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Estimation en temps petit de la densité d’une diffusion hypoelliptique (Estimation in small time of the density of a hypoelliptic diffusion). (French) Zbl 0585.60075
Let \(P_ t\) e a semigroup with generator \(L=2^{- 1}\sum^{m}_{i=1}X^ 2_ i+X_ 0\), where \(X_ i\), \(0\leq i\leq m\), are vector fields in \({\mathbb{R}}^ d\). Let \(p_ t(x,y)\) be the density of the semigroup. Under some conditions it is proved that uniformly on every compact \[ \lim_{t\to o}2t Log p_ t(x,y)=-d^ 2(x,y), \] where d(\(\cdot,\cdot)\) is an appropriately constructed Riemannian distance. Stochastic differential equations of Stratonovich type and Malliavin calculus are involved.
Reviewer: O.Enchev

60J60 Diffusion processes
60F10 Large deviations