0. 11 that VA > c, limp t—oD kiN and ft-iN are bounded in probability when ( sup s Pit2 I RN (s) Aa . At) < exp — 0 Almost all what follows will be generalized in the next section. Nevertheless, we believe that the understanding of the geometry of the free 2-step Carnot group is much more easy to get than the geometry of general Carnot groups and however contains the most important ideas of sub-Riemannian geometry. Let d > 2 and denote ASd the space of d x d skew-symmetric matrices. We consider the group G2 (Rd) defined in the following way G2 (Rd ) = (Rd x ASd, 0) where ® is the group law defined by (4:11,w1) 0 (a2 1 w2) = (cEi +2 1 1 + w2 + —al A). From this, we conclude that V = 0 and thus that C is invertible with probability one. Therefore, the random variable XT° admits a density with respect to the Lebesgue measure. Step 3. To show that the density of for every p> 1, E ( XT° is smooth, we have to show that 1 ) I det r 1P <+00. i. 10-1 t ) t >0; see [Malliavin (1997)], pp. 240, for further details. These equations make easy the proof of the fact that for every p> 1, 1 <+00. I det Jo,i 12P) E Therefore, it remains to show that for every p> 1, E ( 1 ) <+00.
An Introduction to the Geometry of Stochastic Flows by Fabrice Baudoin
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