Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise

Abstract

Existence and uniqueness of a strong solution in $H^{-1}({\mathbb{R}}^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-{\rm div}(DX)dt-Δβ(X)dt=X\,dW \mbox{ in }(0,T)\times\mathbb R^d,\ X(0)=x,$$ via a corresponding random differential equation. Here $d\ge 1$, $W$ 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 $\beta \in L_{\mathrm{loc}}^1(\mathbb{R})$, 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 $dX-\mathrm{div}(a(X))dt-\Delta \beta (X)dt=XdW$ are also considered for Lipschitzian continuous functions $a:\mathbb{R}\to {\mathbb{R}}^d$.