On stochastic stability of Markov dynamical systems

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.