A nonholonomic Moser theorem and optimal transport

TL;DR

This paper extends the classical Moser theorem to nonholonomic settings, establishing a framework for volume form isotopies, mass transport, and defining a nonholonomic Wasserstein metric with applications to gradient flows.

Contribution

It introduces a nonholonomic Moser theorem, formal solutions for nonholonomic mass transport, and a Hamiltonian framework for nonholonomic optimal transport.

Findings

Proves nonholonomic Moser theorem for volume form isotopies.

Defines a nonholonomic Wasserstein metric on density spaces.

Shows subriemannian heat equation as a gradient flow in this setting.

Abstract

We prove the following nonholonomic version of the classical Moser theorem: given a bracket-generating distribution on a connected compact manifold (possibly with boundary), two volume forms of equal total volume can be isotoped by the flow of a vector field tangent to this distribution. We describe formal solutions of the corresponding nonholonomic mass transport problem and present the Hamiltonian framework for both the Otto calculus and its nonholonomic counterpart as infinite-dimensional Hamiltonian reductions on diffeomorphism groups. Finally, we define a nonholonomic analog of the Wasserstein (or, Kantorovich) metric on the space of densities and prove that the subriemannian heat equation defines a gradient flow on the nonholonomic Wasserstein space with the potential given by the Boltzmann relative entropy functional.