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

In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,μ}$ with initial $(t,μ)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,μ}$ 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 $V(t,x,μ)=\mathbb{E}[Φ(X^{t,x,μ}_T,m^{t,μ}(T,\cdot))]$. 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 $μ\mapsto (m^{t,μ}, X^{t,x,μ})$ over $μ\in L^2(\mathbb{R}^d)$. 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 $μ$.

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