Thick points of 4D critical branching Brownian motion

Abstract We study the thick points of branching Brownian motion and branching random walk with a critical branching mechanism, focusing on the critical dimension . We determine the exponent governing the probability to hit a small ball with an exceptionally high number of pioneers, showing that this has a second‐order transition between an exponential phase and a stretched exponential phase at an explicit value () of the thickness parameter . We apply the outputs of this analysis to prove that the associated set of thick points has dimension , so that there is a change in behaviour at but not at in this case. Along the way, we obtain related results for the non‐positive solutions of a boundary value problem associated to the semi‐linear partial differential equation (PDE) and develop a strong coupling between tree‐indexed random walk and tree‐indexed Brownian motion that allows us to deduce analogues of some of our results in the discrete case. We also obtain in each dimension an infinite‐order asymptotic expansion for the probability that critical branching Brownian motion hits a distant unit ball, finding that this expansion is convergent when and divergent when . This reveals a novel, dimension‐dependent critical exponent governing the higher order terms of the expansion, which we compute in every dimension.