Some Results on Lipschitzian Stochastic Differential Equations by Dirichlet Forms Methods

Since the impulse given by P. Malliavin, the stochastic calculus of variations has been mainly applied to stochastic differential equations with C°° coefficients, see Ocone [01] for a comprehensive exposition. But it is also important for applications to get regularity results for solutions of SDE with less smooth coefficients and in particular under Lipschitz hypotheses which axe, in dimension greater than one, the most natural hypotheses of existence and uniqueness of solutions. The celebrated integration by parts method cannot apparently be extended beyond the case of functionals in the domain VL of the Ornstein-Uhlenbeck operator (ID 2 ,2 with the notations of Watanabe [Wl]), so that the regularity of solutions of Lipschitzian SDE must come from specific technics. Especially well adapted are Dirichlet forms methods which allow to exploit intensively the fact that Lipschitz functions operate on D 2 ,i = V\/—L. We give here an account of results already obtained in this direction 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 (cf [B-Hl] [B-H2]) for functionals in ID24 or ^24 an(i 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 XQ has a density, then X t has a density for all t.