Stochastic Patterns: From Translation to Rotation

In this talk, we start with a short summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs) commonly occuring in mathematical biology. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave
problems. Then we are going to explain a recent result on stochastic pattern formation going beyond travelling waves: stochastic rotating waves generated by SPDEs. We establish two different approaches for stochastic rotating waves, the variational phase and the approximated variational phase, which both help us to compute a stochastic ordinary differential equation (SODE), which describes 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.