Stochastic Rotating Waves

TL;DR

This paper introduces mathematical methods to analyze how stochastic perturbations affect rotating wave patterns in SPDEs, combining deterministic PDE techniques with stochastic analysis.

Contribution

It develops two approaches for defining stochastic phase variables along rotating waves, enabling the study of noise effects on these patterns.

Findings

Established variational and approximated variational phase methods.

Proved stability of stochastic rotating waves under small noise.

Showed stochastic waves remain close to deterministic ones over certain time scales.

Abstract

Stochastic dynamics has emerged as one of the key themes ranging from models in applications to theoretical foundations in mathematics. One class of stochastic dynamics problems that has received considerable attention recently are travelling wave patterns occurring in stochastic partial differential equations (SPDEs), i.e., how deterministic travelling waves behave under stochastic perturbations. In this paper, we start the mathematical study of related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic dynamics PDE techniques with methods from stochastic analysis. We establish two different approaches, the variational phase and the approximated variational phase, for defining stochastic phase variables along the rotating wave, which track the effect of noise on neutral spectral modes associated to the special Euclidean symmetry group of rotating waves. Furthermore, we prove transverse stability results for rotating waves showing that over certain time scales and for small noise, the stochastic rotating wave stays close to its deterministic counterpart.