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:**

- 60H07 Stochastic calculus of variations and the Malliavin calculus
- 60H15 Stochastic partial differential equations, See also {35R60}
- 35R60 Partial differential equations with randomness, See Also {

**Résumé:** In this paper we show that the Cahn-Hilliard stochastic SPDE has a
function valued solution in dimension 4 and 5 when the
perturbation is driven by a space-correlated Gaussian noise. This
is done proving general results on SPDEs with globally Lipschitz
coefficients associated with operators on smooth domains of
$\mathbb{R}^d$ which are parabolic in the sense of
Petrovski\u{\i}, and do not necessarily define a semi-group of
operators. We study the regularity of the trajectories of the
solutions and the absolute continuity of the law at some given
time and position.

**Mots Clés:** *Parabolic operators ; Cahn-Hilliard equation ; Green function ;
SPDEs ; Malliavin calculus.*

**Date:** 2001-09-17

**Prépublication numéro:** *PMA-685*

**Postscript file : **PMA-685.ps

**Revised version (Oct 12 2001): **PMA-685bis.ps,
PMA-685bis.dvi