Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations

In this paper the existence of a unique invariant measure for Markov processes satisfying the conditions $1^\circ-9^\circ$ is proved. This result is applied to obtain the asymptotic properties of the solution to the Cauchy problem for the parabolic equation $\partial u/\partial t=Lu$ when $t\to+\infty$. It is established that these properties depend on properties of the solution to the extremal Dirichlet problem for the equations $Lu=0$ and $Lu=-1$. The sufficient conditions for them expressed in terms of the behaviour of the coefficients in the equation $Lu=\partial u/\partial t$ are given in the appendix.