Stability of the global attractor under Markov-Wasserstein noise

TL;DR

This paper introduces a weak Ważewski principle for dynamical systems on metric spaces, demonstrating that global attractors as invariant measures can be continued under small Markov-type Wasserstein noise.

Contribution

It develops a weak Ważewski principle for metric space dynamical systems and establishes conditions for the continuation of global attractors under Markov-Wasserstein noise.

Findings

Global attractors can be continued under small Wasserstein perturbations.

Proper metric spaces have weakly proper Wasserstein spaces.

Small bounded and Gaussian noises satisfy the conditions for attractor continuation.

Abstract

We develop a "weak Wa\.zewski 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.