Publication date:
9 July 2017
Source:Advances in Mathematics, Volume 314
Author(s): Ugo Boscain, Robert Neel, Luca Rizzi
On a sub-Riemannian manifold we define two types of Laplacians. The macroscopic Laplacian ${\mathrm{\Delta }}_{\omega }$, 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 by $L^{\mathbf{c}}$. 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 $\mathcal{P}$) but not a canonical choice of complement. The result depends heavily on the type of structure under investigation:
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On contact structures, for every volume ω, there exists a unique complement c such that ${\mathrm{\Delta }}_{\omega }=L^{\mathbf{c}}$. •
On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ${\mathrm{\Delta }}_H=L^{\mathbf{c}}$. However this complement is not unique in general. •
For quasi-contact structures, in general, ${\mathrm{\Delta }}_{\mathcal{P}}\ne L^{\mathbf{c}}$ for any choice of c. In particular, $L^{\mathbf{c}}$ is not symmetric with respect to Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ${\mathrm{\Delta }}_{\mathcal{P}}$ 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 $L^{\mathbf{c}}$.