Quasi-everywhere regularity of solutions of Brownian stochastic differential equations and their flows
The Wiener space being equipped with the Dirichlet form and the capacity induced by the Ornstein-Uhlenbeck semi-group, we show, through an extension of Kolmogorov's criterion, that the solutions of Lipschitzian S.D.E. admit quasi-continuous versions which are continuous processes outside a polar set. The properties of the flows of S.D.E. (homeomorphism, differentiability) are then established for such quasi- continuous versions under suitable hypotheses. Finally the solutions of S.D.E. are studied under measures which do not charge polar sets and an example of such a measure is given.
scientific article; zbMATH DE number 4054730