Some Results on Lipschitzian Stochastic Differential Equations by Dirichlet Forms Methods.

Nicolas Bouleau (1) , Francis Hirsch (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 Département de Mathématiques (Evry)

We give an account of results already obtained in the direction of regularity of solution of Lipschitzian SDE by Dirichlet forms methods and we present in details a new example which gives rise to an extension of the stochastic calculus. The first part introduces the framework of the Dirichlet space related to the Ornstein-Uhlenbeck semigroup on the Wiener space and recalls the absolute continuity criterion for functionals and some consequences on Lipschitz SDE. The second part is devoted to the regularity of solutions of Lipschitz SDE with respect to initial data. It is shown that the solution is differentiable in a slightly weakened sense. That gives for example the following simple result: under these hypotheses, if the initial variable Xo has a density, then X(t) has a density for all t. It is shown in the third part, that the solutions of Lipschitz SDE can be refined, by taking quasi-continuous versions for each t, into processes with continuous paths outside a polar set and unique up to a quasi- evanescent set. The main tool here is an extension of the Kolmogorov theorem on existence of continuous versions to the case where the measure is changed to a capacity.

Mots clés: stochastic differential equation; integration by part; Dirichlet form; Lipschitz condition; continuity of paths; Ornstein-Uhlenbeck semigroup; flow; capacity; quasi-continuous; Girsanov; polar set

Domaines: Analyse fonctionnelle [math.FA]; Probabilités [math.PR]

Nicolas Bouleau, Francis Hirsch. Some Results on Lipschitzian Stochastic Differential Equations by Dirichlet Forms Methods.. Classical and modern potential theory, 1993, Bonas, France. pp.125-132. ⟨hal-00452279⟩