Conditioned stochastic stability and transient dynamics

Date: Thursday 28 March 2024

Abstract

In dynamical systems theory, the concept of attractors plays a central role,  captivating long-term behaviour. In the presence of (too) many ergodic invariant measures on such attractors, certain stochastically stable ergodic invariant measures have been shown to represent best the statistics of typical observations.

In practice, many important dynamical phenomena of interest are not characterised by dynamics on attractors, but instead is inherently transient.

There is little to no rigorous mathematical theory for transient dynamics. A important special case of transient dynamics is that of orbits with initial conditions close to repellers. Like attractors, repellers also may have (too) many ergodic invariant measures. However, when addressing the question on how to select relevant measures for the statistics of typical observations of transients close to repellers, the traditional notion of stochastic stability cannot be employed, as it inherently does not apply to repellers.

In this talk, we propose the novel notion of conditioned stochastic stability for invariant measures on repellers. This notion refers to a Markov process that is conditioned on survival close to the repeller. We show that conditioned stochastically stable measures capture typical statistical observations of long transients near repellers. For uniformly expanding repellers it is shown that conditioned stochastically stable measures are equal to specific well-known equilibrium measures.

This is joint work with Bernat Bassols-Cornudellla and Matheus Manzatto de Castro.