Université Paris 6
Pierre et Marie Curie | Université Paris 7
Denis Diderot | |

CNRS U.M.R. 7599
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``Probabilités et Modèles Aléatoires''
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**Auteur(s): **

**Code(s) de Classification MSC:**

- 35Q53 KdV-like equations (Korteweg-de Vries, Burgers, sine-Gordon, sinh-Gordon, etc.), See also {58F07}
- 35R60 Partial differential equations with randomness, See Also {
- 60F10 Large deviations
- 60J30 Processes with independent increments

**Résumé:** In this article we look at a one-dimensional infinitesimal particle system governed by the completely inelastic
collision rule. Considering uniformly spread mass, we feed the system with initial velocities, so that when time evolves the
corresponding velocity
field fulfils the inviscid Burgers equation. More precisely, we suppose here that the initial velocities are zero, except for
particles located on a stationary regenerative set for which the velocity is some given constant number. We give results of a
large deviation type. First, we estimate the probability that a typical particle is located at time $1$ at distance at least $D$
from its initial position, when $D$ tends to infinity. Its behaviour is related to the left tail of the gap measure of the
regenerative set. We also show the same asymptotics for the tail of the shock interval length distribution.
Second, we analyse the event that a given particle stands still at time $T$ as $T$ tends to infinity. The data to which
we relate its behaviour are the right tail of the gap measure of the regenerative set. We conclude with some results on the
shock structure.

**Mots Clés:** *Inviscid Burgers equation ; random initial velocity ; regenerative sets ; subordinators ; large deviations*

**Date:** 1999-12-16

**Prépublication numéro:** *PMA-554*