Quantifying the threshold phenomena for propagation in nonlocal diffusion equations

TL;DR

This paper investigates the threshold phenomena for propagation in nonlocal diffusion equations, providing quantitative estimates that depend on the dispersal kernel's tails, using large deviation theory and sub-super solution construction.

Contribution

It offers the first quantitative estimates for threshold phenomena in nonlocal diffusion equations, highlighting the influence of dispersal kernel tails.

Findings

Threshold phenomena depend heavily on dispersal kernel tails.

Quantitative estimates are derived using large deviation theory.

The approach combines tail estimates with sub- and super-solution methods.

Abstract

We are interested in the threshold phenomena for propagation in nonlocal diffusion equations with some compactly supported initial data. In the so-called bistable and ignition cases, we provide the first quantitative estimates for such phenomena. The outcomes dramatically depend on the tails of the dispersal kernel and can take a large variety of different forms. The strategy is to combine sharp estimates of the tails of the sum of i.i.d. random variables (coming, in particular, from large deviation theory) and the construction of accurate sub-and super-solutions.