Tube estimates for diffusions under a local strong Hörmander condition

International audienceWe study lower and upper bounds for the density of a diffusion process in R n in a small (but not asymptotic) time, say δ. We assume that the diffusion coefficients σ 1 ,. .. , σ d may degenerate at the starting time 0 and point x 0 but they satisfy a strong Hörmander condition involving the first order Lie brackets. The density estimates are written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time δ, the diffusion process propagates with speed √ δ in the direction of the diffusion vector fields σ j and with speed δ = √ δ × √ δ in the direction of [σ i , σ j ]. In the second part of this paper, such estimates will be used in order to study lower and upper bounds for the probability that the diffusion process remains in a tube around a skeleton path up to a fixed time