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We give an invariance principle for very general additive functionals of conditioned Bienaymé-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable Lévy tree. This includes the case when the offspring distribution has finite variance (the Lévy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height. [ABSTRACT FROM AUTHOR]

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process. [ABSTRACT FROM AUTHOR]

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest. [ABSTRACT FROM AUTHOR]

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We consider the exit measure of super Brownian motion with a stable branching mechanism of a smooth domain D of ℝ[sup d] . We derive lower bounds for the hitting probability of small balls for the exit measure and upper bounds in the critical dimension. This completes results given by Sheu [22] and generalizes the results of Abraham and Le Gall [2]. Because of the links between exits measure and partial differential equations, those results imply bounds on solutions of elliptic semi-linear PDE. We also give the Hausdorff dimension of the support of the exit measure and show it is totally disconnected in high dimension. Eventually we prove the exit measure is singular with respect to the surface measure on ∂D in the critical dimension. Our main tool is the subordinated Brownian snake introduced by Bertoin, Le Gall and Le Jan [4]. [ABSTRACT FROM AUTHOR]

We study some properties of the exit measure of super Brownian motion from a smooth domain D in R. In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of Δ u= u in D, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of Δ u= u in D that tends to ∞ on an open subset O of ϖ D and to 0 on the complement in ϖ D of the closure of O. Our proofs use the path-valued process studied in [L2, L3]. [ABSTRACT FROM AUTHOR]