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Asymptotic stability of a system of randomly connected transformations on Polish spaces

100%

Horbacz K.

nr 3

197-211

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

Invariant measures related with randomly connected Poisson driven differential equations

100%

Horbacz K.

2002

tom 79

nr 1

31-44

We consider the stochastic differential equation (1) $du(t) = a(u(t),ξ(t))dt + ∫_{Θ} σ(u(t),θ) 𝓝_p(dt,dθ)$ for t ≥ 0 with the initial condition u(0) = x₀. We give sufficient conditions for the existence of an invariant measure for the semigroup ${P^t}_{t≥0}$ 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 ${P^t}_{t≥0}$ describing the evolution of measures along trajectories and vice versa.

Irreducible Markov systems on Polish spaces

63%

Horbacz K. , Szarek T.

nr 3

285-295

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.

Asymptotic stability of dynamical systems with multiplicative perturbations

50%

Horbacz K.

nr 2

209-218