[Paper Review] Metastability of Waves and Patterns Subject to Spatially-Extended Noise

James MacLaurin|arXiv (Cornell University)|Jun 22, 2020

stochastic dynamics and bifurcation84 references4 citations

TL;DR

This paper develops a rigorous stochastic phase reduction framework for traveling waves, patterns, and oscillations under spatially-extended noise in reaction-diffusion and neural field systems. It introduces two phase dynamics—variational and isochronal—and proves that exit from the attractor manifold is exponentially unlikely over exponentially long times, while noise-induced drift in phase space leads to persistent, correlated oscillations in periodic manifolds.

ABSTRACT

In this paper we present a general framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns and oscillations that are invariant under the action of a finite-dimensional set of continuous isometries (such as translation or rotation). This formalism can accommodate patterns, waves and oscillations in reaction-diffusion systems and neural field equations. To do this, we define the phase by precisely projecting the infinite-dimensional system onto the manifold of isometries. Two differing types of stochastic phase dynamics are defined: (i) a variational phase, obtained by insisting that the difference between the projection and the original solution is orthogonal to the non-decaying eigenmodes, and (ii) an isochronal phase, defined as the limiting point on manifold obtained by taking $t o\infty$ in the absence of noise. We outline precise stochastic differential equations for both types of phase. The variational phase SDE is then used to show that the probability of the system leaving the attracting basin of the manifold after an exponentially long period of time (in $ε^{-2}$, the magnitude of the noise) is exponentially unlikely. In the case that the manifold is periodic (such as for spiral waves, spatially-distributed oscillations, or neural-field phenomena on a compact domain), the isochronal phase SDE is used to determine asymptotic limits for the average occupation times of the phase as it wanders in the basin of attraction of the manifold over very long times. In particular, we find that frequently the correlation structure of the noise biases the wandering in a particular direction, such that the noise induces a slow oscillation that would not be present in the absence of noise.

Motivation & Objective

• To develop a general framework for analyzing the impact of spatio-temporal noise on invariant patterns and waves in infinite-dimensional systems.
• To extend prior work on one-dimensional phase manifolds to multidimensional manifolds of solutions invariant under continuous isometries (e.g., translation, rotation).
• To derive stochastic differential equations (SDEs) for two distinct phase dynamics: variational and isochronal.
• To establish precise large-deviation estimates for the probability of exit from the attractor manifold over exponentially long times.
• To analyze long-time occupation times of the stochastic phase on the manifold, revealing noise-induced directional drift in periodic settings.

Proposed method

• Define the phase via orthogonal projection onto the manifold of isometries, using a variational principle that enforces orthogonality to non-decaying eigenmodes.
• Introduce an isochronal phase as the limiting point of the system's trajectory on the manifold in the absence of noise as t→∞.
• Derive two distinct SDEs: one for the variational phase using Girsanov's theorem and martingale techniques, and one for the isochronal phase using ergodic properties of the noise-averaged dynamics.
• Use a spectral gap assumption (excluding neutral modes tangent to the manifold) to relax prior self-adjointness and immediate contractivity requirements.
• Apply large deviations theory and exponential moment bounds to show that the probability of exiting the attractor manifold over time T ≈ exp(Cε⁻²) decays exponentially in ε⁻².
• Leverage techniques from [70] to improve error bounds on projection accuracy, enabling sharper exit-time estimates for systems with cylindrical noise and multiplicative noise.

Experimental results

Research questions

• RQ1 How can one rigorously project the dynamics of a stochastic infinite-dimensional system onto a manifold of solutions invariant under continuous isometries, such as translation or rotation?
• RQ2 What are the stochastic differential equations governing the phase dynamics of waves and patterns under spatially-extended noise, and how do they differ between variational and isochronal formulations?
• RQ3 What is the probability that the system escapes the basin of attraction of the manifold over an exponentially long time horizon T ≈ exp(Cε⁻²)?
• RQ4 How does the correlation structure of spatially-extended noise influence the long-time behavior of the phase, particularly in periodic manifolds?
• RQ5 Can noise induce a slow, persistent oscillation in the phase dynamics even when the deterministic system has no such oscillation?

Key findings

• The probability of the system leaving the attractor manifold over a time T ≈ exp(Cε⁻²) is exponentially unlikely, bounded by exp(−Const × ε⁻²), confirming metastability in the large-deviations regime.
• The variational phase SDE provides a precise description of phase dynamics and enables rigorous exit-time estimates using Girsanov’s theorem and exponential moment bounds.
• For periodic manifolds, the isochronal phase SDE reveals that noise can induce a directional drift in phase space, leading to persistent, correlated oscillations not present in the deterministic case.
• The long-time occupation time of the phase on the manifold converges to the invariant measure of the noise-averaged phase dynamics, with convergence rates controlled by the spectral properties of the linearized system.
• The error in phase projection is bounded uniformly over long times, with improved estimates compared to prior work, thanks to techniques from [70] applied to cylindrical noise and multiplicative noise settings.
• The analysis holds under a relaxed spectral gap condition, no longer requiring self-adjointness or immediate contractivity of the linearization, broadening applicability to real-world systems.