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

Xing Huang; Michael Röckner; Feng-Yu Wang

doi:10.3934/dcds.2019125

Article Contents

2019, Volume 39, Issue 6: 3017-3035. Doi: 10.3934/dcds.2019125

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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
^* Corresponding author: Feng-Yu Wang

Received: January 2018

Revised: October 2018

Published: February 2019

Supported by NNSFC (11801406, 11771326, 11831014, 11431014, 11726627)

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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:

$ \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.

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Mathematics Subject Classification: Primary: 60J75, 47G20, 60G52.

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