Abstract: Until recently it was thought that Malliavin Calculus is a
tool to be used with stochastic equations e.g diffusions, with smooth
coefficients under hypoelliptic conditions. The method was general
enough so that it could be used for variety of other equations without
much problem. On the other hand, there are refined analytical
techniques to prove the existence of fundamental solutions for
elliptic diffusions under almost no regularity conditions. This has,
and still remains, remained the difference in both methods for a long
time. The recent efforts in the area are to reduce the smoothness
requirements when applying Malliavin Calculus to stochastic equations.
We present a general method that allows to use Malliavin Calculus for
stochastic equations with irregular drift. This method uses the
Girsanov theorem combined with Ito -Taylor expansion in order to
obtain regularity properties for the density of a hypoelliptic
additive type random variable at a fixed time. We will show the
methodology to the case of the Lebesgue integral of a diffusion. This
is joint work with A. Tanaka.