Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence, Regularity, and a Strong Itô Formula for the Isochronal Phase of SPDE

Zachary P. Adams

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 Itô 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.

Existence, Regularity, and a Strong Itô Formula for the Isochronal Phase of SPDE

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 Itô 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.

Paper Structure

This paper contains 3 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Regularity of the Isochron Map & a Strong Itô Formula

Key Result

Theorem \oldthetheorem

Under Assumptions assn:PDE & assn:Manifold, there exists a unique function $\pi:B(\Gamma)\rightarrow\mathcal{S}\subset\mathbb{R}^m$ satisfying eq:pidefinition. This function is twice continuously Fréchet differentiable at each $x_0\in B(\Gamma)\subset E$ in the topology of $E$.

Theorems & Definitions (13)

  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 3 more