INTEGRAL IDENTITY AND MEASURE ESTIMATES FOR STATIONARY FOKKER-PLANCK EQUATIONS

We consider a Fokker-Planck equation in a general domain in R-n with L-loc(p) drift term and W-loc(1,p) diffusion term for any p > n. By deriving an integral identity, we give several measure estimates of regular stationary measures in an exterior domain with respect to diffusion and Lyapunov-like or anti-Lyapunov-like functions. These estimates will be useful to problems such as the existence and nonexistence of stationary measures in a general domain as well as the concentration and limit behaviors of stationary measures as diffusion vanishes.