Asymptotic Radial Speed of the Support of Supercritical Branching Brownian Motion and Super-Brownian Motion in R

It has long been known that the left-most or right-most particle in a one dimensional dyadic branching Brownian motion with constant branching rate β > 0 has almost sure asymptotic speed √ 2β, (cf. [18]). Recently similar results for higher dimensional branching Brownian motion and super-Brownian motion have also been established but in the weaker sense of convergence in probability; see [20] and [8]. In this short note we confirm the ‘folklore’ for higher dimensions and establish an asymptotic radial speed of the support of the latter two processes in the almost sure sense. The proofs rely on Pinsky’s local extinction criterion, martingale convergence, projections of branching processes from higher to one dimensional spaces together with simple geometrical considerations.