# American Institute of Mathematical Sciences

June  2019, 39(6): 3017-3035. doi: 10.3934/dcds.2019125

## Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs

 1 Center for Applied Mathematics, Tianjin University, Tianjin 300072, China 2 Department of Mathematics, Bielefeld University, D-33501 Bielefeld, Germany 3 Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author: Feng-Yu Wang

Received  January 2018 Revised  October 2018 Published  February 2019

Fund Project: Supported by NNSFC (11801406, 11771326, 11831014, 11431014, 11726627).

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:
 $\partial_t \mu(t) = L_{t, \mu_t}^*\mu_t, \ \ t\ge 0,$
where
 $\mu(t)$
is the image of
 $\mu_t$
under the projection
 ${\scr {C}}\ni\xi\mapsto \xi(0)\in\mathbb R^d$
, and
 \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*}
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.
Citation: Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125
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