Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion

Abstract

This paper investigates the probability distribution of solutions to McKean--Vlasov stochastic differential equations driven by fractional Brownian motion with Hurst parameter H>1/2. Our main contribution is the derivation of the associated Fokker--Planck equation, which characterizes the time evolution of the law of the solution in a suitable distributional framework. Under mild assumptions, we show that the law-valued process is absolutely continuous in time and provide an explicit weak formulation of the corresponding fractional McKean--Vlasov Fokker--Planck equation. In the case where the law admits a density, we obtain a more explicit partial differential equation with time-dependent diffusion coefficients induced by the fractional noise. We further establish a fractional Feynman--Kac representation, linking the forward Fokker--Planck equation with a backward Kolmogorov equation for functionals of the solution process. This result extends the classical Feynman--Kac framework to mean--field dynamics driven by fractional Brownian motion. To illustrate the theory, we analyze several explicit examples, including the law of fractional Brownian motion itself and linear McKean--Vlasov fractional SDEs. These examples highlight how fractional noise and mean--field interactions jointly affect the probabilistic and analytic structure of the system.