Diffusions under a local strong Hörmander condition. Part I: density estimates

Vlad Bally 1, 2 Lucia Caramellino 3 Paolo Pigato 4, 5
1 MATHRISK - Mathematical Risk Handling
UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris
4 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We 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.
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Vlad Bally, Lucia Caramellino, Paolo Pigato. Diffusions under a local strong Hörmander condition. Part I: density estimates. 2016. ⟨hal-01413546⟩

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