## Annals of Probability

### Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model

#### 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.

#### Article information

Source
Ann. Probab., Volume 48, Number 1 (2020), 94-146.

Dates
Revised: January 2019
First available in Project Euclid: 25 March 2020

https://projecteuclid.org/euclid.aop/1585123324

Digital Object Identifier
doi:10.1214/19-AOP1347

Mathematical Reviews number (MathSciNet)
MR4079432

Zentralblatt MATH identifier
07206754

#### Citation

Černý, Jiří; Drewitz, Alexander. Quenched invariance principles for the maximal particle in branching random walk in random environment and the parabolic Anderson model. Ann. Probab. 48 (2020), no. 1, 94--146. doi:10.1214/19-AOP1347. https://projecteuclid.org/euclid.aop/1585123324

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