## 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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