Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry
Résumé
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 geodesic random walk. 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 w.r.t. 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 smaller or equal than 4, and in particular in the 4-dimensional quasi-contact structure mentioned above.
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