The Isochronal Phase of Stochastic PDE and Integral Equations: Metastability and Other Properties

Abstract

We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighbourbhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T.~Winfree and J.~Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than $O(\sigma^{-2})$, but less than $O(\exp(C\sigma^{-2}))$, where $\sigma\ll1$ is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold.