Fokker-Planck equations with terminal condition and related McKean probabilistic representation

TL;DR

This paper investigates Fokker-Planck equations with terminal conditions, establishing existence and uniqueness, and provides a probabilistic McKean representation related to the time-reversal of diffusion processes.

Contribution

It introduces conditions for well-posedness of Fokker-Planck equations with prescribed final data and links them to McKean-type stochastic processes.

Findings

Established existence and uniqueness conditions for PDEs with terminal data

Derived a probabilistic representation via McKean equations

Connected time-reversal dynamics of diffusions to PDE solutions

Abstract

Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process.