DH 3rd floor SR

We review the analytic transformations allowing to construct standard bridges from a semistable Markov process, with indec 1/2, enjoying the time inversion property. These are generalized and some of there properties are studied. The new family maps the space of continuous real-valued functions into a family which is the topic of our focus. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these by the considered semi-stable process

When two single server queues have the same arrivals process, this is said to be a `fork-join queue'. In the case where the arrivals and service processes are Brownian motions, the queue lengths process is a reflecting Brownian motion in the nonnegative orthant. Tan and Knessl [1996] have given a simple explicit formula for the stationary distribution for this queueing system in a symmetric case, which they obtain as a heavy traffic limit of the classical discrete model. With this as a starting point, we analyse the Brownian model directly in further detail, and consider some related exit problems.

We obtain a new identity giving a quintuple law of overshoot, time of

overshoot, undershoot, last maximum, and time of last maximum of a general Levy

process at ?rst passage. The identity is a simple product of the jump measure

and its ascending and descending bivariate renewal measures. With the help of

this identity, we consider applications for passage problems of stable

processes, recovering and extending results of V. Vigon on the bivariate jump

measure of the ascending ladder process of a general Levy process and present

some new results for asymptotic overshoot distributions for Levy processes with

regularly varying jump measures.

(Parts of this talk are based on joint work with Ron Doney and Claudia

Kluppelberg)

In recent years, the use of random planar maps as discretized random surfaces has received a considerable attention in the physicists community. It is believed that the large-scale properties, or the scaling limit of these objects should not depend on the local properties of these maps, a phenomenon called universality.

By using a bijection due to Bouttier-di Francesco-Guitter between certain classes of planar maps and certain decorated trees, we give instances of such universality

phenomenons when the random maps follow a Boltzmann distribution where each face with degree $2i$ receives a nonnegative weight $q(i)$. For example, we show that under

certain regularity hypothesis for the weight sequence, the radius of the random map conditioned to have $n$ faces scales as $n^{1/4}$, as predicted by physicists and shown in the case of quadrangulations by Chassaing and Schaeffer. Our main tool is a new invariance principle for multitype Galton-Watson trees and discrete snakes.

The aging of spin-glasses has been of much interest in the last decades. Since its explanation in the context of real spin-glass models is out of reach, several effective models were proposed in physics literature. In my talk I will present how aging can be rigorously proved in so called trap models and what is the mechanism leading to it. In particular I will concentrate on conditions leading to the fact that one of usual observables used in trap models converges to arc-sine law for Levy processes.