Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise

Existence and uniqueness of a strong solution in $H^{-1}({\mathbb{R}}^d)$ is proved for the stochastic nonlinear Fokker-Planck equation $$dX-\mathrm{d}\mathrm{i}\mathrm{v}(DX)dt-\mathrm{\Delta }\beta (X)dt=XdWin(0,T)\times {\mathbb{R}}^d,X(0)=x,$$ 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 $x\in L^1\cap L^{\mathrm{\infty }}$ 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 $a:\mathbb{R}\to {\mathbb{R}}^d$.