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 $({\mu }_t{)}_{t\ge 0}$ on the path space $\mathcal{C}:=C([-r_0,0];{\mathbb{R}}^d),$ is analyzed: $${\partial }_t\mu (t)=L_{t,{\mu }_t}^{\ast }{\mu }_t,t\ge 0,$$ 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.

22 pages

References in corpus (2)

• Harnack inequality and applications for stochastic generalized porous media equations
• Harnack Inequalities for Functional SDEs with Multiplicative Noise and Applications

Cited by in corpus (4)

• Distribution Dependent SDEs with Singular Coefficients
• Least squares estimation for path-distribution dependent stochastic differential equations
• Order Preservation for Path-Distribution Dependent SDEs
• Comparison Theorem for Distribution Dependent Neutral SFDEs