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

On a sub-Riemannian manifold we define two types of Laplacians. The macroscopic Laplacian Δω, 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 Lc. 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 Δω=Lc.• On Carnot groups, if H is the Haar volume, then there always exists a complement c such that ΔH=Lc. However this complement is not unique in general.• For quasi-contact structures, in general, ΔP≠Lc for any choice of c. In particular, Lc is not symmetric with respect to Popp's measure. This is surprising especially in dimension 4 where, in a suitable sense, ΔP 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 Lc.