Density and tube estimates for diffusion processes under Hormander-type conditions

We recall some classic results on the regularity of solutions of stochastic differential equations. Then we consider two specific diffusion processes satisfying hypoellipticity conditions of Hormander type. Using Malliavin Calculus techniques recently developed to deal with degenerate problems, we find estimates for the density of the law of the solution, which we use to prove exponential bounds for the probability that the diffusion remains in a small tube, around a deterministic path, up to a given time. We then present some work in progress on asymptotic sharp estimates for the density and its derivatives for a similar, higher dimensional system.