Integration by Parts and Absolute Continuity of Probability Laws

In this chapter, we give some general results on the existence of density for probability laws and properties of these densities. There are different approaches depending on whether one wishes to compute the densities —and even their derivatives-or not. The criteria proved by Malliavin in reference [33] establish existence and smoothness of the density (see Proposition 1.2). The approach by Watanabe (ref. [68]) yields —under stronger assumptions-a description of the densities and their derivatives.