# Seminar on Stochastic Analysis, Random Fields and Applications VIEntropic Measure on Multidimensional Spaces

Seminar on Stochastic Analysis, Random Fields and Applications VI: Entropic Measure on... [We construct the entropic measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{P}^\beta$$ \end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{C} : \mathcal{P}(M) \longrightarrow \mathcal{P}(M).$$ \end{document} 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.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# Seminar on Stochastic Analysis, Random Fields and Applications VIEntropic Measure on Multidimensional Spaces

Part of the Progress in Probability Book Series (volume 63)
Editors: Dalang, Robert; Dozzi, Marco; Russo, Francesco
16 pages

/lp/springer-journals/seminar-on-stochastic-analysis-random-fields-and-applications-vi-h6oDPrPKpL

# References (19)

Publisher
Springer Basel
© Springer Basel AG 2011
ISBN
978-3-0348-0020-4
Pages
261 –277
DOI
10.1007/978-3-0348-0021-1_17
Publisher site
See Chapter on Publisher Site

### Abstract

[We construct the entropic measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{P}^\beta$$ \end{document} 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{C} : \mathcal{P}(M) \longrightarrow \mathcal{P}(M).$$ \end{document} 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.]

Published: Feb 4, 2011

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

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