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Entropic Measure on Multidimensional Spaces

  • Karl-Theodor SturmEmail author
Conference paper
Part of the Progress in Probability book series (PRPR, volume 63)

Abstract

We construct the entropic measure \(\mathbb{P}^\beta\) on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (a random probability measure, well-known to exist on spaces of any dimension) under the conjugation map
$$\mathfrak{C} : \mathcal{P}(M) \longrightarrow \mathcal{P}(M).$$
This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher-dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of μ and C(μ) are inverse to each other.
We also present a heuristic interpretation of the entropic measure as
$$d \mathbb{P}^\beta(\mu) = \frac{1}{\rm Z} {\rm exp} (- \beta \cdot {\rm Ent}(\mu | m)) \cdot d \mathbb{P}^0(\mu).$$

Keywords

Optimal transport entropic measure Wasserstein space entropy gradient flow Brenier map Dirichlet distribution random probability measure 

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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikBonnGermany

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