A note on stable point processes occurring in branching Brownian motion

By P. Maillard. 2011

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We call a point process $Z$ on $\mathbb R$ 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 LePage decomposition of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb R$.

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Citations in the corpus

Freezing and Decorated Poisson Point Processes (2014, methodology): “By the result of [44], this yields the required structure, without, however, saying anything about the decoration process.”
Branching random walks (2015, background): “The basic framework is set up in Jagers [148] and Chauvin [84]; for a sample of interesting applications, see Biggins and Kyprianou [56], [57], [58], Kyprianou [161], Maillard [186], Olofsson [205].”
Soft local times and decoupling of random interlacements (2012, background): “Recent works have also shown a connection between random interlacements, the Gaussian free field [28], [27] and cover times of random walks [2].”
Convergence in Law for the Branching Random Walk Seen from Its Tip (2011, background): “” We refer to [22] for a more complete description.”