The stability of almost homogeneous in time Markov semigroups of operators
Author:
M. V. Kartashov
Translated by:
Oleg Klesov
Journal:
Theor. Probability and Math. Statist. 71 (2005), 119-128
MSC (2000):
Primary 60J45; Secondary 60A05
DOI:
https://doi.org/10.1090/S0094-9000-06-00652-1
Published electronically:
January 4, 2006
MathSciNet review:
2144325
Full-text PDF Free Access
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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
- N. V. Kartashov, Strong stable Markov chains, VSP, Utrecht; TBiMC Scientific Publishers, Kiev, 1996. MR 1451375
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- 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
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- Onésimo Hernández-Lerma and Jean Bernard Lasserre, Markov chains and invariant probabilities, Progress in Mathematics, vol. 211, Birkhäuser Verlag, Basel, 2003. MR 1974383
- 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
- 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
References
- N. V. Kartashov, Strong Stable Markov Chains, VSP/TBiMC, Utrecht/Kiev, 1996. MR 1451375 (99e:60150)
- T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1965. MR 0203473 (34:3324)
- D. Revuz, Markov Chains, North-Holland Elsevier, Amsterdam, 1975. MR 0415773 (54:3852)
- E. B. Dynkin, Markov Processes, Fizmatlit, Moscow, 1962; English transl., Springer-Verlag, Berlin–Göttingen–Heidelberg, 1965.
- O. Hernandez-Lerma and J. B. Lasser, Markov Chains and Invariant Probabilities, Birkhäuser, Basel–Boston–Berlin, 2003. MR 1974383 (2004b:60002)
- M. V. Kartashov, Uniformly ergodic jump Markov processes with bounded intensities, Teor. Imovir. Matem. Statist. 52 (1995), 86–98; English transl. in Theory Probab. Math. Statist. 52 (1996), 91–103. MR 1445541 (97m:60104)
- M. V. Kartashov, Computation and estimation of the exponential ergodicity exponent for general Markov processes and chains with recurrent kernels, Teor. Imovir. Matem. Statist. 54 (1996), 47–57; English transl. in Theory Probab. Math. Statist. 54 (1997), 49–60. MR 1644643 (99e:60199)
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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
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
