We consider random dynamical systems with randomly chosen jumps on Polish spaces. They generalize Markov processes corresponding to iterated function systems, Poisson driven stochastic differential equations, and irreducible Markov systems. We formulate criteria for the existence of an invariant measure and asymptotic stability for these systems. Estimates of the lower pointwise and concentration dimension of invariant measures are also given.

We give sufficient conditions for asymptotic stability of a Markov operator governing the evolution of measures due to the action of randomly chosen dynamical systems. We show that the existence of an invariant measure for the transition operator implies the existence of an invariant measure for the semigroup generated by the system.

We give sufficient conditions for the strong asymptotic stability of the distributions of dynamical systems with multiplicative perturbations. We apply our results to iterated function systems.

We consider the stochastic differential equation
(1) $du\left(t\right)=a(u\left(t\right),\xi \left(t\right))dt+{\int}_{\Theta}\sigma {(u\left(t\right),\theta )}_{p}(dt,d\theta )$ for t ≥ 0
with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${{P}^{t}}_{t\ge 0}$ corresponding to (1). We show that the existence of an invariant measure for a Markov operator P corresponding to the change of measures from jump to jump implies the existence of an invariant measure for the semigroup ${{P}^{t}}_{t\ge 0}$ describing the evolution of measures along trajectories and vice versa.

We give sufficient conditions for the existence of a matrix of probabilities ${\left[{p}_{ik}\right]}_{i,k=1}^{N}$ such that a system of randomly chosen transformations ${\Pi}_{k}$, k = 1,...,N, with probabilities ${p}_{ik}$ is asymptotically stable.

We consider the stochastic differential equation $$dX(t)=a(X(t);\xi (t))dt+{\int}_{\mathrm{\Theta}}b(X(t);\theta ){\mathcal{N}}_{p}(dt;d\theta )$$ for $t\ge 0$ with the initial condition $X(0)={x}_{0}$. We give sufficient conditions for the asymptotic stability of the semigroup ${\{{P}^{t}\}}_{t\ge 0}$ generated by the stochastic differential equation (1).

Contractive Markov systems on Polish spaces which arise from graph directed constructions of iterated function systems with place dependent probabilities are considered. It is shown that their stability may be studied using the concentrating methods developed by the second author [Dissert. Math. 415 (2003)]. In this way Werner's results obtained in a locally compact case [J. London Math. Soc. 71 (2005)] are extended to a noncompact setting.

Download Results (CSV)