On stochastic stability of Markov dynamical systems. (English) Zbl 1164.37333

Carkovs, Je.; Vernigora, I.; Yasinski, V.

Teor. Jmovirn. Mat. Stat. 75, 155-164 (2006).

The authors deal with the dynamical system in the form of quasilinear differential equation \({dx\over dt}=A(y(t))x+f(x,y(t))\), where \(A(y)\) is continuous bounded marix-valued function and \(y(t)\) is stochastically continuous Feller Markov process; \(f(x,y)\) is continuous function with uniformly bounded \(x\)-derivative, satisfying condition \(f(0,y)=0\). The authors prove that for linear Markov dynamical systems equilibrium asymptotical stability with probability one is equivalent to exponential decreasing of \(p\)-moment with sufficiently small \(p\). Validity of equilibrium stability analysis of Markov dynamical systems applying a linear approximation of vector field is discussed. Semigroup approach for mean square stability analysis of linear Markov dynamical systems is studied.

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

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

Keywords:

Markov dynamical systems; mean square stability; Lyapunov methods; stochastic differential equations