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 $(\mu_t)_{t \geq 0}$ on the path space $\mathcal C:=C([-r_0,0];\mathbb R^d),$ is analyzed: $$\partial_t \mu(t)=L_{t,\mu_t}^*\mu_t,\ \ t\ge 0,$$ where $\mu(t)$ is the image of $\mu_t$ under the projection $\mathcal C\ni\xi\mapsto \xi(0)\in\mathbb R^d$, and $$L_{t,\mu}(\xi):= \frac 1 2\sum_{i,j=1}^d a_{ij}(t,\xi,\mu)\frac{\partial^2} {\partial_{\xi(0)_i} \partial_{\xi(0)_j}} +\sum_{i=1}^d b_i(t,\xi,\mu)\frac{\partial}{\partial_{\xi(0)_i}},\ \ t\ge 0, \xi\in \mathcal C, \mu\in \mathcal P^{\mathcal C}.$$ 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.