Communications in Analysis and Geometry

Volume 27 (2019)

Number 8

On the regular representation of measures

Pages: 1799 – 1823

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n8.a5

Authors

Jürgen Jost (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Rostislav Matveev (Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany)

Jacobus W. Portegies (Eindhoven University of Technology, Eindhoven, The Netherlands)

Christian S. Rodrigues (Institute of Mathematics, Universidade Estadual de Campinas, SP, Brazil)

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.

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The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 267087. C.S.R. has also been supported by the Brazilian agency: grant #2015/02230-9 and grant #2016/00332-1, S˜ao Paulo Research Foundation (FAPESP).

Received 17 November 2016

Accepted 20 August 2017

Published 21 January 2020