Nonlinear Fokker-Planck equations driven by Gaussian linear multiplicative noise

Existence of a strong solution in H-1 (R-d) is proved for the stochastic nonlinear Fokker-Planck equation dX - div(DX)dt - Delta beta(X)dt = X dW in (0, T) x R-d, X(0) = x, respectively, for a corresponding random differential equation. Here d >= 1, W is a Wiener process in H-1(R-d), D is an element of C-1(R-d, R-d) and beta is a continuous monotonically increasing function satisfying some appropriate sublinear growth conditions which are compatible with the physical models arising in statistical mechanics. The solution exists for x is an element of L-1 boolean AND L-infinity and preserves positivity. If beta is locally Lipschitz, the solution is unique, pathwise Lipschitz continuous with respect to initial data in H-1(R-d). Stochastic Fokker-Planck equations with nonlinear drift of the form dX - div(a(X))dt - Delta beta(X)dt = X dW are also considered for Lipschitzian continuous functions a : R -> R-d. (C) 2018 Elsevier Inc. All rights reserved.