Abstract
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.
Article PDF
Similar content being viewed by others
References
Ambrosio, L.: Lecture Notes on Optimal Transport Problems, in Mathematical Aspects of Evolving Interfaces, LNS 1812. Springer, Berlin (2003)
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)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows. Birkhäuser, Basel (2005)
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)
Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces (2017). (preprint)
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163–7, 1405–1490 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)
Bessi, U.: The stochastic value function in metric measure spaces. Discrete Contin. Dyn. Syst. 37–4, 1839–1919 (2017)
Brezis, H.: Analisi Funzionale. Liguori, Napoli (1986)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces, GAFA. Geom. Funct. Anal. 9, 428–517 (1999)
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)
Fleming, W.H.: The cauchy problem for a nonlinear first order partial differential equation. JDE 5, 515–530 (1969)
Gigli, N., Han, B.: The continuity equation on metric measure spaces. Calc. Var. Partial Differ. Equ. 53, 149–177 (2015)
Lisini, S.: Characterisation of absolutely continuous curves in Wasserstein space. Calc. Var. Partial Differ. Equ. 28, 85–120 (2007)
Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)
Nelson, E.: Dynamical Theories of Brownian Motion. Princeton University Press, Princeton (1967)
Villani, C.: Topics in Optimal Transportation. American Mathematical Society, Providence (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Work partially supported by the PRIN2009 Grant “Critical Point Theory and Perturbative Methods for Nonlinear Differential Equations”.
Rights and permissions
About this article
Cite this article
Bessi, U. An entropy generation formula on \(\varvec{RCD(K,\infty )}\) spaces. Nonlinear Differ. Equ. Appl. 25, 26 (2018). https://doi.org/10.1007/s00030-018-0518-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-018-0518-6