A convergence to Brownian motion on sub-Riemannian manifolds

Gordina, Maria; Laetsch, Thomas

doi:10.1090/tran/6831

Cited by 18 publications

(26 citation statements)

References 21 publications

(20 reference statements)

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“…Another approach is to choose a k-dimensional linear subspace h q in T * q M transverse to the non compact directions of the cylinder (playing the role of the "most horizontal" geodesics) and to average on h q ∩ q . This last approach is essentially the one followed in [16,15,17,18]. Certainly the problem of finding a canonical subspace h q in T * q M is a non-trivial question and in general it is believed that it is not possible.…”

Section: The Sub-riemannian Microscopic Laplacianmentioning

confidence: 99%

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

Boscain

Neel²,

Rizzi³

2017

Advances in Mathematics

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On a sub-Riemannian manifold we define two type of Laplacians. The macroscopic Laplacian ∆ω, as the divergence of the horizontal gradient, once a volume ω is fixed, and the microscopic Laplacian, as the operator associated with a sequence of geodesic random walks. We consider a general class of random walks, where all sub-Riemannian geodesics are taken in account. This operator depends only on the choice of a complement c to the sub-Riemannian distribution, and is denoted L c .We address the problem of equivalence of the two operators. This problem is interesting since, on equiregular sub-Riemannian manifolds, there is always an intrinsic volume (e.g. Popp's one P) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation:• On contact structures, for every volume ω, there exists a unique complement c such that ∆ω = L c .• On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ∆H = L c . However this complement is not unique in general.• For quasi-contact structures, in general, ∆P = L c for any choice of c. In particular, L c is not symmetric with respect to Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ∆P is the unique intrinsic macroscopic Laplacian.A crucial notion that we introduce here is the N-intrinsic volume, i.e. a volume that depends only on the set of parameters of the nilpotent approximation. When the nilpotent approximation does not depend on the point, a N-intrinsic volume is unique up to a scaling by a constant and the corresponding N-intrinsic sub-Laplacian is unique. This is what happens for dimension less than or equal to 4, and in particular in the 4-dimensional quasi-contact structure mentioned above. Finally, we prove a general theorem on the convergence of families of random walks to a diffusion, that gives, in particular, the convergence of the random walks mentioned above to the diffusion generated by L c .

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Section: The Sub-riemannian Microscopic Laplacianmentioning

confidence: 99%

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

Boscain

Neel²,

Rizzi³

2017

Advances in Mathematics

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“…The operator L V has been introduced in [7], where it is shown that L V is the generator of a process which is the limit of a naturally constructed horizontal random walk. The operator L V can be viewed as the generator of a horizontal Brownian motion on M , the role played by the Laplace-Beltrami operator on Riemannian manifolds.…”

Section: The Operator L Vmentioning

confidence: 99%

“…Our approach is to compare two operators on a sub-Riemannian manifold M that can be thought of as geometrically canonical to the sub-Riemannian structure we have on M . One of these operators, L V , is a sub-Laplacian we constructed previously in [7]. The advantage of this construction is that while it is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure.…”

Section: Introductionmentioning

confidence: 99%

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Sub-Laplacians on Sub-Riemannian Manifolds

Gordina

Laetsch

2016

Potential Anal

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Abstract. We consider different sub-Laplacians on a sub-Riemannian manifold M . Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a sub-Laplacian we constructed previously in [7]. This operator is canonical with respect to the horizontal Brownian motion, we are able to define the sub-Laplacian without some a priori choice of measure. The other operator is div ω grad H for some volume form ω on M . We illustrate our results by examples of three Lie groups equipped with a sub-Riemannian structure: SU (2), the Heisenberg group and the affine group.

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“…(3) A sub-Riemannian example is given by the Heisenberg group H, [10,27,38,41], realized as R 3 together with the non-commutative multiplication…”

Section: Finite Energy Coordinatesmentioning

confidence: 99%

Finite Energy Coordinates and Vector Analysis on Fractals

Hinz

Teplyaev

2015

Fractal Geometry and Stochastics V

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Abstract. We consider (locally) energy finite coordinates associated with a strongly local regular Dirichlet form on a metric measure space. We give coordinate formulas for substitutes of tangent spaces, for gradient and divergence operators and for the infinitesimal generator. As examples we discuss Euclidean spaces, Riemannian local charts, domains on the Heisenberg group and the measurable Riemannian geometry on the Sierpinski gasket.

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A convergence to Brownian motion on sub-Riemannian manifolds

Cited by 18 publications

References 21 publications

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry

Sub-Laplacians on Sub-Riemannian Manifolds

Finite Energy Coordinates and Vector Analysis on Fractals

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