A notion of semi-selfsimilarity of R d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: 1. The existence of the exponent of semi-selfsimilar processes. 2. Characterization for semi-selfsimilar processes as scaling limits. 3. Relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions. 4. Construction of semi-selfsimilar processes with stationary increments. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes.

We consider increasing self--similar Markov processes (X t , t 0) on ]0, #[.

Abstract: We establish integral tests and laws of the iterated logarithm for the lower envelope of positive self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments. These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdös [11], Motoo [17] and Rivero [18].

Abstract We establish integral tests and laws of the iterated logarithm for the upper envelope of the future infimum of positive self-similar Markov processes and for increasing self-similar Markov processes at 0 and +∞. Our proofs are based on the Lamperti representation and time reversal arguments due to Chaumont and Pardo [9]. These results extend laws of the iterated logarithm for the future infimum of Bessel processes due to Khoshnevisan et al. [11].

Abstract: We establish integral tests and laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation, time reversal arguments and on the study of the upper envelope of their future infimum due to Pardo [19]. These results extend integral test and laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdös [10] and stable Lévy processes conditioned to stay positive with no positive jumps due to Bertoin [1]. Key words: Self-similar Markov process, Self-similar additive processes, Future infimum process, Lévy process, Lamperti representation, First and last passage time, integral test, law of the iterated

E l e c t r o n J o u r n a l o f

Abstract. We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian height fragmentation when it is about to reduce to dust are described in a limit theorem. Résumé. Une étude additionnelle de la fragmentation de hauteur brownienne est présentée. Plus précisément, une représentation de la masse du fragment marqué en termes d’une transformation de Doob du subordinateur stable d’indice 1/2 est décrite puis utilisée pour étudier les sauts du processus de masse; ceci nous renseigne sur la façon dans laquelle un fragment typique se casse. Ces résultats se généralisent au cadre des fragmentations de hauteur de l’arbre stable. Enfin, nous donnons un théorème limite de la fragmentation de l’excursion Brownienne par les hauteurs, centrée autour du dernier fragment qui se décompose en poussière.

A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton–Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions φ according to whether φ-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes’s conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved. 1. Introduction. An