The differentiation of hypoelliptic diffusion semigroups

Marc Arnaudon (1) , Anton Thalmaier (2)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]
2 Institute of Mathematics

Basic derivative formulas are presented for hypoelliptic heat semigroups and harmonic functions extending earlier work in the elliptic case. Emphasis is placed on developing integration by parts formulas at the level of local martingales. Combined with the optional sampling theorem, this turns out to be an efficient way of dealing with boundary conditions, as well as with finite lifetime of the underlying diffusion. Our formulas require hypoellipticity of the diffusion in the sense of Malliavin calculus (integrability of the inverse Malliavin covariance) and are formulated in terms of the derivative flow, the Malliavin covariance and its inverse. Finally some extensions to the nonlinear setting of harmonic mappings are discussed.

Mots clés: Diffusion semigroup; hypoelliptic operator; integration by parts; Malliavin calculus; Malliavin covariance

Domaines: Probabilités [math.PR]

Marc Arnaudon, Anton Thalmaier. The differentiation of hypoelliptic diffusion semigroups. Illinois Journal of Mathematics, 2010, 54, pp.1285-1311. ⟨hal-00472696⟩