SPATIAL GROWTH PROCESSES WITH LONG RANGE DISPERSION: MICROSCOPICS, MESOSCOPICS AND DISCREPANCY IN SPREAD RATE

• 成果类型: Article
• 署名作者: Bezborodov, Viktor; Di Persio, Luca; Krueger, Tyll; Tkachov, Pasha
• 署名单位: Wroclaw University of Science & Technology; University of Verona; Gran Sasso Science Institute (GSSI)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/19-AAP1524
• 发表日期: 2020
• 页码: 1091-1129
• 关键词: branching random-walk; asymptotic shape; CONVERGENCE; sums

摘要

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 vertical bar x vertical bar(-2 alpha) , x -> infinity. We show that if alpha > 2, then the system spreads at a linear speed, while for alpha is an element of (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 > 1/2.