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Large deviations and the emergence of a logarithmic delay in a nonlocal linearised Fisher-KPP equation

Nathanaël Boutillon

Abstract

We study a variant of the Fisher-KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation. Previous papers found very precise results for the nonlinear equation with strong assumptions on the decay of the kernel. Our results are less precise, but they are valid for all continuous symmetric thin-tailed kernels.

Large deviations and the emergence of a logarithmic delay in a nonlocal linearised Fisher-KPP equation

Abstract

We study a variant of the Fisher-KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude that the logarithmic delay emerges also for the solutions of the nonlinear equation. Previous papers found very precise results for the nonlinear equation with strong assumptions on the decay of the kernel. Our results are less precise, but they are valid for all continuous symmetric thin-tailed kernels.
Paper Structure (12 sections, 8 theorems, 77 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 77 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminary results about large deviations
  4. Proofs of the main results
  5. Proof of Proposition \ref{['ppn:rep']}
  6. Proof of Theorem \ref{['thm:main']}
  7. Step 1: Application of \ref{['eq:property_bahadur']}.
  8. Step 2: Estimation of $\Lambda^*$.
  9. Step 3: Estimation of each single term of the sum $T_{a,b}(t,x_t)$.
  10. Step 1: Estimation of the partial sum $S_{\alpha-\varepsilon,\alpha+\varepsilon}(t,x_t)$; simple integrals arise.
  11. Step 2: Estimation of the integral arising in \ref{['eq:theta_sepsilon-']} and conclusion.
  12. Proof of Corollary \ref{['cor:main']}

Key Result

Proposition 2.1

Let $(S_n)_{n\geqslant 1}$ be the random walk defined above. Let $u(t,x)$ be a solution of the linear Cauchy problem eq:lin. Then, for all $t\in\left[0,+\infty\right)$, for all $x\in\mathds{R}$,

Figures (1)

  • Figure 1: Graphical interpretation of the Legendre-Fenchel transform of $\Lambda$ (inspired from DemZei10). The curve in blue looks like $\Lambda$, the dotted line in red is the line tangent to the blue curve and with slope $z$. The dotted line in red crosses the vertical axis at $-\Lambda^*(z)$, the opposite of the Legendre-Fenchel transform of $\Lambda$ at $z$.

Theorems & Definitions (19)

  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Theorem 3.2: Bahadur-Rao BahRao60
  • proof : Proof of Proposition \ref{['ppn:rep']}
  • Remark 4.1
  • ...and 9 more