Critical branching random walks with small drift

Abstract

We study critical branching random walks (BRWs) U ( n ) on Z + where the displacement of an offspring from its parent has drift 2 β / n towards the origin and reflection at the origin. We prove that for any α > 1 , conditional on survival to generation [ n α ] , the maximal displacement is ∼ ( α − 1 ) / ( 4 β ) n log n . We further show that for a sequence of critical BRWs with such displacement distributions, if the number of initial particles grows like y n α for some y > 0 , α > 1 , and the particles are concentrated in [ 0 , O ( n ) ] , 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 R + like an exponential distribution.