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Theory of Probability and Mathematical Statistics

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The stability of almost homogeneous in time Markov semigroups of operators

Author: M. V. Kartashov
Translated by: Oleg Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 71 (2004).
Journal: Theor. Probability and Math. Statist. 71 (2005), 119-128
MSC (2000): Primary 60J45; Secondary 60A05
Published electronically: January 4, 2006
MathSciNet review: 2144325
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Abstract: A homogeneous in time semigroup of Markov operators defined by its infinitesimal operator with a dense domain is considered. The operator is perturbed by another bounded operator that depends on time, and this results in a nonhomogeneous semigroup. Under certain assumptions, we prove that the perturbed semigroup is a unique solution of a weak integral equation determined by the initial semigroup and an operator perturbation function; this equation is an integral analog of the perturbed Kolmogorov equation. We find explicit estimates for the stability of the perturbed semigroup in the case where the perturbation operator is uniformly small.

References [Enhancements On Off] (What's this?)

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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Keywords: Semigroup of operators, infinitesimal operator, uniform convergence
Received by editor(s): November 4, 2003
Published electronically: January 4, 2006
Article copyright: © Copyright 2006 American Mathematical Society