Moment asymptotics for branching random walks in random environment

2012/08/01 by Gün, Onur, König, Wolfgang, Sekulović, Ozren
#60F10 #60J55 #60J80 #60K37 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1208.0306

Abstract

We consider the long-time behaviour of a branching random walk in random environment on the lattice \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 < m_n^p > , 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 \citeGM98, as m₁ is closely connected with the parabolic Anderson model. For some special distributions, \citeA00 extended this to n≥2, 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 < m_n^p > and < m₁^np > 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 the spine techniques developed in \citeHR11.

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