Moment asymptotics for branching random walks in random environment

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 m_np , 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 m_1 is closely connected with the parabolic Anderson model. For some special distributions, citeA00 extended this to ngeq2, 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 m_1^np are asymptotically equal, up to an error e^o(t). The cornerstone of our method is a direct Feynman-Kac-type formula for mn, which we establish using the spine techniques developed in citeHR1.1