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Carnot groups have their own concept of differentiation. In order to present this concept introduced in [Pansu (1989)], we first have to define a concept of linear maps between two Carnot groups. 41 SDE's and Carnot Groups Let Gi and G2 be two Carnot groups with Lie algebras fli and g2. A Lie group morphism 4> : Gi —> G2 is said to be a Carnot group morphism if for any t > 0, g £ G\, where A Gl (resp. AG2) denote the canonical dilations on Gi (resp. G2). In the same way, a Lie algebra morphism a : gi —> 92 is said to be a Carnot algebra morphism if for any t > 0, x e gi, Cxi V* dj \ — u* ix\djy, where 5Gl (resp.
The number M(e) of these balls has the lower bound: We deduce that for every s > 0, which shows that if s < D then the Hausdorff s-dimensional measure of • B5(0,1) is +00. 6 Observe, and this is typical in sub-Riemannian geometry, that the Hausdorff dimension of G is therefore strictly greater than the topological dimension. Carnot groups have their own concept of differentiation. In order to present this concept introduced in [Pansu (1989)], we first have to define a concept of linear maps between two Carnot groups.
These groups called the free Carnot groups are introduced and their geometries are discussed. When the Lie algebra £ is not nilpotent, this representation does not hold anymore but provides a good approximation for Xx° in small times. We conclude the chapter with an introduction to the rough paths theory of [Lyons (1998)], in which Carnot groups also play a fundamental role. e. that [V;,Vj] = 0 for 1 < i,ji < d. This is therefore the simplest possible case: it has been first studied by [Doss (1977)] and [Siissmann (1978)].