State-Density Flows of Non-Degenerate Density-Dependent Mean Field SDEs and Associated PDEs

TL;DR

This paper establishes the regularity and uniqueness of solutions for a coupled system of a Fokker-Planck equation and a mean-field SDE, linking probabilistic and PDE methods under non-degeneracy conditions.

Contribution

It introduces a novel combined probabilistic and analytical approach to prove regularity and uniqueness of solutions for nonlocal PDEs associated with density-dependent mean-field SDEs.

Findings

Proves the function V is the unique classical solution of a nonlocal PDE.

Develops a method to analyze the flow's differential properties in L^2 space.

Provides an example illustrating the main theoretical results.

Abstract

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,\mu}$ with initial $(t,\mu)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,\mu}$ with initial $(t,x)\in[0,T]\times \mathbb{R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,\mu)=\mathbb{E}[\Phi(X^{t,x,\mu}_T,m^{t,\mu}(T,\cdot))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V$ is the unique classical solution of a nonlocal partial differential equation of mean-field type. The proof depends heavily on the differential properties of the flow $\mu\mapsto (m^{t,\mu}, X^{t,x,\mu})$ over $\mu\in L^2(\mathbb{R}^d)$. We also give an example to illustrate the role of our main result. Finally, we give a discussion on the $L^1$ choice case in the initial $\mu$.