Nonlinear Fokker–Planck equations driven by Gaussian linear multiplicative noise

Abstract

Existence of a strong solution in H−1(Rd) is proved for the stochastic nonlinear Fokker–Planck equationdX−div(DX)dt−Δβ(X)dt=XdW in (0,T)×Rd,X(0)=x, respectively, for a corresponding random differential equation. Here d≥1, W is a Wiener process in H−1(Rd), D∈C1(Rd,Rd) and β 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∈L1∩L∞ and preserves positivity. If β is locally Lipschitz, the solution is unique, pathwise Lipschitz continuous with respect to initial data in H−1(Rd). Stochastic Fokker–Planck equations with nonlinear drift of the form dX−div(a(X))dt−Δβ(X)dt=XdW are also considered for Lipschitzian continuous functions a:R→Rd.