Stationary diffusion processes with discontinuous drift coefficients

Abstract:

The paper is devoted to the stationary Fokker–Planck equation $\Delta u - \mathrm {div} (u \mathbf {f})=0$ with a locally bounded measurable vector field $\mathbf {f}$ defined on the entire $\mathbb {R}^n$. The existence of a positive (not necessarily integrable) solution is proved. Various conditions on the vector field $\mathbf {f}$ are deduced that suffice for the existence of a solution that is a probability density. Under these conditions, the corresponding diffusion has an invariant probability measure with density $u$. In particular, the assumptions of the theorems proved here are fulfilled for certain vector fields $\mathbf {f}$ with trajectories tending to infinity. In this situation, the approach in question turns out to be more efficient than the earlier methods based on the construction of a Lyapunov function.