Nonlinear Fokker-Planck equations with time-dependent coefficients

TL;DR

This paper proves existence, uniqueness, and properties of solutions to nonlinear time-dependent Fokker-Planck equations, linking them to McKean-Vlasov SDEs, under specific conditions.

Contribution

It introduces an operatorial approach to establish strong solutions for nonlinear Fokker-Planck equations with time-dependent coefficients and connects these solutions to associated stochastic differential equations.

Findings

Existence and uniqueness of strong solutions in Sobolev spaces.

Preservation of probability density over time.

Construction of weak solutions linked to McKean-Vlasov SDEs.

Abstract

An operatorial based approach is used here to prove the existence and uniqueness of a strong solution $u$ to the time-varying nonlinear Fokker--Planck equation $u_t(t,x)-\Delta(a(t,x,u(t,x))u(t,x))+{\rm div}(b(t,x,u(t,x))u(t,x))=0$ in $(0,\infty)\times \mathbb{R}$ $u(0,x)=u_0(x),\ x\in\mathbb{R}^d$ in the Sobolev space $H^{-1}(\mathbb{R}^d)$, under appropriate conditions on the $a:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}$ and $b:[0,T]\times\mathbb{R}^d\times\mathbb{R}\to\mathbb{R}^d.$ It is proved also that, if $u_0$ is a density of a probability measure, so is $u(t,\cdot)$ for all $t\ge0$. Moreover, we construct a weak solution to the McKean-Vlasov SDE associated with the Fokker-Planck equation such that $u(t)$ is the density of its time marginal law. MSC: 60H15, 47H05, 47J05. Keywords: Fokker--Planck equation, Cauchy problem, stochastic differential equation, Sobolev space, periodic solution.