State-density flows of non-degenerate density-dependent mean field SDEs and associated PDEs

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 $\mu$.