Let H be a dense Hilbertian subspace of a l.c. H.s. X. The stochastic calculus of variations relative to a Gaussian measure g on X with reproducing space H can be extended in the vectorial case and if g is replaced by an \(H-C^{\infty}\) probability measure M on X. This means that for all j, the absolute derivative \(\nabla^ jM\) induces a polynomial function \(C_ j:H\to \cap_{p}L^ p_ M(X).\)
Theorem 1 gives the H-quasi-invariance of M if: \[ (C)\quad \forall p\in]1,\infty [,\quad \forall h\in H,\quad \sum^{\infty}_{j=0}\| C_ j.h^ j\|_ p/j!<\infty. \] Theorem 2 gives the inclusion: \[ (IP)\quad W^{p,k+1}(X\times Y,G)\subset W^{p,k}(X,W^{2,\ell}(Y,G)). \] Assuming the continuity of divergence, theorem 3 gives the smoothness of laws and the lifting of temperate distributions.