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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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. IntroductionThe dynamical system we shall deal with in this paper has the form of a quasilinear n-dimensional differential equationwhere y(t) is a homogeneous ergodic Markov process on metric compact phase space Y with weak infinitesimal operator Q [2] and f (x, y) is a continuous function with uniformly bounded x derivative satisfying the condition f (0, y) ≡ 0. By the definition of Markov process, for any s ≥ 0 and y ∈ Y there exists a unique stochastic process y (t, s, y) satisfying the initial condition y(s) = y. Substituting any realization of this process in place of y(t) into (1), we can analyze the resulting equation as an ordinary differential equation in R n with dependent on t (nonstationary) right hand side. It is not difficult to prove that for any above-mentioned realization and any s ∈ R and x ∈ R n there exists a unique solution x(t, s, x, y) of the Cauchy problem x(s) = 0 for (1). The trivial solution x(t) ≡ 0 is the equilibrium point of (1), and we will discuss the behavior of other solutions starting in a sufficiently small vicinity of this equilibrium. We will say that equilibrium is(1) locally almost surely stable if for any s ∈ R, η > 0, and β > 0 there exists δ > 0 such that the inequality sup y∈Y P(sup t≥s |x(t, s, x, y)| > η) < β follows from the condition x ∈ B δ (0), where B δ (0) := {x ∈ R n : |x| < δ};2000 Mathematics Subject Classification. Primary 37H10, 34D20. Key words and phrases. Markov dynamical systems, mean square stability, Lyapunov methods, limit theorems for random dynamical systems, stochastic differential equations.