On Some Diffusion Processes with Stationary Distributions

Abstract

This paper provides a sufficient condition (on the vector field $\bff(x)$), under which the stationary Fokker–Planck equation $\Delta u-{\rm div}\,(u{\bf f})=0$ has a solution in the form of probability density. The stochastic differential equation with the drift ${\bf f}(x)$ has an invariant probabilistic measure under this condition. This condition is satisfied by some vector fields ${\bf f}$ (drifts) having a sequence of fixed locally stable points, which tends to infinity. Sometimes the proposed method turns out to be more effective than previously known methods based on the Lyapunov function.