Stochastic rotating waves

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 recently received considerable attention are traveling wave patterns occurring in stochastic partial differential equations (SPDEs). Here, one is interested in how deterministic traveling waves behave under stochastic perturbations. In this paper, we start the mathematical study of a related class of problems: stochastic rotating waves generated by SPDEs. We combine deterministic partial differential equation (PDE) dynamics 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.