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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1959, Volume 4, Issue 3, Pages 332–341 (Mi tvp4893)

On Positive Solutions of the Equation $\mathfrak Au+Vu=0$

R. Z. Khas'minskii

Moscow

Abstract: Let $X_t$ be a path of the continuous Markov process in the domain $D$ with boundary $\Gamma$ in a metric space, $\tau$ is the moment of reaching $\Gamma$; $\mathfrak{A}$ is the extended infinitesimal operator of the process and $V$ is a continuous non-negative function at $D$. The theorem reads as follows.
Let $\Gamma$ be regular in the sense of (1), $X_t$ be a strongly Feller process; then $\mathbf M_x\exp\{ \int_0^\tau{V(X_t ) dt}\}<+\infty$ if and only if the equation (2) has a positive and continuous solution in $D\cup\Gamma$.
This theorem is applied to obtain different conditions, which guarantee the existence of the unique solution of the first boundary value problem for (2). Stability of the maximal eigenvalue of the operator $\mathfrak{A}u+Vu$ ($u|_\Gamma=0$) by some global changes of domain is proved also.

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English version:
Theory of Probability and its Applications, 1959, 4:3, 309–318

Citation: R. Z. Khas'minskii, “On Positive Solutions of the Equation $\mathfrak Au+Vu=0$”, Teor. Veroyatnost. i Primenen., 4:3 (1959), 332–341; Theory Probab. Appl., 4:3 (1959), 309–318

Citation in format AMSBIB
\Bibitem{Kha59} \by R.~Z.~Khas'minskii \paper On Positive Solutions of the Equation $\mathfrak Au+Vu=0$ \jour Teor. Veroyatnost. i Primenen. \yr 1959 \vol 4 \issue 3 \pages 332--341 \mathnet{http://mi.mathnet.ru/tvp4893} \transl \jour Theory Probab. Appl. \yr 1959 \vol 4 \issue 3 \pages 309--318 \crossref{https://doi.org/10.1137/1104030} 

• http://mi.mathnet.ru/eng/tvp4893
• http://mi.mathnet.ru/eng/tvp/v4/i3/p332

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This publication is cited in the following articles:
1. V. V. Sarafyan, “On the limit behavior of the largest eigenvalue of an elliptic operator with a small parameter”, Math. USSR-Sb., 55:2 (1986), 529–545