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Fluctuations for fully pushed stochastic fronts

Alison Etheridge, Raphaël Forien, Thomas Hughes, Sarah Penington

Abstract

We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.

Fluctuations for fully pushed stochastic fronts

Abstract

We study the asymptotic behaviour, in the small noise limit, of stochastic travelling wave solutions to reaction-diffusion equations perturbed by Wright-Fisher noise. Such equations are predicted to display three distinct responses to noise in three parametric regimes: fully pushed, semi-pushed, and pulled. We prove, for the entire fully pushed regime, that solutions are asymptotically close to a stochastic shift of the deterministic travelling wave, and characterize the limiting shift process as a Brownian motion with drift. This gives the first full fluctuation theorem demonstrating fully pushed phenomenology for a non-linear stochastic reaction-diffusion equation and verifies a physical conjecture of Birzu, Hallatschek and Korolev [BHK18]. The proof uses an infinite-dimensional version of a method introduced by Katzenberger [Kat91], as pioneered by Funaki [Fun95]. This approach views the dynamics as a stochastic perturbation of a dynamical system (the PDE) with strong drift towards an invariant manifold, in our case the set of shifts of the travelling wave profile, and gives an expression for the stochastic motion "along" this manifold. Implementing this method in our setting requires many ingredients, including a close analysis of the dynamics of the corresponding PDE, integrability and regularity properties of solutions to the SPDE, and sharp control of the position of the right endpoint of the solution's support.

Paper Structure

This paper contains 41 sections, 110 theorems, 1430 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Summary of the main result
  4. Relation to existing literature
  5. Coordinates along the stable manifold
  6. Proof of the main result
  7. Stability of the invariant manifold
  8. The energy functional
  9. Stability of the manifold under the stochastic flow
  10. The right interface
  11. The walk process and proof of Proposition \ref{['prop_R_main']}
  12. Interface estimates
  13. Some moment bounds
  14. Analysis of $X^{N,S}_t$
  15. Proof of Proposition \ref{['prop_SN_inc']}
  16. ...and 26 more sections

Key Result

lemma 1

Suppose $u^N_0:\R\to [0,1]$ is measurable with Then spde_u_N with initial condition $u_0^N$ has a mild solution $(u^N_t,t\ge 0)$, which is unique in law. Moreover, $x \mapsto u^N_t(x)$ is a continuous real-valued function for each $t> 0$, and, almost surely, for all $t\ge 0$, $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Values of $\mu_{(\alpha)}\xspace$ and $\sigma_{(\alpha)}\xspace^2$ as a function of $\alpha$, for $\alpha \in (0,3/2)$, when $f$ is given by \ref{['f']}. These values were computed numerically by approximating the deterministic equation with a finite difference method on a discretised domain and solving the equation for $p_{0,t}(m,\cdot,\cdot)$ on the resulting domain. When $\alpha \downarrow 0$, we have $\mu_{(\alpha)}\xspace \to 0$ (as expected by symmetry, since in this case $1-u^N_t(-x)$ has the same dynamics as $u^N_t$) and $\sigma_{(\alpha)}\xspace \to \sigma_{(0)} > 0$, where $\sigma_{(0)}$ can be obtained by adapting Funaki's result in funaki_scaling_1995. The plot suggests that $\mu_{(\alpha)}\xspace > 0$, as we conjecture. We also see that $\mu_{(\alpha)}\xspace$ and $\sigma_{(\alpha)}\xspace^2$ both appear to be increasing as functions of $\alpha$.

Theorems & Definitions (227)

  • remark 1
  • definition 1
  • lemma 1: Theorem 1.1 and Lemma 2.2 in mueller_speed_2021
  • theorem 1
  • lemma 2
  • theorem 2
  • definition 2
  • lemma 3
  • proposition 1
  • lemma 4
  • ...and 217 more