Existence, Regularity, and a Strong It\^o Formula for the Isochronal Phase of SPDE

TL;DR

This paper establishes the existence and regularity of the isochron map for stable invariant manifolds in evolution equations, enabling a strong Itô formula for the isochronal phase in stochastic PDEs with applications to reaction-diffusion and neural field equations.

Contribution

It introduces new regularity results for the isochron map in a broad class of evolution equations, facilitating advanced stochastic calculus for pattern phases in SPDEs.

Findings

Proves existence and regularity of the isochron map for stable invariant manifolds.

Derives a strong Itô formula for the isochronal phase in SPDEs.

Applies results to reaction-diffusion and neural field equations.

Abstract

We prove the existence and regularity of the isochron map for stable invariant manifolds of a large class of evolution equations. Our results apply in particular to the isochron map of reaction-diffusion equations and neural field equations. Using the regularity properties proven here, we are able to obtain a strong It\^o formula for the isochronal phase of stochastically perturbed travelling waves, spiral waves, and other patterns appearing in SPDEs driven by white noise, even for SPDEs that only admit mild solutions.