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:**

- 35K55 Nonlinear PDE of parabolic type
- 60K35 Interacting random processes; statistical mechanics type models; percolation theory, See also {82B43, 82C43}

**Résumé:** We study a class of nonlinear martingale problems in one
dimension, that involve a singular integral of the density in the
drift term, and are related to systems of particles with singular
interactions. First we prove existence and uniqueness of regular
solutions of the associated nonlinear evolution equation. Then, we
establish a suitable framework and conditions where the martingale
problem is well posed. This extends the results of Bonami,
Bouchut, C\'epa and L\'epingle \cite{BBCL} to a wide class of
coefficients and initial conditions. Finally, we obtain our
solution of the martingale problem as the chaotic limit of some
systems of particles interacting through regular approximating
kernels.

**Mots Clés:** *McKean-Vlasov equation ; singular integrals ; nonlinear martingale problem ;
Monte Carlo approximation*

**Date:** 2002-03-27

**Prépublication numéro:** *PMA-718*

**Pdf file : **PMA-718.pdf