Derivative martingale of the branching Brownian motion in dimension $d \geq 1$

TL;DR

This paper studies the behavior of the derivative martingale in branching Brownian motion in multiple dimensions, establishing the existence of a random measure on the sphere based on the martingale's limits in certain directions.

Contribution

It proves the existence of a directional derivative martingale limit in any dimension $d \\geq 1$, and constructs a related random measure on the sphere.

Findings

Existence of a random subset of directions with martingale limits

Construction of a random measure on the sphere from the martingale

Approximation method using truncated processes for convergence

Abstract

We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $\Theta$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $\theta \in \Theta$ almost surely. This allows us to define a random measure on $\mathbb{S}^{d-1}$ whose density is given by the derivative martingale. The proof is based on first moment arguments: we approximate the martingale of interest by a series of processes, which do not take into account the particles that travelled too far away. We show that these new processes are uniformly integrable martingales whose limits can be made to converge to the limit of the original martingale.