Piecewise Convex Deterministic Dynamical Systems and Weakly Convex Random Dynamical Systems and Their Invariant Measures

We consider random dynamical systems with randomly chosen jumps on infinite-dimensional spaces. The choice of deterministic dynamical systems and jumps depends on a position. The system generalizes dynamical systems corresponding to learning systems, Poisson driven stochastic differential equations, iterated function system with infinite family of transformations and random evolutions. We will show that distributions which describe the dynamics of this system converge to an invariant distribution. We use recent results concerning asymptotic stability of Markov operators on infinite-dimensional spaces obtained by T. Szarek.

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The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy formula

Discrete and Continuous Dynamical Systems ◽

10.3934/dcds.2013.33.4123 ◽

2013 ◽

Vol 33 (9) ◽

pp. 4123-4155 ◽

Cited By ~ 6

Author(s):

Zhiming Li ◽

◽

Lin Shu ◽

Keyword(s):

Hilbert Space ◽

Dynamical Systems ◽

Invariant Measures ◽

Random Dynamical Systems ◽

Metric Entropy ◽

Entropy Formula ◽

Pesin’S Entropy Formula ◽

Space Characterization

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On the Computation of Invariant Measures in Random Dynamical Systems

Stochastics and Dynamics ◽

10.1142/s0219493703000711 ◽

2003 ◽

Vol 03 (02) ◽

pp. 247-265 ◽

Cited By ~ 4

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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