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Theory of Probability and Mathematical Statistics

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On stochastic stability of Markov dynamical systems

Authors: Je. Carkovs, I. Vernigora and V. Yasinskii
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 179-188
MSC (2000): Primary 37H10, 34D20
Published electronically: January 25, 2008
MathSciNet review: 2321191
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Abstract: This paper aims at discussing methods and results of Lyapunov stability theory for dynamical systems with vector field subjected to permanent Markov type perturbations. The paper is organized as follows. Section 1 introduces the model of Markov dynamical system (MDS) and suggests different possible definitions of equilibrium stochastic stability, which are under discussion in the next sections. It is proven that for linear Markov dynamical systems equilibrium asymptotical stability with probability one is equivalent to the exponential decreasing of the $ p$-moment with sufficiently small $ p$. In Section 3 we will discuss validity of equilibrium stability analysis of Markov dynamical systems applying a linear approximation of a vector field. Section 4 is devoted to a semigroup approach for mean square stability analysis of linear Markov dynamical systems. It permits us to write the Lyapunov matrix in an explicit form and to reduce the equilibrium stability problem to real spectrum analysis of a specially constructed closed operator.

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Additional Information

Je. Carkovs
Affiliation: Department of Theory Probability and Mathematical Statistics, Riga Technical University, Meza Street, 1/4, Riga, LV-1048, Latvia

I. Vernigora
Affiliation: Department of Computer Sciences, Chernivtsi National University, Kotsyubyns’kiĭ Street, 2, Chernivtsi, 58000, Ukraine

V. Yasinskii
Affiliation: Department of Computer Sciences, Chernivtsi National University, Kotsyubyns’kiĭ Street, 2, Chernivtsi, 58000, Ukraine

Keywords: Markov dynamical systems, mean square stability, Lyapunov methods, limit theorems for random dynamical systems, stochastic differential equations
Received by editor(s): November 24, 2004
Published electronically: January 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society