Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent sdes

Abstract

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker-Planck equations for probability measures (μ t ) t≥0 on the path space C := C([-r 0 , 0];ℝ d ); is analyzed: (Equation Presented) where μ(t) is the image of μ t under the projection C ∋ ξ → ξ(0)∈ ℝ 2 , and (Equation presented) Under reasonable conditions on the coefficients a ij and b i , the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.