Branching Random Walk in Random Environment: Fully Quenched Case
2001, v.7, №2, 349-353
The purpose of this short report is to introduce a branching random walk in random environment on $Z^d$ where particles perform independent simple random walks and branch according to a law that is obtained by fixing branching numbers at each point of $Z^d$. These numbers represent a realization of an integer-valued random field on $Z^d$ with the value at each point being independent of those at other points. With just one particle starting at the origin, we identify the conditions which separate transience and recurrence, i.e., the progeny hits the origin with probability less than 1, respectively, equal to 1, in the same manner as it was done by Menshikov and Volkov in [Branching Markov chains: qualitative characteristics. Markov Processes Relat. Fields, 1997, v.3, pp.225-241], and den Hollander, Menshikov and Popov in [A note on transience vs. recurrence for a branching random walk in random environment. J. Stat. Phys., 1999, v.95, pp.587-614].
Keywords: branching random walk,random environment,quenched and annealedproblems