Research-Stochastic Differential Equations versus Fokker-Planck-Kolmogorov Equations: a Short Review and Some Recent Progress

Existence of a strong solution in

is proved for the stochastic nonlinear FokkerPlanck equation

, via a corresponding random differential equation. Here d ≥ 1, W is a Wiener process in

and β is a continuous monotonically increasing function satisfying some appropriate polynomial growth conditions. The solution exists for

and preserves positivity. If β is locally Lipschitz, the solution is unique, path-wise Lipschitz continuous with respect to initial data in

. Stochastic Fokker-Planck equations with nonlinear drift of the form

are also considered for Lipschitzian continuous functions

. Joint work with Viorel Barbu (Romanian Academy of Sciences, Iasi).