Geodesic
Principle of Averaging for Parabolic and Elliptic Differential Equations and for Markov Processes with Small Diffusion
Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 1, pp. 3-25

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In §§ 1–3 of the present paper we prove N. N. Bogolyubov's principle of averaging [1] for parabolic equations (theorems 1,2.2'). Lemma 2 is of most importance for the proof. Kolmogorov's theorem ([14], lemma 2.2) is essentially used for the proof of this lemma. In § 4 theorem 1 is used for studying more general parabolic and elliptic equations. The theorem of the convergence of an invariant measure of a Markov process on a torus to an invariant measure of the flow on a torus (Theorem 3) is proved in § 5. 
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     title = {Principle of {Averaging} for {Parabolic} and {Elliptic} {Differential} {Equations} and for {Markov} {Processes} with {Small} {Diffusion}},
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R. Z. Khas'minskii. Principle of Averaging for Parabolic and Elliptic Differential Equations and for Markov Processes with Small Diffusion. Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 1, pp. 3-25. http://geodesic.mathdoc.fr/item/TVP_1963_8_1_a0/