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.