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

Abstract

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 .

Short Geodesics Losing Optimality in Contact Sub-Riemannian Manifolds and Stability of the 5-Dimensional Caustic

“…Recall the power series notation f (ητ, h(0)) = η k f (k) (τ, h(0)). As a consequence of Proposition 2.1, we can give a new expansion of the Hamiltonian flow for the special class of initial covectors of type (15) in terms of coefficients of the power series of x, z, h, w. (Recall that for all R > 0, B R denotes the set…”

Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling

“…26]. First note that a random walk q ε t as described here corresponds to a random walk X h t in the notation of [10], with h = ε 2 /2k, and with each step being given either by a continuous curve (which may or may not be a geodesic), as addressed in Remark 26. Every random walk in our class has the property that, during any step, the path never goes more than distance κε from the starting point of the step for some fixed κ > 0, by construction, and this immediately shows that every random walk in our class satisfies Eq.…”

“…On the stochastic side, in Section 2, we introduce a general scheme for the convergence of random walks of a sufficiently general class to include all our constructions, based on the results of [10]. Further, in the process of developing the random walks just described, we naturally obtain an intuitively appealing description of the solution to a Stratonovich SDE on a manifold as a randomized flow along the vector fields V 1 , .…”

On the Essential Self-Adjointness of Singular Sub-Laplacians

“…The Popp sub-Laplacian is not the only intrinsic second order diffusion operator associated with a sub-Riemannian structure. See for example [10,4,20] for intrinsic operators associated with sub-Riemannian random walks. Other possible sub-Laplacians are related with different choices of intrinsic measures.…”