An entropy generation formula on \(\varvec{RCD(K,\infty )}\) spaces

  • Ugo BessiEmail author


J. Feng and T. Nguyen have shown that the solutions of the Fokker–Planck equation in \(\mathbf{R}^d\) satisfy an entropy generation formula. We prove that, in compact metric measure spaces with the \(RCD(K,\infty )\) property, a similar result holds for curves of measures whose density is bounded away from zero and infinity. We use this fact to show the existence of minimal characteristics for the stochastic value function.

Mathematics Subject Classification

49L25 58J35 


  1. 1.
    Ambrosio, L.: Lecture Notes on Optimal Transport Problems, in Mathematical Aspects of Evolving Interfaces, LNS 1812. Springer, Berlin (2003)Google Scholar
  2. 2.
    Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with \(\sigma \)-finite measure. Trans. Am. Math. Soc. 367, 4661–4701 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows. Birkhäuser, Basel (2005)zbMATHGoogle Scholar
  4. 4.
    Ambrosio, L., Gigli, N., Savaré, G.: Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case. Analysis and Numerics of Partial Differential Equations, pp. 63–115. Springer, Milano (2013)zbMATHGoogle Scholar
  5. 5.
    Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces (2017). (preprint) Google Scholar
  6. 6.
    Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163–7, 1405–1490 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bessi, U.: The stochastic value function in metric measure spaces. Discrete Contin. Dyn. Syst. 37–4, 1839–1919 (2017)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Brezis, H.: Analisi Funzionale. Liguori, Napoli (1986)Google Scholar
  10. 10.
    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces, GAFA. Geom. Funct. Anal. 9, 428–517 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feng, J., Nguyen, T.: Hamilton–Jacobi equations in space of measures associated with a system of conservation laws. Journal de Mathématiques pures et Appliquées 97, 318–390 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fleming, W.H.: The cauchy problem for a nonlinear first order partial differential equation. JDE 5, 515–530 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gigli, N., Han, B.: The continuity equation on metric measure spaces. Calc. Var. Partial Differ. Equ. 53, 149–177 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lisini, S.: Characterisation of absolutely continuous curves in Wasserstein space. Calc. Var. Partial Differ. Equ. 28, 85–120 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)zbMATHGoogle Scholar
  17. 17.
    Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità Roma TreRomaItaly

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