Diffusions under a local strong Hörmander condition. Part II: tube estimates
Abstract
We study lower and upper bounds for the probability that a diffusion process in R^n remains in a tube around a skeleton path up to a fixed time. We assume that the diffusion coefficients σ_1 ,. .. , σ_d may degenerate but they satisfy a strong Hörmander condition involving the first order Lie brackets around the skeleton of interest. The tube is 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 ]. The proof consists in a concatenation technique which strongly uses the lower and upper bounds for the density proved in the part I.
Domains
Probability [math.PR]
Origin : Files produced by the author(s)
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