On the regular representation of measures

TL;DR

This paper establishes conditions under which families of probability measures on Riemannian manifolds with boundary can be represented by smooth random maps, even when densities vanish at the boundary, and discusses obstructions to such representations.

Contribution

It provides new sufficient conditions for regular representation of measures by smooth maps on manifolds with boundary, including cases with zero densities at the boundary.

Findings

Sufficient conditions for $C^k$-regular representation of measures.

Identification of two obstructions to regular representability.

Applicability to measures with densities approaching zero at the boundary.

Abstract

We give sufficient conditions for a parametrised family of probability measures on a Riemannian manifold with boundary to be represented by random maps of class $C^k$. The conditions allow for the probability densities to approach zero towards the boundary of the manifold. We also formulate two obstructions to regular representability.