Quantifying the Threshold Phenomenon for Propagation in Nonlocal Diffusion Equations

Abstract

We are interested in the threshold phenomenon 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 forms. The strategy is to combine sharp estimates of the tails of the sum of independent and identically random variables (coming, in particular, from large deviation theory) and the construction of accurate sub- and supersolutions.