ABSTRACT
In this chapter, we shall consider random vectors defined on the probability space
( R m,B(Rm), µm
) , where µm is the standard Gaussian
measure, that is
µm(dx) = (2π)− m 2 exp
( −|x|
) dx.
We denote by Em the expectation with respect to the measure µm. Consider a random vector F : Rm → Rn. The purpose is to find sufficient conditions ensuring absolute continuity with respect to the Lebesgue measure on Rn of the probability law of F and the smoothness of the density. More precisely, we would like to obtain expressions such as (1.1). This will be done in a quite sophisticated way as a prelude to the methodology to be applied in the infinite dimensional case. For the sake of simplicity, we shall only deal with multiindices α of order one. That means that we shall only address the problem of existence of density for the random vector F .
