Julien Sohier

Levy multiplicative chaos and star scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Levy multiplicative chaos that generalizes the already existing constructions. Keywords: Random measure, star equation, scale invariance, multiplicative chaos, uniqueness, infinitely divisible processes, multifractal processes.

Metastability for general dynamics with rare transitions : escape time and critical configurations

Metastability is a physical phenomenon ubiquitous in first order phase transitions. A fruitful mathematical way to approach this phenomenon is the study of rare transitions Markov chains. For Metropolis chains associated with statistical mechanics systems, this phenomenon has been described in an elegant way in terms of the energy landscape associated to the Hamiltonian of the system. In this paper, we provide a similar description in the general rare transitions setup. Beside their theoretical content, we believe that our results are a useful tool to approach metastability for non-Metropolis systems such as Probabilistic Cellular Automata.

The Scaling limits of a heavy tailed Markov renewal process.

Dans cet article, nous considerons des processus de renouvellement markovien a queues lourdes. Nous montrons que, convenablement renormalises, ils convergent vers l’ensemble regeneratif d’indice α. Nous appliquons ces resultats a un modele d’accrochage dans une bande. Dans ce modele, une marche aleatoire S, contrainte a rester au-dessus d’un mur, est recompensee ou penalisee lorsqu’est atteinte la bande [0,∞)×[0,a] ou a est un reel strictement positif. La convergence que nous etablissons permet de caracteriser les limites d’echelle de ce modele au point critique.