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Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with Hölder continuous paths

Amjad Saef, Wilhelm Stannat

TL;DR

The paper analyzes the stability of travelling wave solutions to reaction-diffusion equations under infinite-dimensional additive noise with Hölder paths, including self-similar drivers like fractional Brownian motion. It develops a pathwise, phase-adaptive framework that decomposes perturbations into a first-order stochastic component and a nonlinear residual, leveraging a dissipative linearisation and Young integration against sectorial operators. The authors derive short- and long-time error bounds for the distance to the travelling-wave orbit, revealing how stability depends on the noise’s Hurst index and Hölder regularity, and they extend the results to self-similar, possibly non-Gaussian drivers with explicit tail estimates. This framework provides quantitative, pathwise stability results that can be applied to a broad class of infinite-dimensional noise processes, with potential implications for modelling noise-driven front propagation in biological systems and other spatiotemporal phenomena.

Abstract

In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with Hölder continuous paths, covering in particular fractional Brownian motion with general Hurst index. We obtain long- and short time asymptotic error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts. These bounds, in terms of Hurst index and Hölder exponent, apply to a large class of infinite-dimensional self-similar drivers with Hölder continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, a smaller gap between Hurst index and Hölder exponent implies stability for larger noise amplitudes.

Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with Hölder continuous paths

TL;DR

The paper analyzes the stability of travelling wave solutions to reaction-diffusion equations under infinite-dimensional additive noise with Hölder paths, including self-similar drivers like fractional Brownian motion. It develops a pathwise, phase-adaptive framework that decomposes perturbations into a first-order stochastic component and a nonlinear residual, leveraging a dissipative linearisation and Young integration against sectorial operators. The authors derive short- and long-time error bounds for the distance to the travelling-wave orbit, revealing how stability depends on the noise’s Hurst index and Hölder regularity, and they extend the results to self-similar, possibly non-Gaussian drivers with explicit tail estimates. This framework provides quantitative, pathwise stability results that can be applied to a broad class of infinite-dimensional noise processes, with potential implications for modelling noise-driven front propagation in biological systems and other spatiotemporal phenomena.

Abstract

In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with Hölder continuous paths, covering in particular fractional Brownian motion with general Hurst index. We obtain long- and short time asymptotic error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts. These bounds, in terms of Hurst index and Hölder exponent, apply to a large class of infinite-dimensional self-similar drivers with Hölder continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, a smaller gap between Hurst index and Hölder exponent implies stability for larger noise amplitudes.
Paper Structure (16 sections, 23 theorems, 235 equations)

This paper contains 16 sections, 23 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setting
  4. Travelling wave solutions
  5. Existence of Solutions
  6. Stochastic Nagumo equation driven by Hölder noise
  7. Pathwise stability for small perturbations
  8. Deterministic phase adaptation
  9. First order approximation of residual
  10. Stability for small noise amplitudes
  11. Growth estimates on the first order approximation
  12. Application to self-similar stochastic processes with Hölder continuous paths
  13. Stability of travelling waves for self-similar noise with Hölder continuous paths
  14. Identities for convolutions against Hölder paths
  15. Tail estimates on Hölder norms of self-similar processes
  16. ...and 1 more sections

Key Result

Proposition 3.3

Suppose that $L \colon D(L) \subset \mathcal{B} \rightarrow \mathcal{B}$ is an injective sectorial operator. Let $(\mathcal{B}_\gamma)_{\gamma \in \mathbb R}$ be the induced scale of Banach spaces (cf. Assumption AnaSem) and $(P(t))_{t \geq 0}$ be the analytic semigroup generated by $L$. Let $X \in Further, for all $T > 0$, there exist constants $C_1$ and $C_2$ only dependent on $\eta, \gamma, \d

Theorems & Definitions (67)

  • Remark 2.1
  • Remark 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7
  • Example 2.8
  • Remark 2.9
  • Remark 3.1
  • ...and 57 more