Geodesic
Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations
Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 2, pp. 196-214 
R. Z. Khas'minskii. Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations. Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 2, pp. 196-214. http://geodesic.mathdoc.fr/item/TVP_1960_5_2_a2/
@article{TVP_1960_5_2_a2,
     author = {R. Z. Khas'minskii},
     title = {Ergodic {Properties} of {Recurrent} {Diffusion} {Processes} and {Stabilization} of the {Solution} to the {Cauchy} {Problem} for {Parabolic} {Equations}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {196--214},
     year = {1960},
     volume = {5},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1960_5_2_a2/}
}
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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.