Dimensions of invariant measures for continuous random dynamical systems

2015 ◽  
Author(s):  
Tomasz Bielaczyc ◽  
Katarzyna Horbacz
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Random Dynamical Systems

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.

Atomic operators, random dynamical systems and invariant measures

2015 ◽  
Vol 26 (4) ◽  
pp. 607-642
Author(s):  
A. Ponosov ◽  
E. Stepanov
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Random Dynamical Systems
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