By Fabrice Baudoin
This e-book goals to supply a self-contained advent to the neighborhood geometry of the stochastic flows. It experiences the hypoelliptic operators, that are written in Hörmander’s shape, through the use of the relationship among stochastic flows and partial differential equations.
The e-book stresses the author’s view that the neighborhood geometry of any stochastic move is set very accurately and explicitly through a common formulation often called the Chen-Strichartz formulation. The normal geometry linked to the Chen-Strichartz formulation is the sub-Riemannian geometry, and its major instruments are brought during the textual content.
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Example text
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)].