A note on stable point processes occurring in branching Brownian motion

We call a point process $Z$ on $\mathbb R$ \emph{exp-1-stable} if for every $\alpha,\beta\in\mathbb R$ with $e^\alpha+e^\beta=1$, $Z$ is equal in law to $T_\alpha Z+T_\beta Z'$, where $Z'$ is an independent copy of $Z$ and $T_x$ is the translation by $x$. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process $D$ on $\mathbb R$ such that $Z$ is equal in law to $\sum_{i=1}^\infty T_{\xi_i} D_i$, where $(\xi_i)_{i\ge1}$ are the atoms of a Poisson process of intensity $e^{-x}\,\mathrm d x$ on $\mathbb R$ and $(D_i)_{i\ge 1}$ are independent copies of $D$ and independent of $(\xi_i)_{i\ge1}$. In this note, we show how this decomposition follows from the classic \emph{LePage decomposition} of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb R$

Slowdown in branching Brownian motion with inhomogeneous variance

We consider a model of Branching Brownian Motion with time-inhomogeneous variance of the form \sigma(t/T), where \sigma is a strictly decreasing function. Fang and Zeitouni (2012) showed that the maximal particle's position M_T is such that M_T-v_\sigma T is negative of order T^{-1/3}, where v_\sigma is the integral of the function \sigma over the interval [0,1]. In this paper, we refine we refine this result and show the existence of a function m_T, such that M_T-m_T converges in law, as T\to\infty. Furthermore, m_T=v_\sigma T - w_\sigma T^{1/3} - \sigma(1)\log T + O(1) with w_\sigma = 2^{-1/3}\alpha_1 \int_0^1 \sigma(s)^{1/3} |\sigma'(s)|^{2/3}\,\dd s. Here, -\alpha_1=-2.33811... is the largest zero of the Airy function. The proof uses a mixture of probabilistic and analytic arguments.Comment: A proof of convergence added in v2; details added and minor typos corrected in v