Two-Sided Bounds for Degenerate Processes with Densities Supported in Subsets of ℝ ^N

We obtain bounds for the density of (Formula presented) where (W_t)_t≥0 is a standard Brownian motion of ℝⁿ, k∈ℕ^∗ is even and x=(x_1,n,x_n+1)∈ℝⁿ⁺¹. This process satisfies a weak Hörmander 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 Hörmander’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.