Thick points of 4D critical branching Brownian motion

TL;DR

This paper investigates the properties of thick points in 4D critical branching Brownian motion, revealing phase transitions, dimensional characteristics, and asymptotic behaviors, with implications for related PDEs and discrete models.

Contribution

It provides explicit analysis of thick points in 4D critical branching Brownian motion, including phase transition thresholds, dimension of thick point sets, and asymptotic expansions, extending understanding to PDEs and discrete analogues.

Findings

Probability of hitting small balls exhibits a second-order phase transition at a=2.

Dimension of thick point set (a) is (4-a)_+ for a in [0,4].

Asymptotic expansion for hitting probabilities diverges at dimension 4.

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 $d\geq 1$ 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 $d\neq 4$ 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.