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Invariant measures for random dynamical systems

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.

Randomly connected dynamical systems - asymptotic stability

Katarzyna Horbacz — 1998

Annales Polonici Mathematici

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.

Invariant measures related with randomly connected Poisson driven differential equations

Katarzyna Horbacz — 2002

Annales Polonici Mathematici

We consider the stochastic differential equation (1) d u ( t ) = a ( u ( t ) , ξ ( t ) ) d t + Θ σ ( u ( t ) , θ ) p ( d t , d θ ) 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.

Asymptotic stability of a semigroup generated by randomly connected Poisson driven differential equations

Katarzyna Horbacz — 2006

Bollettino dell'Unione Matematica Italiana

We consider the stochastic differential equation d X ( t ) = a ( X ( t ) ; ξ ( t ) ) d t + Θ b ( X ( t ) ; θ ) 𝒩 p ( d t ; d θ ) for t 0 with the initial condition X ( 0 ) = x 0 . We give sufficient conditions for the asymptotic stability of the semigroup { P t } t 0 generated by the stochastic differential equation (1).

Irreducible Markov systems on Polish spaces

Katarzyna HorbaczTomasz Szarek — 2006

Studia Mathematica

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.

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