

Feynman-Kac formula for Schrödinger-type PDEs on probability spaces

Ergodicity of certain measure-valued diffusion processes

Abstract

We consider a special class of mean field SDEs with common noise which depend on the image of the solution (i.e. the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games, the Feynman-Kac formula is established to solve Schr\"ordinegr type PDEs on $\scr P_2$, and the ergodicity is proved for a class of measure-valued diffusion processes.