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 {
- 60J30 Processes with independent increments
- 60J40 Right processes

**Résumé:** We study the infinite time shock limits given certain Markovian initial velocities
to the inviscid Burgers
turbulence. Specifically, we consider the one-sided case where
initial velocities are zero on the negative half-line and follow a
time-homogeneous nice Markov process $X$ on the positive
half-line. Finite shock limits occur if the Markov process is
transient tending to infinity. They form a Poisson point process if
$X$ is spectrally negative. We give an explicit description when $X$
is furthermore spatially homogeneous (a L\'evy process) or a
self-similar process on $(0,\infty)$. We also consider the
two-sided case where we suppose an independent dual process in
the negative spatial direction. Both spatial homogeneity and
an exponential L\'evy condition lead to stationary shock limits.

**Mots Clés:** *Inviscid Burgers equation ; random initial velocity ; shock structure ;
Markov processes ; self-similar processes ; spectrally negative Lévy processes *

**Date:** 2001-05-14

**Prépublication numéro:** *PMA-657*