## U. Boscain - R. Neel - L. Rizzi

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

created by rizzi1 on 08 Feb 2016

modified on 15 May 2017

[

BibTeX]

*Accepted Paper*

**Inserted:** 8 feb 2016

**Last Updated:** 15 may 2017

**Journal:** Adv. Math.

**Year:** 2015

**Abstract:**

On a sub-Riemannian manifold we define two type of Laplacians. The
\emph{macroscopic Laplacian} $\Delta_\omega$, as the divergence of the
horizontal gradient, once a volume $\omega$ is fixed, and the \emph{microscopic
Laplacian}, as the operator associated with a sequence of geodesic random
walks. We consider a general class of random walks, where \emph{all}
sub-Riemannian geodesics are taken in account. This operator depends only on
the choice of a complement $\mathbf{c}$ 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 $\omega$, there exists a
unique complement $c$ such that $\Delta_\omega=L^c$. On Carnot groups, if $H$
is the Haar volume, then there always exists a complement $c$ such that
$\Delta_H=L^c$. However this complement is not unique in general. For
quasi-contact structures, in general, $\Delta_P \neq L^c$ 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, $\Delta_P$ 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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