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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]

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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]

Consider the catalytic super-Brownian motion X[sup ϱ] (reactant) in ℝ[sup d] , d≤3, which branching rates vary randomly in time and space and in fact are given by an ordinary super-Brownian motion ϱ (catalyst). Our main object of study is the collision local time L = L[sub [ϱ,X[sup ϱ] ]] (d(s,x) )of catalyst and reactant. It determines the covariance measure in themartingale problem for X[sup ϱ] and reflects the occurrence of “hot spots” of reactant which can be seen in simulations of X[sup ϱ] . In dimension 2, the collision local time is absolutely continuous in time, L(d(s,x) ) = ds K[sub s] (dx). At fixed time s, the collision measures K[sub s] (dx) of ϱ[sub s] and X[sub s] [sup ϱ] have carrying Hausdorff dimension 2. Spatial marginal densities of L exist, and, via self-similarity, enter in the long-term randomergodic limit of L (diffusiveness of the 2-dimensional model). We alsocompare some of our results with the case of super-Brownian motions withdeterministic time-independent catalysts. [ABSTRACT FROM AUTHOR]

Abstract. We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the epsilon-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of X[sub t] is capacity-equivalent to [0, 1][sup 2] in R[sup d], d is greater than or equal to 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1][sup 4] in R[sup d] d less than or equal to 5. [ABSTRACT FROM AUTHOR]