Representation of Markov chains by random maps: existence and regularity conditions

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• 作者：Jürgen Jost ; Martin Kell…
• 关键词：Primary 37C40 ; 49K45 ; 49N60 ; Secondary 37H10 ; 37C05
• 刊名：Calculus of Variations and Partial Differential Equations
• 出版年：2015
• 出版时间：November 2015
• 年：2015
• 卷：54
• 期：3
• 页码：2637-2655
• 全文大小：500 KB
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• 作者单位：Jürgen Jost (1) (2)
  Martin Kell (1) (3)
  Christian S. Rodrigues (1)
  
  1. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103, Leipzig, Germany
  2. Department of Mathematics, University of Leipzig, 04081, Leipzig, Germany
  3. Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440, Bures-sur-Yvette, France
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Systems Theory and Control
  Calculus of Variations and Optimal Control
  Mathematical and Computational Physics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0835

文摘

We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms. Mathematics Subject Classification Primary 37C40 49K45 49N60 Secondary 37H10 37C05