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June 2020 Spatial growth processes with long range dispersion: Microscopics, mesoscopics and discrepancy in spread rate
Viktor Bezborodov, Luca Di Persio, Tyll Krueger, Pasha Tkachov
Ann. Appl. Probab. 30(3): 1091-1129 (June 2020). DOI: 10.1214/19-AAP1524

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 $\alpha \in (\frac{1}{2},2]$ 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 $\alpha >\frac{1}{2}$.

Citation

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Viktor Bezborodov. Luca Di Persio. Tyll Krueger. Pasha Tkachov. "Spatial growth processes with long range dispersion: Microscopics, mesoscopics and discrepancy in spread rate." Ann. Appl. Probab. 30 (3) 1091 - 1129, June 2020. https://doi.org/10.1214/19-AAP1524

Information

Received: 1 August 2018; Revised: 1 August 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133369
Digital Object Identifier: 10.1214/19-AAP1524

Subjects:
Primary: 60K35
Secondary: 60J80

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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