A simple flattening lower bound for solutions to some linear integro-differential equations

Estimates on the asymptotic behaviour of solution to linear integro-differential equations are fundamental in understanding the dynamics occurring in many nonlocal evolution problems. They are usually derived by using precise decay estimates on the heat kernel of the considered diffusion process. In this note, we show that for some generic jump diffusion and particular initial data, one can derive a lower bound of the asymptotic behaviour of the solution using a simple PDE argument. This is viewed as an independent preliminary brick to study invasion phenomena in nonlinear reaction diffusion problems.