Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise

TL;DR

This paper proves existence, uniqueness, and regularity of solutions for a class of nonlinear stochastic Fokker-Planck equations driven by Gaussian noise, extending understanding of their mathematical properties.

Contribution

It establishes the well-posedness and regularity results for nonlinear stochastic Fokker-Planck equations with multiplicative Gaussian noise, including positivity preservation and Lipschitz continuity.

Findings

Existence and uniqueness of strong solutions in $H^{-1}$

Positivity preservation of solutions

Pathwise Lipschitz continuity with respect to initial data

Abstract

Existence and uniqueness of a strong solution in $H^{-1}(\mathbb R^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\rm div}(DX)dt-\Delta\beta(X)dt=X\,dW \mbox{ in }(0,T)\times\mathbb R^d,\ X(0)=x,$$ via a corresponding random differential equation. Here $d\geq 1$, $W$ is a Wiener process in $H^{-1}(\mathbb R^d)$, $D\in C^1(\mathbb R^d,\mathbb R^d)$ and $\beta$ is a continuous monotonically increasing function. The solution exists for $x\in L^1\cap L^\infty$ and preserves positivity. If $\beta \in L^1_{\rm loc}(\mathbb R)$, the solution is pathwise Lipschitz continuous with respect to initial data in $H^{-1}(\mathbb R^d)$. Stochastic Fokker-Planck equations with nonlinear drift of the form $dX-{\rm div}(a(X))dt-\Delta\beta(X)dt=X\,dW$ are also considered for Lipschitzian continuous functions $a:\mathbb R\to\mathbb R^d$.