Spatial growth processes with long range dispersion: microscopics, mesoscopics, and discrepancy in spread rate

TL;DR

This paper investigates how the spread rate of a branching random walk with polynomially decaying dispersion kernel depends on the decay parameter, revealing linear or superlinear growth in microscopic models and exponential growth in mesoscopic equations.

Contribution

It provides a detailed analysis of the spread rate depending on the decay parameter, contrasting microscopic and mesoscopic behaviors.

Findings

For α > 2, the spread is linear.

For α in (1/2, 2], the spread exceeds linear.

Mesoscopic equations exhibit exponential spread for all α > 1/2.

Abstract

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2\alpha}$, $x \to \infty$. We show that if $\alpha > 2$, then the system spreads at a linear speed, {while for $\alpha \in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha > \frac 12$.