Logarithmic Estimates for the Density of Hypoelliptic Two-Parameter Diffusions

We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)∈[0, 1]2 for the Lp-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of2ε2 log ps, t(x, y), as ε↓0, where x is the initial condition of the diffusion, ε=st, and ps, t(x, y) is the density of the hypoelliptic two-parameter diffusion.