Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

2018 ◽  
Vol 36 (3) ◽  
pp. 521-533 ◽  
Author(s):  
Paweł Płonka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Invariant Measures ◽  
Random Dynamical Systems ◽  
Polish Spaces

scholarly journals Random Dynamical Systems with Jumps and with a Function Type Intensity

2016 ◽  
Vol 30 (1) ◽  
pp. 63-87
Author(s):  
Joanna Kubieniec
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Invariant Measure ◽  
Lipschitz Function ◽  
Random Dynamical Systems ◽  
Function Type ◽  
Polish Spaces

AbstractIn paper [4] there are considered random dynamical systems with randomly chosen jumps acting on Polish spaces. The intensity of this process is a constant λ. In this paper we formulate criteria for the existence of an invariant measure and asymptotic stability for these systems in the case when λ is not constant but a Lipschitz function.

scholarly journals Strong Unique Ergodicity of Random Dynamical Systems on Polish Spaces

2016 ◽  
Vol 30 (1) ◽  
pp. 129-142
Author(s):  
Paweł Płonka
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Stability ◽  
Unique Ergodicity ◽  
Random Dynamical Systems ◽  
Random Measures ◽  
Polish Spaces

AbstractIn this paper we want to show the existence of a form of asymptotic stability of random dynamical systems in the sense of L. Arnold using arguments analogous to those presented by T. Szarek in [6], that is showing it using conditions generalizing the notion of tightness of measures. In order to do that we use tightness theory for random measures as developed by H. Crauel in [2].

scholarly journals On the Computation of Invariant Measures in Random Dynamical Systems

2003 ◽  
Vol 03 (02) ◽  
pp. 247-265 ◽  
Author(s):  
Peter Imkeller ◽  
Peter Kloeden
Keyword(s):  
Dynamical Systems ◽  
Markov Chains ◽  
State Space ◽  
Difference Equations ◽  
Invariant Measures ◽  
Random Dynamical Systems ◽  
Continuum State ◽  
Transition Mechanism ◽  
Definition Of ◽  
Stationary Random Processes

Invariant measures of dynamical systems generated e.g. by difference equations can be computed by discretizing the originally continuum state space, and replacing the action of the generator by the transition mechanism of a Markov chain. In fact they are approximated by stationary vectors of these Markov chains. Here we extend this well-known approximation result and the underlying algorithm to the setting of random dynamical systems, i.e. dynamical systems on the skew product of a probability space carrying the underlying stationary stochasticity and the state space, a particular non-autonomous framework. The systems are generated by difference equations driven by stationary random processes modelled on a metric dynamical system. The approximation algorithm involves spatial discretizations and the definition of appropriate random Markov chains with stationary vectors converging to the random invariant measure of the system.

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