@article{EJP2212,
author = {Wolfgang König and Onur Gün and Ozren Sekulović},
title = {Moment asymptotics for branching random walks in random environment},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {18},
year = {2013},
keywords = {branching random walk, random potential, parabolic Anderson model, Feynman-Kac-type formula, annealed moments, large deviations},
abstract = {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 $\e^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac type formula for $m_n$, which we establish using known spine techniques.},
pages = {no. 63, 1-18},
issn = {1083-6489},
doi = {10.1214/EJP.v18-2212},
url = {http://ejp.ejpecp.org/article/view/2212}}