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

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.