Non-Archimedean Analysis: A Systematic Approach to Rigid by S. Bosch, U. Güntzer, R. Remmert

By S. Bosch, U. Güntzer, R. Remmert

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Famous problems of elementary geometry: the duplication of the cube, the trisection of an angle, the quadrature of the circle: an authorized translation of F. Klein's Vorträge

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Step 1. Let u) : R>o —> Md be an absolutely continuous path. Let t > 0 and n £ N*. We consider a regular subdivision 0 = to < *i < ••• < tn = t 36 An Introduction to the Geometry of Stochastic Flows of the time interval [0,t]. ,d. If (xs)o*, 0 * ~)

2 The wave front in the Heisenberg group. , Vd) is two-step nilpotent. 4) can be written where (B^)t>o is the lift of (Bt)t>o in the group i Proof. 4). We conclude by the pathwise uniqueness property. 2(Md). The next step for us is now to generalize the previous study to any nilpotent stochastic differential equation. 3 N-step nilpotent SDE's We introduce now the notion of Carnot group. Carnot groups are to subRiemannian geometry what Euclidean spaces are to Riemannian geometry. g. [Bellaiche (1996)], [Folland and Stein (1982)], [Goodman (1976)], [Gromov (1996)]).

SDE's and Carnot Groups 43 Proof. , Ud be a basis of the basis of G. , d. We can now define a surjective Carnot group morphism n : GN(Rd) -» G by 7r(es) = ed7r(fl), # £ fljv(Kd). Observe that it defines n in a unique way because in Carnot groups the exponential map is a diffeomorphism. ,Dd). These vector fields agree at the origin with (af~'''' ' all) • ^ o m a ^ e o u r approach essentially frame independent, it is important to relate the horizontal lifts of the same Brownian motion with respect to two different basis.

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