论文标题
Dimension Brownian Motion $ d \ geq 1 $
Derivative martingale of the branching Brownian motion in dimension $d \geq 1$
论文作者
论文摘要
我们考虑$ \ mathbb {r}^d $中的一个分支布朗尼运动。我们证明存在$ \ mathbb {s}^{d-1} $的随机子集$θ$，使得几乎肯定在所有方向$θ\ inθ$的所有方向上同时存在衍生品the的极限。这使我们能够在$ \ mathbb {s}^{d-1} $上定义一个随机度量，其密度由导数martingale给出。 证明是基于第一刻论点：我们通过一系列过程近似关注的群岛，这些过程没有考虑到太远的粒子。我们表明，这些新过程是统一的可整合的捕食者，它们可以使其限制融合到原始的Martingale的极限。
We consider a branching Brownian motion in $\mathbb{R}^d$. We prove that there exists a random subset $Θ$ of $\mathbb{S}^{d-1}$ such that the limit of the derivative martingale exists simultaneously for all directions $θ\in Θ$ 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.
