Fokker-Planck equations for conditional McKean-Vlasov systems driven by Brownian sheets

TL;DR

This paper develops a mathematical framework for analyzing conditional McKean-Vlasov equations driven by space-time white noise, establishing existence, uniqueness, and deriving associated stochastic Fokker-Planck equations with applications to control problems.

Contribution

It introduces a novel approach using stochastic calculus of white noise to study conditional McKean-Vlasov systems and derives integral stochastic Fokker-Planck equations for their laws.

Findings

Proved existence and uniqueness of solutions for the SPDEs.

Derived stochastic Fokker-Planck equations for conditional laws.

Applied the framework to a partial observation control problem.

Abstract

We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus of time-space white noise, utilizing tools such as the two-parameter Ito formula, Malliavin calculus, and orthogonal decompositions to analyze convergence and stochastic properties. Existence and uniqueness of solutions for the associated stochastic partial differential equations (SPDEs) are rigorously established. Additionally, an integral stochastic Fokker-Planck equation is derived for the conditional law, employing Fourier transform methods and stochastic analysis in the plane. The framework is further applied to a partial observation control problem, showcasing its potential for analyzing stochastic systems with conditional dynamics.