Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise

摘要

Existence and uniqueness of a strong solution in $H^{-1}(\mathbb R^d)$ isproved for the stochastic nonlinear Fokker-Planck equation $$dX-{\rmdiv}(DX)dt-\Delta\beta(X)dt=X\,dW \mbox{ in }(0,T)\times\mathbb R^d,\ X(0)=x,$$via a corresponding random differential equation. Here $d\geq 1$, $W$ is aWiener 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 solutionexists for $x\in L^1\cap L^\infty$ and preserves positivity. If $\beta \inL^1_{\rm loc}(\mathbb R)$, the solution is pathwise Lipschitz continuous withrespect to initial data in $H^{-1}(\mathbb R^d)$. Stochastic Fokker-Planckequations with nonlinear drift of the form $dX-{\rmdiv}(a(X))dt-\Delta\beta(X)dt=X\,dW$ are also considered for Lipschitziancontinuous functions $a:\mathbb R\to\mathbb R^d$.