Non-autonomous stochastic lattice systems with Markovian switching

Abstract

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic lattice systems with Markovian switching. We first show existence of an evolution system of measures of the stochastic system. We then study the pullback (or forward) asymptotic stability in distribution of the evolution system of measures. We finally prove that any limit point of a tight sequence of an evolution system of measures of the stochastic lattice systems must be an evolution system of measures of the corresponding limiting system as the intensity of noise converges zero.In particular, when the coefficients are periodic with respect to time, we show every limit point of a sequence of periodic measures of the stochastic system must be a periodic measure of the limiting system as the noise intensity goes to zero.

Non-autonomous hybrid stochastic systems with delays

“…(4.3) has a unique ρ-periodic evolution system of measures {µ t } t∈R , which is asymptotic stability in distribution, i.e., for any (ξ, j) ∈ H Proof. By Lemma 4.3 and 4.4, we get (4.5) from Theorem 2.10 in[6] immediately.Remark 4.8. In[9], the authors also investigated the asymptotic stability in distribution of the solutions of Eq.…”

“…A controlled system is obtained: Moreover, if the intervals are not equal, the controlled system (4.3) is not periodic. By Theorem 2.10 in [6], we can get Eq. ( 4.3) has a unique evolution system of measures {µ t } t∈R , which is asymptotic stability in distribution.…”

“…The asymptotic stability in distribution of invariant measures was also obtained in [25,4,20] for the hybrid stochastic differential equations with constant delays. In [6], the authors obtained the sufficient conditions of existence, stability and convergence of evolution system of measures of non-autonomous hybrid stochastic evolution systems.…”