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Teor. Veroyatnost. i Primenen., 1966, Volume 11, Issue 3, Pages 463–471 (Mi tvp613)  

The exterior Dirichlet problem for the class of bounded functions

M. I. Freidlin

Moscow

Abstract: We consider different settings of the exterior Dirichlet problem for the class of bounded functions for an elliptic operator of the second order. It is known that if the Markov process corresponding to a given operator is a non return one the solution of the exterior problem may not be unique when no additional conditions are imposed at infinity. We study the conditions at infinity which secure the uniqueness of the solution in the class of bounded functions. It comes out that there is a nontrivial boundary at infinity. This boundary is constructed as the set of equilibrium points of some vector field on the unit sphere. The aforementioned vector field is constructed from the coefficients of the operator. All the results are obtained by investigating the behaviour of the trajectories of the corresponding Markov process at $t\to\infty$.

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English version:
Theory of Probability and its Applications, 1966, 11:3, 407–414

Bibliographic databases:

Received: 06.06.1965

Citation: M. I. Freidlin, “The exterior Dirichlet problem for the class of bounded functions”, Teor. Veroyatnost. i Primenen., 11:3 (1966), 463–471; Theory Probab. Appl., 11:3 (1966), 407–414

Citation in format AMSBIB
\Bibitem{Fre66}
\by M.~I.~Freidlin
\paper The exterior Dirichlet problem for the class of bounded functions
\jour Teor. Veroyatnost. i Primenen.
\yr 1966
\vol 11
\issue 3
\pages 463--471
\mathnet{http://mi.mathnet.ru/tvp613}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=201812}
\zmath{https://zbmath.org/?q=an:0202.47002}
\transl
\jour Theory Probab. Appl.
\yr 1966
\vol 11
\issue 3
\pages 407--414
\crossref{https://doi.org/10.1137/1111039}


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