Annealed Moment Lyapunov Exponents for a Branching Random Walk in a Homogeneous Random Branching Environment

#### S. Albeverio, L.V. Bogachev, S.A. Molchanov, E.B. Yarovaya

2000, v.6, Issue 4, 473-516

ABSTRACT

We consider a continuous-time branching random walk on the lattice $\bZ^{d}$ ($d\ge1$) evolving in a random branching environment. The motion of particles proceeds according to the law of a simple symmetric random walk. The branching medium formed of Markov birth-and-death processes at the lattice sites is assumed to be spatially homogeneous. We are concerned with the long-time behavior of the quenched moments $m_n$ ($n\in\bN$) for the local and total particle populations in a frozen'' medium. We pursue the moment approach via studying the asymptotics of the annealed moments $\langle m_n^p\rangle$ ($p\ge1$), obtained by averaging over the medium realizations. Under the assumption that the random branching potential has a Weibull type upper tail, we compute the corresponding Lyapunov exponents $\lambda_{n,\myp p}$. Our results show that the quenched moments of all orders grow in a non-regular, intermittent fashion. The proofs are based on the study of a Cauchy problem for the Anderson operator with random potential and random source. In particular, we derive the Feynman - Kac representation for the solution of the inhomogeneous problem.

Keywords: branching random walk,random branching environment,quenched moments,annealed moments,backward equations,Feynman - Kac representation,Lyapunov exponents,intermittency