Noncommutative orbital stability of stochastic patterns in Banach spaces

TL;DR

This paper establishes the orbital stability of stochastic pattern solutions in Banach spaces, accounting for noncommutative symmetries and introducing a novel phase tracking method that handles lower regularity noise.

Contribution

It introduces a new phase tracking approach for stability analysis of stochastic patterns in Banach spaces with noncommutative symmetries, extending previous methods.

Findings

Stability persists on exponential timescales relative to noise amplitude.

Noncommutativity influences pattern motion and stability.

The method applies to lower regularity noise without Hilbert space dependence.

Abstract

We consider stochastic perturbations of PDEs which have special pattern solutions, such as (nonlinear) travelling waves, solitons, and spiral waves. We show orbital stability of these patterns on a timescale which is exponential in the inverse square of the noise amplitude. We systematically treat equations with noncommutative symmetry groups, and show how the noncommutativity affects the motion of the pattern. This is done by introducing a new method to track the (generalized) phase of the pattern. Furthermore, we demonstrate how orbital stability arises from a mismatch of symmetry between the pattern and the equation. Our phase tracking method does not rely on a Hilbert space structure. This allows us to show stability in general Banach spaces, and to treat noise with lower regularity than before.