On reducibility of linear markov switched difference equations

The paper deals with the system of linear difference equations in ℝd with the right part switched by homogeneous ergodic Markov chain {yt,t∈ℤ} on the compact phase space 𝕐. We prove that the shift operator family for the conditional first moments of the solutions E{xt+s/ys=y}:=mt(y) possess a semigroup property and derive the infinitesimal generator A for this semigroup. This approach permits to propose convenient to application algorithm for asymptotic analysis of the equations with near to constant coefficients. We have proved that for sufficiently small Markov type perturbations there exists such a matrix Λ that the first moments of the solutions have an asymptotic close to Λtx.