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.