Random processes, random media

On considère une marche aléatoire dans Z^dxZ, avec un champ de lois de transition invariant par les translations de Z^d. Un premier modèle de ce type (dans le plan) a été introduit par Campanino et Petritis en 2003. Une condition nécessaire et suffisante de récurrence est présentée. L'asymptotique de la marche s'avère liée à une notion de variance de flux directionnel, mesurant le pouvoir dispersif du milieu. Nous présentons des exemples d'application et évoquons quelques questions en suspens.

We consider the long Brownian bridge started from the origin in hyperbolic space H_d and show that its range, after being suitably renormalised, converges in law to a Brownian continuum tree in the sense of Gromov-Hausdorff. The rough idea of the proof will be talked about, by presenting the convergence of the radial part, the invariance property of re-rooting and the hyperbolicity property. The similar idea will be applied to obtain the local convergence of the rescaled Brownian loop in hyperbolic space.

We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the

It is well-known that the first hitting time of 0 by a negatively drifted Brownian motion starting at $a>0$ has the inverse Gaussian law. Moreover, conditionally on this first hitting time, the BM up to that time has the law of a 3-dimensional Bessel bridge. In this talk, we will give a generalization of this result to a familly of Brownian motions with interacting drifts. The law of the hitting times will be given by the inverse of the random potential that appear in the context of the self-interacting process called the Vertex Reinforced Jump Process. We will also explain some "commutativity" property of these BM and its relation with the martingale that appeared in previous work on the VRJP. Work in progress with Xiaolin Zeng.

Je ferai quelques discussions élémentaires sur la façon dont le mouvement brownien branchant reste bien inférieur à ses valeurs usuelles. Travail en collaboration avec B. Derrida.