Sub-Laplacians on Sub-Riemannian Manifolds

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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“…Plugging the explicit expression of g into Eq. (16) gives the condition of our statement. Uniqueness follows from Lemma 57.…”

Section: Integrability Conditionsmentioning

confidence: 96%

“…Remark 16. The compatibility condition χ (c,ω) = 0 is the same appearing in [16,Theorem 5.13], written in a different form. We call compatible the pairs (c, ω) solving the compatibility condition.…”

Section: The Equivalence Problemmentioning

confidence: 99%

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

Boscain

Neel²,

Rizzi³

2017

Advances in Mathematics

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“…where the X i are C ∞ vector fields. We refer the reader to [15] for more details on operators of the form given in (3.10) in the context of sub-Riemannian manifolds. Consider the following assumption.…”

Section: 21mentioning

confidence: 99%

On the Cheng-Yau gradient estimate for Carnot groups and sub-Riemannian manifolds

Baudoin

Gordina

Mariano

2019

Proc. Amer. Math. Soc.

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“…Once the measure is chosen, an operator defined as a divergence of the horizontal gradient is the usual sum of squares operator. Note that the argument in [1] has a mistake, and for more detailed discussion of related issues we refer to [16,17]. But no matter which point of view we use, in the case of a nilpotent Lie group all these approaches give the same result: the sum of squares operator.…”

Section: Hypoelliptic Heat Equation On a Sub-riemannian Manifoldmentioning

confidence: 99%

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

Asaad

Gordina

2016

Potential Anal

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The starting point of our analysis is an old idea of writing an eigenfunction expansion for a heat kernel considered in the case of a hypoelliptic heat kernel on a nilpotent Lie group G. One of the ingredients of this approach is the generalized Fourier transform. The formula one gets using this approach is explicit as long as we can find all unitary irreducible representations of G. In the current paper we consider an n-step nilpotent Lie group Gn as an illustration of this techinique. First we apply Kirillov's orbit method to describe these representations for Gn. This allows us to write the corresponding hypoelliptic heat kernel using an integral formula over a Euclidean space. As an application, we describe a short-time behaviour of the hypoelliptic heat kernel in our case.

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Sub-Laplacians on Sub-Riemannian Manifolds

Cited by 32 publications

References 12 publications

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

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

On the Cheng-Yau gradient estimate for Carnot groups and sub-Riemannian manifolds

Hypoelliptic Heat Kernels on Nilpotent Lie Groups

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