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

TL;DR

This paper develops a framework to analyze the effects of stochastic perturbations on wave patterns in SPDEs and integral equations by reducing the dynamics to a finite-dimensional SDE on an invariant manifold using the isochronal phase.

Contribution

It introduces a novel method to approximate stochastic dynamics of patterns via the isochronal phase, extending finite-dimensional oscillator concepts to infinite-dimensional stochastic systems.

Findings

Probability measure accurately describes pattern wandering over the manifold.

The measure is valid on time-scales greater than O(σ^{-2}) and less than exponential in σ^{-2}.

Expected velocity of stochastic deviation from deterministic motion is determined.

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.