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

 Teor. Veroyatnost. i Primenen., 1973, Volume 18, Issue 1, Pages 3–28 (Mi tvp2677)

Formulas for eigen-functions and eigen-measures associated with Markov process

A. D. Wentzell

Moscow

Abstract: Let ($x_t$, $\zeta$, $\mathscr M_t$, $\mathbf P_x$) be a strong Markov process, $\tau_1$ a Markov time, $a=a_1+ia_2$ a complex number, $\mathbf M_xe^{a_1\tau_1}<\infty$. We can consider two operators with the kernel
$$q_a(x,\Gamma)=\mathbf M_xe^{a_1\tau_1}\chi_\Gamma(x_{\tau_1})$$
($\chi_\Gamma$ stands for the indicator function of the set $\Gamma$, $\mathbf M_x$ for the expectation corresponding to the probability measure $\mathbf P_x$), one acting upon functions, the other upon measures. Let us call $a$-eigen-functions (measures) eigen-function (measures) of the semi-group generator connected with the Markov process that correspond to the eigen-value $-a$. For certain classes of Markov times, there is a one-to-one correspondence between $a$-eigen-functions (measures) and eigen-functions (measures) of $q_a$ with the eigen-value 1. As for functions, this correspondence is expressed in an obvious way, but for measures the following holds: If $\nu=\nu q_a$, then $\mu$ is an $a$-eigen-measure,
$$\mu(\Gamma)=\int\nu (dx)\mathbf M_x\int_0^{\tau_1}e^{at}\chi_\Gamma(x_t) dt$$
in continuous parameter case; for Markov chains the inner integral is replaced by a sum.
This relation is a generalization of a formula for invariant measures (i.e. $a=0$) which was introduced in many papers ([2]–[8]).
The class of admissible Markov times includes times when a motion cycle between two disjoint sets ends; for these Markov times, the whole construction is a generalization of an approach to eigen-functions of a differential operator based on Schwartz alternating method.
The results concerning the correspondence between $a$-eigen functions (measures) and eigen-functions (measures) of $q_a$ can be applied to investigate the asymptotical behaviour of eigen-values and eigen-functions of a differential operator with small parameter (cf. [13]).

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English version:
Theory of Probability and its Applications, 1973, 18:1, 1–26

Bibliographic databases:

Citation: A. D. Wentzell, “Formulas for eigen-functions and eigen-measures associated with Markov process”, Teor. Veroyatnost. i Primenen., 18:1 (1973), 3–28; Theory Probab. Appl., 18:1 (1973), 1–26

Citation in format AMSBIB
\Bibitem{Ven73} \by A.~D.~Wentzell \paper Formulas for eigen-functions and eigen-measures associated with Markov process \jour Teor. Veroyatnost. i Primenen. \yr 1973 \vol 18 \issue 1 \pages 3--28 \mathnet{http://mi.mathnet.ru/tvp2677} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=321200} \zmath{https://zbmath.org/?q=an:0287.60067} \transl \jour Theory Probab. Appl. \yr 1973 \vol 18 \issue 1 \pages 1--26 \crossref{https://doi.org/10.1137/1118001} 

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• http://mi.mathnet.ru/eng/tvp/v18/i1/p3

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. M. I. Freidlin, “The averaging principle and theorems on large deviations”, Russian Math. Surveys, 33:5 (1978), 107–160
2. 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
3. X. Decombes, E. A. Zhizhina, “Application of Gibbs Random Fields Methods to Image Denoising Problems”, Problems Inform. Transmission, 40:3 (2004), 279–295