Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below View Full Text


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Article Info

DATE

2014-02

AUTHORS

Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré

ABSTRACT

This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces . Our main results are: A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).The equivalence of the heat flow in generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures .The proof of density in energy of Lipschitz functions in the Sobolev space .A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality. A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d). The equivalence of the heat flow in generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures . The proof of density in energy of Lipschitz functions in the Sobolev space . A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. More... »

PAGES

289-391

References to SciGraph publications

  • 1998-09. Quasiconformal maps in metric spaces with controlled geometry in ACTA MATHEMATICA
  • 1999-06. Differentiability of Lipschitz Functions on Metric Measure Spaces in GEOMETRIC AND FUNCTIONAL ANALYSIS
  • 2006-07. On the geometry of metric measure spaces in ACTA MATHEMATICA
  • 2012-07. Local Poincaré inequalities from stable curvature conditions on metric spaces in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2004-03. Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space in PROBABILITY THEORY AND RELATED FIELDS
  • 2010-03. Wasserstein space over the Wiener space in PROBABILITY THEORY AND RELATED FIELDS
  • 2009-10. Finsler interpolation inequalities in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2009. Optimal Transport, Old and New in NONE
  • 2006-07. On the geometry of metric measure spaces. II in ACTA MATHEMATICA
  • 2010-09. On the heat flow on metric measure spaces: existence, uniqueness and stability in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • 2013. A User’s Guide to Optimal Transport in MODELLING AND OPTIMISATION OF FLOWS ON NETWORKS
  • 2007. Measure Theory in NONE
  • 2009-11. Existence and stability for Fokker–Planck equations with log-concave reference measure in PROBABILITY THEORY AND RELATED FIELDS
  • 2007-01. Characterization of absolutely continuous curves in Wasserstein spaces in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00222-013-0456-1

    DOI

    http://dx.doi.org/10.1007/s00222-013-0456-1

    DIMENSIONS

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    37 schema:description This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces . Our main results are: A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d).The equivalence of the heat flow in generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures .The proof of density in energy of Lipschitz functions in the Sobolev space .A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper. Our results apply in particular to spaces satisfying Ricci curvature bounds in the sense of Lott and Villani (Ann. Math. 169:903–991, 2009) and Sturm (Acta Math. 196: 65–131, 2006, and Acta Math. 196:133–177, 2006) and require neither the doubling property nor the validity of the local Poincaré inequality. A general study of the relations between the Hopf–Lax semigroup and Hamilton–Jacobi equation in metric spaces (X,d). The equivalence of the heat flow in generated by a suitable Dirichlet energy and the Wasserstein gradient flow of the relative entropy functional in the space of probability measures . The proof of density in energy of Lipschitz functions in the Sobolev space . A fine and very general analysis of the differentiability properties of a large class of Kantorovich potentials, in connection with the optimal transport problem, is the fourth achievement of the paper.
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