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) is an element of [0, 1](2) for the L-p-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of 2epsilon(2) log p(s, t) (x, y), as epsilon down arrow 0, where x is the initial condition of the diffusion, epsilon = rootst, and p(s,t) (x,y) is the density of the hypoelliptic two-parameter diffusion. (C) 2002 Elsevier Science (USA) .