The contour of splitting trees is a Lévy process
Résumé
Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump-Mode-Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {infinity}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure.
A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (nu, tau) of some individual nu (vertex) in a discrete tree where tau is a nonnegative real number called chronological level (time). We introduce a total-order on existence points, called linear order, and a mapping phi from the tree into the real line which preserves this order. The inverse of phi is called the exploration process, and the projection of this inverse on chronological levels the contour process.
For splitting trees truncated up to level tau, we prove that a thus defined contour process is a Levy process reflected below tau and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall-Le Jan's theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.