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Moment asymptotics for branching random walks in random environment. (English) Zbl 1286.60086

Summary: We consider the long-time behaviour of a branching random walk in random environment on the lattice \(\mathbb{Z}^d\). 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 \(\langle m_n^p \rangle \), 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, as \(m_1\) is closely connected with the parabolic Anderson model. For some special distributions, this was extended to \(n\geq2\), 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 \(\langle m_n^p \rangle\) and \(\langle m_1^{np} \rangle\) are asymptotically equal up to an error \(\text e^{o(t)}\). The cornerstone of our method is a direct Feynman-Kac type formula for \(m_n\), which we establish using known spine techniques.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J55 Local time and additive functionals
60F10 Large deviations
60K37 Processes in random environments
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