Critical branching random walks with small drift

We study critical branching random walks (BRWs) U(n) on  where the displacement of an offspring from its parent has drift  towards the origin and reflection at the origin. We prove that for any [alpha]>1, conditional on survival to generation [n[alpha]], the maximal displacement is . We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like yn[alpha] for some y>0, [alpha]>1, and the particles are concentrated in , then the measure-valued processes associated with the BRWs converge to a measure-valued process, which, at any time t>0, distributes its mass over  like an exponential distribution.