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Random dynamical systems with jumps. (English) Zbl 1091.47012

The author studies the asymptotic distribution of infinite-dimensional systems with randomly chosen jumps on a separable Banach space \(X\). These systems generalize some widely studied systems such as learning systems, dynamical systems generated by Poisson driven stochastic differential equations, iterated systems with an infinite family of transformations, and random evolutions. The author proves that, under certain conditions, an invariant distribution \(\mu\) exists which is the weak limit of finite-dimensional distributions. The proof is based on Szarek’s theory of concentrating Markov operators, which provides conditions ensuring the existence of an invariant measure for nonexpansive Markov operators.

MSC:

47A35 Ergodic theory of linear operators
37L55 Infinite-dimensional random dynamical systems; stochastic equations
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