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January 2020 Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model
Jiří Černý, Alexander Drewitz
Ann. Probab. 48(1): 94-146 (January 2020). DOI: 10.1214/19-AOP1347

Abstract

We consider branching random walk in spatial random branching environment (BRWRE) in dimension one, as well as related differential equations: the Fisher–KPP equation with random branching and its linearized version, the parabolic Anderson model (PAM). When the random environment is bounded, we show that after recentering and scaling, the position of the maximal particle of the BRWRE, the front of the solution of the PAM, as well as the front of the solution of the randomized Fisher–KPP equation fulfill quenched invariance principles. In addition, we prove that at time $t$ the distance between the median of the maximal particle of the BRWRE and the front of the solution of the PAM is in $O(\ln t)$. This partially transfers classical results of Bramson (Comm. Pure Appl. Math. 31 (1978) 531–581) to the setting of BRWRE.

Citation

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Jiří Černý. Alexander Drewitz. "Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model." Ann. Probab. 48 (1) 94 - 146, January 2020. https://doi.org/10.1214/19-AOP1347

Information

Received: 1 January 2018; Revised: 1 January 2019; Published: January 2020
First available in Project Euclid: 25 March 2020

zbMATH: 07206754
MathSciNet: MR4079432
Digital Object Identifier: 10.1214/19-AOP1347

Subjects:
Primary: 60G70, 60J80, 82B44

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • January 2020
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