Two-Sided Bounds for Degenerate Processes with Densities Supported in Subsets of RN

We obtain bounds for the density of where (W (t) ) (ta parts per thousand yen0) is a standard Brownian motion of a"e (n) , kaa"center dot (au) is even and x=(x (1,n) ,x (n+1))aa"e (n+1). This process satisfies a weak Hormander condition but the support of its density is not the whole space. Also, the Density has various asymptotic regimes depending on the starting/final points considered (which are as well related to the number of brackets needed to span the space in Hormander's theorem). The proofs of lower and upper bounds are based on Harnack inequalities and Malliavin calculus respectively. The case of the joint law of Brownian motion and the integral of odd powers of its coordinates is also considered.