Theory of Probability and Mathematical Statistics

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On stochastic stability of Markov evolution associated with impulse Markov dynamical systems


Authors: V. Korolyuk and Je. Carkovs
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 65-69
MSC (2000): Primary 37H10, 34D20
Published electronically: January 24, 2008
MathSciNet review: 2321181
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper deals with the family of Cauchy matrices of a linear differential equation dependent on a step Markov process and an impulse type dynamical system rapidly switched by the above process. Applying the stochastic and deterministic averaging procedures according to the invariant measures of the Markov process one achieves a simpler linear differential equation dependent on simpler dynamical systems such as an ordinary differential equation, a differential equation with the right hand side switched by a merger Markov process or a stochastic Itô differential equation. It is proved that under some hypotheses one may successfully apply these resulting evolution families not only to analyzing the initial family on an arbitrary finite time interval but also to describing a time asymptotic of this family.


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Additional Information

V. Korolyuk
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv 4, Ukraine

Je. Carkovs
Affiliation: Department of Probability Theory and Mathematical Statistics, Riga Technical University, Meza Street, 1/4, Riga, Latvia
Email: carkovs@livas.lv

DOI: http://dx.doi.org/10.1090/S0094-9000-08-00714-X
Received by editor(s): November 3, 2005
Published electronically: January 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society