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Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems

Mark van den Bosch, Hermen Jan Hupkes

TL;DR

The paper addresses the persistence of planar travelling waves in multidimensional stochastic reaction-diffusion systems with multiplicative noise on long timescales, leveraging orbital stability in one dimension. It introduces a local phase-tracking approach in the transverse directions, showing that noise energy dissipates predominantly in the transverse sector and leaves localized phase shifts, avoiding the need for global phase tracking. The authors derive detailed moment bounds for the running suprema of perturbations, establish maximal-regularity bounds for stochastic convolutions with both transverse heat and full diffusion semigroups, and obtain dimension-dependent timescales: polynomial for $d\in\{2,3,4\}$ and exponential for $d\ge5$. The analysis rests on a variational framework, spectral-gap assumptions, and careful control of nonlinear terms, yielding explicit tail estimates and nonlinear stability results that quantify metastability of planar waves under stochastic forcing with spatially coloured noise.

Abstract

Planar travelling waves on $\mathbb R^d,$ with $ d\geq 2,$ are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally stable in dimension one ($d=1$). While a global phase tracking mechanism is required to determine the location of the stochastically perturbed wave in one dimension, or on a cylindrical domain, we show that the travelling wave on the full unbounded space can be controlled by keeping track of local deviations only. In particular, the energy infinitesimally added to or withdrawn from the system by noise dissipates almost fully into the transverse direction, leaving behind small localised phase shifts. The noise process considered is white in time and coloured in space, possibly weighted, and either translation invariant or trace class.

Local Phase Tracking and Metastability of Planar Waves in Stochastic Reaction-Diffusion Systems

TL;DR

The paper addresses the persistence of planar travelling waves in multidimensional stochastic reaction-diffusion systems with multiplicative noise on long timescales, leveraging orbital stability in one dimension. It introduces a local phase-tracking approach in the transverse directions, showing that noise energy dissipates predominantly in the transverse sector and leaves localized phase shifts, avoiding the need for global phase tracking. The authors derive detailed moment bounds for the running suprema of perturbations, establish maximal-regularity bounds for stochastic convolutions with both transverse heat and full diffusion semigroups, and obtain dimension-dependent timescales: polynomial for and exponential for . The analysis rests on a variational framework, spectral-gap assumptions, and careful control of nonlinear terms, yielding explicit tail estimates and nonlinear stability results that quantify metastability of planar waves under stochastic forcing with spatially coloured noise.

Abstract

Planar travelling waves on with are shown to persist in systems of reaction-diffusion equations with multiplicative noise on significantly long timescales with high probability, provided that the wave is orbitally stable in dimension one (). While a global phase tracking mechanism is required to determine the location of the stochastically perturbed wave in one dimension, or on a cylindrical domain, we show that the travelling wave on the full unbounded space can be controlled by keeping track of local deviations only. In particular, the energy infinitesimally added to or withdrawn from the system by noise dissipates almost fully into the transverse direction, leaving behind small localised phase shifts. The noise process considered is white in time and coloured in space, possibly weighted, and either translation invariant or trace class.

Paper Structure

This paper contains 39 sections, 71 theorems, 456 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Deterministic setting
  3. Evolution of the perturbation
  4. Numerical toy problem
  5. Algebraic decay and stochastic convolutions
  6. Moments of maximal processes
  7. Outlook
  8. Organisation
  9. Main results
  10. Deterministic assumptions
  11. Stochastic assumptions
  12. Characterisation of solutions
  13. Moment bounds and probability tail estimates
  14. Nonlinearities
  15. Preliminaries
  16. ...and 24 more sections

Key Result

Theorem 1.1

Suppose $k > d/2+1$. Under technical yet rather mild assumptions, there are (small) constants $\kappa_b> 0,$$\kappa_c> 0,$$\delta_\eta>0$, and $\delta_\sigma > 0$ so that for all $0<\eta<\delta_\eta$, $0 < \sigma < \delta_\sigma$, andWe view the parameter $r$ as a means to balance the length of the holds, where the timescales are given by whenever $u(x,y,0)$ is $\mathcal{O}(\eta)$-close to $\Phi

Figures (3)

  • Figure 1: Average over 200 realisations of the quantity $\Theta(t)$ defined in \ref{['eq:timescale:rel']}. Since the curves exhibit super-linear growth when viewed in this double logarithmic scale, these computations suggest that $\mathbb E[\Theta(T)]$ grows faster than any power of $T$.
  • Figure 2: Power law behaviour of $\tau_{\rm avg}(\eta)$ for $d=2$ in terms of $\sigma^{-1}$, obtained by averaging over 200 realisations of $\Theta(t)$ with: $\eta=1$ (diamond); $\eta=\exp(-2)\approx 0.1$ (triangle); and $\eta=\exp(-4)\approx 0.02$ (circle). (a) We start with $\sigma=0.075$ and decrease $\sigma$ until $\tau_{\rm avg}(\eta)$ is beyond the threshold 5000. A standard regression is performed on all collected data points for fixed $\eta$. (b) For each $\sigma$, we plot the negative exponent of the power law obtained from a standard regression on the subset of data points in Figure \ref{['fig:datafit']} that lie in in the horizontal range $[\sigma, 0.075]$. It would be beneficial to acquire more data with smaller values of $\sigma$ to decrease the spread in the exponents, but the required running times increase prohibitively. Nevertheless, these results provide clear evidence that $\hat{\vartheta}(t)$ experiences significant stochastic forcing well before the timescale $t\sim \sigma^{-2}.$
  • Figure 3: On the growth of stochastic convolutions and weighted integrals. (a) Average over 200 realisations of $Y^*(t)=\sup_{0\leq s\leq t} Y(s)^2$ and $Y^*_{\rm ou}(t)=\sup_{0\leq s\leq t} Y_{\rm ou}(s)^2$; see \ref{['eq:Y(t)']} and \ref{['eq:int:def:Y:ou']}. The purple markers represent the estimates \ref{['eq:int:stoch:conv:growth']} with associated constants $C_2=1.1$ and $C_3=1.2$. (b) Average over 200 realisations of $I(t)=\sup_{0\leq s\leq t}[\int_0^s (1+s-r)^{-\frac{d-1}{4}}|Y_{\rm ou}(r)|\,\mathrm dr]^2$. The green markers represent the $T$-dependence in \ref{['eq:int:crude:est:theta:mix']} with associated constants $K_{2,1}=0.175$ and $K_{4,1}=1.0$. The purple markers omit the logarithmic terms and feature the constants $K_{2,0}=0.75$ and $K_{4,0}=5.5.$ Notice that the latter do not appear to fully capture the growth of $I(t)$, showing that the logarithmic factors are indeed essential.

Theorems & Definitions (140)

  • Theorem 1.1: see §\ref{['sec:preliminaries']}
  • Proposition 2.1: Perturbation system; see §\ref{['sec:evol:pert']}
  • Proposition 2.2: Original system; see §\ref{['sec:evol:pert']}
  • Theorem 2.3: see §\ref{['sec:stability']}
  • Remark 2.4
  • Corollary 2.5
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Proposition 3.1
  • Proposition 3.2
  • ...and 130 more