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

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.