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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**1.**N. V. Kartashov,*Strong stable Markov chains*, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR**1451375****2.**Tosio Kato,*Perturbation theory for linear operators*, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR**0203473****3.**D. Revuz,*Markov chains*, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 11. MR**0415773****4.**E. B. Dynkin,*Markov Processes*, Fizmatlit, Moscow, 1962; English transl., Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.**5.**Onésimo Hernández-Lerma and Jean Bernard Lasserre,*Markov chains and invariant probabilities*, Progress in Mathematics, vol. 211, Birkhäuser Verlag, Basel, 2003. MR**1974383****6.**M. V. Kartashov,*Uniformly ergodic jump Markov processes with bounded intensities*, Teor. Ĭmovīr. Mat. Stat.**52**(1995), 86–98 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**52**(1996), 91–103. MR**1445541****7.**M. V. Kartashov,*Computation and estimation of the exponential ergodicity exponent for general Markov processes and chains with recurrent kernels*, Teor. Ĭmovīr. Mat. Stat.**54**(1996), 47–57 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist.**54**(1997), 49–60. MR**1644643**

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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

Email:
winf@carrier.kiev.ua

DOI:
http://dx.doi.org/10.1090/S0094-9000-06-00652-1

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