WIAS Preprint No. 1729, (2012)

Moment asymptotics for branching random walks in random environment



Authors

  • Gün, Onur
  • König, Wolfgang
  • Sekulović, Ozren

2010 Mathematics Subject Classification

  • 60J80 60J55 60F10 60K37

Keywords

  • branching random walk, random potential, parabolic Anderson model, Feynman-Kac-type formula, annealed moments, large deviations

DOI

10.20347/WIAS.PREPRINT.1729

Abstract

We consider the long-time behaviour of a branching random walk in random environment on the lattice ℤd. 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 ⟨ mnp ⟩, 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 [GM98], as m1 is closely connected with the parabolic Anderson model. For some special distributions, [ABMY00] 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 mn. In this work we derive also the second term of the asymptotics, for a much larger class of distributions. In particular, we show that ⟨ mnp ⟩ and ⟨ m1np ⟩ are asymptotically equal, up to an error eo(t). The cornerstone of our method is a direct Feynman-Kac-type formula for mn, which we establish using the spine techniques developed in [HR12].

Appeared in

  • Electron. J. Probab., 18 (2013) pp. 1--18.

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