Francis Hirsch

Formes de Dirichlet générales et densité des variables aléatoires réelles sur l'espace de Wiener

Abstract It is shown that the main properties of Dirichlet spaces can be proved without the classical local compacity hypothesis. To this purely measurable framework we extend the functional calculus and the “energy-image density property.” These results are then applied to the Ornstein-Uhlenbeck semi-group on Wiener space, where it is obtained that the real random variables, which are solutions of lipschitzian stochastic differential equations, always have probability densities after perhaps an initial deterministic phase.

Propriété d'absolue continuité pour les équations différentielles stochastiques dépendant du passé

Abstract Stochastic differential equations with coefficients which depend on the past are considered. It is shown that under weak assumptions on the coefficients (lipschitz regularity and partial ellipticity) the laws of some projections of the solution are absolutely continuous with respect to the Lebesgue measure.