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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

##### MSC:
 60J60 Diffusion processes 60F10 Large deviations