An averaging principle for dynamical systems in Hilbert space with Markov random perturbations
It is studied the asymptotic behavior of solutions of differential equations \({dx\over dt}= A(y(t/\varepsilon)) x\), \(x(0)= x_0\), where \(A(y)\), \(y\in Y\), is a family of operators generating semigroups of bounded linear operators in a Hilbert space \(H\), and \(y(t)\) is an ergodic jump Markov process in \(Y\). It is shown the convergence of the process \(x_\varepsilon(t)\), as \(\varepsilon\to 0\), to the nonrandom function \(x(t)\) fulfilling an averaged differential equation. There are proved two theorems, and an application to randomly perturbed partial differential equations with nonrandom initial and boundary conditions is included. It is recommended to those interested in asymptotic behavior of solution and of deviation for perturbed infinite-dimensional systems. The exposition is clear and selfcontained.
averaging principle for Markov perturbations
scientific article; zbMATH DE number 894789
scientific article