Moment asymptotics for branching random walks in random environment
- Gün, Onur
- König, Wolfgang
- Sekulović, Ozren
2010 Mathematics Subject Classification
- 60J80 60J55 60F10 60K37
- branching random walk, random potential, parabolic Anderson model, Feynman-Kac-type formula, annealed moments, large deviations
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].
- Electron. J. Probab., 18 (2013) pp. 1--18.