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Wave propagation for 1-dimensional reaction-diffusion equation with nonzero random drift

Dihang Guan, Hui He, Wenqing Hu, Jiaojiao Yang

TL;DR

The paper studies traveling-wave behavior for a one-dimensional FKPP-type reaction-diffusion equation with a stationary ergodic random drift $b(x)$ and positive mean drift ${\mathbb E}[b]>0$. It develops a fully probabilistic framework based on the Large Deviations Principle for diffusion in random environments and the Feynman-Kac representation to balance reaction heating at rate $\beta=f'(0)$ against diffusion-driven cooling, yielding a complete characterization of wave propagation. Key contributions include a full LDP for hitting times, a precise relation between forward and backward cumulant functions (showing a constant shift due to the drift), and explicit classifications of wavefront behavior and exact front shapes as functions of $\beta$ and drift strength. The findings reveal that the drift acts as an external field shifting the quenched free-energy reference without altering fluctuation structures, with implications for turbulent combustion and population dynamics in random media.

Abstract

We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the asymptotic wave fronts propagating to the positive and negative directions are both pushed in the negative direction, leading to the possibility that both wave fronts propagate toward negative infinity. Our proof is based on the Large Deviations Principle for diffusion processes in random environments, as well as an analysis of the Feynman-Kac formula. Such probabilistic arguments also reveal the underlying physical mechanism of the wave fronts formation: the drift acts as an external field that shifts the (quenched) free-energy reference level without altering the intrinsic fluctuation structure of the system.

Wave propagation for 1-dimensional reaction-diffusion equation with nonzero random drift

TL;DR

The paper studies traveling-wave behavior for a one-dimensional FKPP-type reaction-diffusion equation with a stationary ergodic random drift and positive mean drift . It develops a fully probabilistic framework based on the Large Deviations Principle for diffusion in random environments and the Feynman-Kac representation to balance reaction heating at rate against diffusion-driven cooling, yielding a complete characterization of wave propagation. Key contributions include a full LDP for hitting times, a precise relation between forward and backward cumulant functions (showing a constant shift due to the drift), and explicit classifications of wavefront behavior and exact front shapes as functions of and drift strength. The findings reveal that the drift acts as an external field shifting the quenched free-energy reference without altering fluctuation structures, with implications for turbulent combustion and population dynamics in random media.

Abstract

We consider the wave propagation for a reaction-diffusion equation on the real line, with a random drift and Fisher-Kolmogorov-Petrovskii-Piscounov (FKPP) type nonlinear reaction. We show that when the average drift is positive, the asymptotic wave fronts propagating to the positive and negative directions are both pushed in the negative direction, leading to the possibility that both wave fronts propagate toward negative infinity. Our proof is based on the Large Deviations Principle for diffusion processes in random environments, as well as an analysis of the Feynman-Kac formula. Such probabilistic arguments also reveal the underlying physical mechanism of the wave fronts formation: the drift acts as an external field that shifts the (quenched) free-energy reference level without altering the intrinsic fluctuation structure of the system.
Paper Structure (9 sections, 28 theorems, 276 equations, 2 figures)

This paper contains 9 sections, 28 theorems, 276 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Basic Idea of the Proof and Main Results
  3. Probabilistic Behavior of the Process $X^x(t)$ at large time $t\rightarrow \infty$
  4. Large Deviations for the Hitting Time
  5. Relation between $\mu^\rightarrow(\eta)$ and $\mu^\leftarrow(\eta)$
  6. Large Deviations for the process $X^x(t)$
  7. Other Auxiliary Lemmas
  8. Wave Propagation
  9. Exact shape of the asymptotic wave front

Key Result

Lemma 3.1

Let $\eta\in\mathbb{R}$ be such that Assume $0<c<v$. Then $\mathbf{P}$-almost surely the limit (Eq:LDPHittingTimeFreeEnergy) holds and we have where $\mu(\eta)$ is defined in (Eq:LyapunovFunctionHittingTime:Backward). Thus $H(\eta)<\infty$ when $\eta<\eta_c$ and $H(\eta)=\infty$ when $\eta>\eta_c$, where $\eta_c$ is defined in (Eq:CriticalEta:Backward).

Figures (2)

  • Figure 2: Shape of $I(a)\equiv I^\leftarrow(a)$ and $I^\rightarrow(a)$. (a) when $\eta_c>0$ and $\mu'(\eta_c-)=+\infty$; (b) when $\eta_c>0$ and $\mu'(\eta_c)<+\infty$.
  • Figure 3: Shape of $I(a)\equiv I^\leftarrow(a)$ and $I^\rightarrow(a)$. (c) when $\eta_c=0$ and $\mu'(\eta_c-)=+\infty$; (b) when $\eta_c=0$ and $\mu'(\eta_c)<+\infty$.

Theorems & Definitions (55)

  • Lemma 3.1: Lyapunov Exponent Identity
  • proof
  • Theorem 1: LDP for the Hitting Time
  • Remark 1
  • proof : Proof of Theorem \ref{['Thm:LDPHittingTime']}
  • Lemma 3.2: Properties of $\mu(\eta)$, $\mu^M(\eta)$ and $\mu^{|M}(\eta)$
  • proof
  • Lemma 3.3: Lyapunov Exponent Identity for the Forward Hitting Time
  • Lemma 3.4
  • Theorem 2: LDP for the Hitting Time in the Forward case
  • ...and 45 more