  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 $a:\mathbb R\to\mathbb R^d$.
