On stochastic stability of Markov evolution associated with impulse Markov dynamical systems. (English) Zbl 1164.37334

Korolyuk, V.; Carkovs, Je.

Teor. Jmovirn. Mat. Stat. 75, 57-60 (2006).

The authors study the impulse Markov dynamical system with rapid switching \(\{x^{\varepsilon}(t), t\geq0\}\) satisfying a differential equation for \(t\notin S\): \({dx^{\varepsilon}\over dt}=f(x^{\varepsilon}(t),y^{\varepsilon}(t),{\varepsilon})\), a jump condition for \(t\in S\): \(x^{\varepsilon}=x^{\varepsilon}(t-0)+ {\varepsilon}g(x^{\varepsilon}(t-0),y^{\varepsilon}(t-0),{\varepsilon})\), and Markov evolution family given as two parametric Cauchy matrix family \(\{X^{\varepsilon}(t,s), t\geq s\geq0\}\) satisfying a linear differential equation in \(\mathbb{R}^{n}\): \({d\over dt}X^{\varepsilon}(t,s)= A(x^{\varepsilon}(t),y^{\varepsilon}(t),{\varepsilon})X^{\varepsilon}(t,s)\), where \(\{y^{\varepsilon}(t),t\geq0\}\) is a right continuous step Markov process with switching times \(S\) given on discrete metric space \(Y\) by a weak infinitesimal operator \(Q^{\varepsilon}v(y):={1\over{\varepsilon}}Q_1v(y)+Q_2v(y)\), where \({\varepsilon}\) is a small positive parameter; \(Q_{j}v(y)= a(y)\sum_{z\in Y}[v(z)-v(y)]p_{j}(y,z)\), \(j=1,2\). Using the stochastic and deterministic averaging procedures according to invariant measures of Markov process, the authors obtain more simple linear differential equation dependent on more simple dynamical systems such as an ordinary differential equation, a differential equation with a right part switched by a merger Markov process or a stochastic Ito differential equation.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

37H10	Generation, random and stochastic difference and differential equations
34D20	Stability of solutions to ordinary differential equations

Keywords:

stochastic stability; Markov evolution; impulse Markov dynamical systems; averaging procedure; Ito differential equation