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Moment asymptotics for branching random walks in random environment

Wolfgang König, Onur Gün, Ozren Sekulović

2013
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Electronic Journal of Probability
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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 p n , 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.
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... population sizes. For n = 1, this is well-understood [GM98], as m 1 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 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 p n and m np 1 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 [HR12]. 2 BRWRE, i.e., the p-th moments over the medium of the n-th moment over the killing/branching and migration of the total and local population size. It is the aim of the present paper to significantly increase the validity and the deepness of the results of [ABMY00] and to reveal the general mechanism that leads to the moment asymptotics. In contrast with [ABMY00], we will be using probabilistic methods rather than PDE methods. Branching random walk in random environment Let us describe the model in more detail. The branching random environment on the lattice Z d is a pair Ξ = (ξ 0 , ξ 2 ) of two independent i.i.d. fields ξ 0 = (ξ 0 (y)) y∈Z d and ξ 2 = (ξ 2 (y)) y∈Z d of positive numbers. Indeed, ξ 0 (y) and ξ 2 (y) play the rôle of the rate of the replacement of a particle at y ∈ Z d with 0 or 2 particles, respectively. For n = 0, this is a killing, for n = 2, this is a binary splitting. (See Section 1.4 for more general branching mechanisms.)

doi:10.1214/ejp.v18-2212
fatcat:uqfrt3lv45eanpqisyawf4nw5e