Moment asymptotics for branching random walks in random environment

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 $< m_n^p >$, 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 \cite{GM98}, as $m_1$ is closely connected with the parabolic Anderson model. For some special distributions, \cite{A00} extended this 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 $< m_n^p >$ and $< m_1^{np} >$ are asymptotically equal, up to an error $\e^{o(t)}$. The cornerstone of our method is a direct Feynman-Kac-type formula for $m_n$, which we establish using the spine techniques developed in \cite{HR11}.