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 $ {\scr {C}}: = C([-r_0, 0];\mathbb R^d), $ is analyzed: \begin{document}$ \partial_t \mu(t) = L_{t, \mu_t}^*\mu_t, \ \ t\ge 0, $\end{document} where $ \mu(t) $ is the image of $ \mu_t $ under the projection $ {\scr {C}}\ni\xi\mapsto \xi(0)\in\mathbb R^d $, and \begin{document}$ \begin{align*} L_{t, \mu}(\xi)&: = \frac 1 2\sum\limits_{i, j = 1}^d a_{ij}(t, \xi, \mu)\frac{\partial^2} {\partial_{\xi(0)_i} \partial_{\xi(0)_j }} \\\; &\quad +\sum\limits_{i = 1}^d b_i(t, \xi, \mu)\frac{\partial}{\partial_{\xi(0)_i}}, \ \ t\ge 0, \xi\in {\scr {C}}, \mu\in \scr P^{\scr {C}}. \end{align*} $\end{document} 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.