Stability of the global attractor under Markov-Wasserstein noise

2011/03/17 by Kell, Martin
#34D23 #37B35 #60B05 #60B10 #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1103.3401

Abstract

We develop a "weak Ważewski principle" for discrete and continuous time dynamical systems on metric spaces having a weaker topology to show that attractors can be continued in a weak sense. After showing that the Wasserstein space of a proper metric space is weakly proper we give a sufficient and necessary condition such that a continuous map (or semiflow) induces a continuous map (or semiflow) on the Wasserstein space. In particular, if these conditions hold then the global attractor, viewed as invariant measures, can be continued under Markov-type random perturbations which are sufficiently small w.r.t. the Wasserstein distance, e.g. any small bounded Markov-type noise and Gaussian noise with small variance will satisfy the assumption.

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