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 $d=4$. 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 ($a=2$) of the thickness parameter $a$. We apply the outputs of this analysis to prove that the associated set of thick points $\mathcal{T}(a)$ has dimension $(4-a{)}_{+}$, so that there is a change in behaviour at $a=4$ but not at $a=2$ in this case. Along the way, we obtain related results for the nonpositive solutions of a boundary value problem associated to the semilinear PDE $\Delta v=v^2$ 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 $d\ge 1$ 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 $d\ne 4$ and divergent when $d=4$. This reveals a novel, dimension-dependent critical exponent governing the higher-order terms of the expansion, which we compute in every dimension.

78 pages, 8 figures

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