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2013 Moment asymptotics for branching random walks in random environment
Wolfgang König, Onur Gün, Ozren Sekulović
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Electron. J. Probab. 18: 1-18 (2013). DOI: 10.1214/EJP.v18-2212


We consider the long-time behaviour of a branching random walk in random environment on the lattice $\mathbb{Z}^d$. The migration of particles proceeds according to simple random walk in continuous time, while the medium is given as a random potential of spatially dependent killing/branching rates. The main objects of our interest are the annealed moments $\langle m_n^p \rangle $, i.e., the $p$-th moments over the medium of the $n$-th moment over the migration and killing/branching, of the local and global population sizes. For $n=1$, this is well-understood, as $m_1$ is closely connected with the parabolic Anderson model. For some special distributions, this was extended to $n\geq2$, but only as to the first term of the asymptotics, using (a recursive version of) a Feynman-Kac formula for $m_n$. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that $\langle m_n^p \rangle$ and $\langle m_1^{np} \rangle$ are asymptotically equal, up to an error $\mathrm{e}^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac type formula for $m_n$, which we establish using known spine techniques.


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Wolfgang König. Onur Gün. Ozren Sekulović. "Moment asymptotics for branching random walks in random environment." Electron. J. Probab. 18 1 - 18, 2013.


Accepted: 21 June 2013; Published: 2013
First available in Project Euclid: 4 June 2016

zbMATH: 1286.60086
MathSciNet: MR3078022
Digital Object Identifier: 10.1214/EJP.v18-2212

Primary: 60J80
Secondary: 60F10 , 60J55 , 60K37

Keywords: annealed moments , Branching random walk , Feynman-Kac-type formula , large deviations , Parabolic Anderson model , Random potential

Vol.18 • 2013
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