Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces

TL;DR

This paper establishes a general criterion for the density of Lipschitz subalgebras in metric Sobolev spaces and applies it to Wasserstein Sobolev spaces, demonstrating their Hilbertian structure and providing explicit gradient characterizations.

Contribution

It introduces a new criterion for Lipschitz subalgebra density in metric Sobolev spaces and applies it to Wasserstein spaces, showing their Hilbertian nature and explicit gradient formulas.

Findings

Wasserstein Sobolev spaces are always Hilbertian.

The Cheeger energy in these spaces is a Dirichlet form.

Explicit characterization of the Wasserstein gradient and calculus rules.

Abstract

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a separable and complete metric space $(X,\mathsf{d})$. We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,2}(\mathcal{P}_2(\mathbb{M}),W_{2},\mathfrak{m})$ arising from a positive and finite Borel measure $\mathfrak{m}$ on the Kantorovich-Rubinstein-Wasserstein space $(\mathcal{P}_2(\mathbb{M}),W_{2})$ of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space $\mathbb{M}$. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure $\mathfrak{m}$ so that the resulting Cheeger energy is a Dirichlet form. We will eventually provide an explicit characterization for the corresponding notion of $\mathfrak{m}$-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the $\Gamma$-calculus inherited from the Dirichlet form.