By Fabrice Baudoin

This ebook goals to supply a self-contained advent to the neighborhood geometry of the stochastic flows. It experiences the hypoelliptic operators, that are written in Hormander'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 circulation is decided very accurately and explicitly via a common formulation known as the Chen-Strichartz formulation. The normal geometry linked to the Chen-Strichartz formulation is the sub-Riemannian geometry, and its major instruments are brought in the course of the textual content.

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**Example text**

And moreover that Tv, commutes with the dilations A c , e> O. 0 The maps Tv give the formulas for a change of basis in the theory of free Carnot groups. Notice that without the freeness assumption, Tv, may fail to exist. 9 Notice that the map Aut (Rd) is a group morphism. e. every commutator constructed from the Vi 's with length greater than N is O. 1) can be written Xf° = F(x0,B:), where (Bn t >0 is the lift of (Bt )t>0 in the group GN(Rd) Proof. 3. 1). „,ik ) The definition of GN(Rd ) shows that we can therefore write XT° = F(xo, Br).

22 An Introduction to the Geometry of Stochastic Flows Even though this is only a formal development, it clearly shows how the dependance between B and Xx° is related to the structure of the Lie algebra e generated by the vector fields Ws. If we want to understand more deeply how the properties of this Lie algebra determine the geometry of Xx°, it is wiser to begin with the simplest cases. In a way, the most simple Lie algebras are the nilpotent ones. 1) can be represented from the lift of the Brownian motion (B) >o in a graded free nilpotent Lie group with dilations.

Therefore lln is a Carnot group of depth 2. We consider throughout this section a Lie group G which satisfies the hypothesis of the above definition. Notice that the vector space V1, which is called the basis of G, Lie generates g, where g denotes the Lie algebra of G. Since G is step N nilpotent and simply connected, the exponential map is a diffeomorphism and the Baker-Campbell-Hausdorff formula therefore completely characterizes the group law of G because for U,V E g, exp U exp V = exp (P(U,V)) for some universal Lie polynomial P whose first terms are given by P(U,V) =U +V +1[U,V1+ — 112 HU,V1,V1 — —112 [[U,V],U] 418 gr, [17, gr, viii + • • • .