Abstract
We introduce Brownian motions on time-dependent metric measure spaces, proving their existence and uniqueness. We prove contraction estimates for their trajectories assuming that the time-dependent heat flow satisfies transport estimates with respect to every \(L^p\)-Kantorovich distance, \(p\in [1,\infty ]\). These transport estimates turn out to characterize super-Ricci flows, introduced by Sturm (J Funct Anal 275(12):3504–3569, 2015.)
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Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195(2), 289–391 (2013)
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(1), 339–404 (2015)
Arnaudon, M., Coulibaly, K.A., Thalmaier, A.: Horizontal diffusion in \(C^1\) path space. In: Séminaire de Probabilités XLIII, volume 2006 of Lecture Notes in Mathematics, pp. 73–94. Springer, Berlin (2011)
Bacher, K., Sturm, K.-T.: Ricci Bounds for Euclidean and Spherical Cones. Singular Phenomena and Scaling in Mathematical Models, pp. 3–23. Springer, Cham (2014)
Bakry, D.: Transformations de Riesz pour les semi-groupes symétriques. II. étude sous la condition \(\Gamma _2\ge 0\). In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Mathematics, pp. 145–174. Springer, Berlin (1985)
Bauer, H.: Probability theory and elements of measure theory. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. Second edition of the translation by R. B. Burckel from the third German edition, Probability and Mathematical Statistics
Bauer, H.: Wahrscheinlichkeitstheorie. de Gruyter Lehrbuch. [de Gruyter Textbook]. Walter de Gruyter & Co., Berlin, 5th edn (2002)
Bogachev, V.: Measure Theory, vol. 1. Springer, Berlin (2007)
Bolley, F., Gentil, I., Guillin, A., Kuwada, K.: Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition. Ann. Sc. Norm. Super. Pisa. Cl. Sci. (5) 18(3), 845–880 (2018)
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)
Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series, vol. 35. Princeton University Press, Princeton (2012)
Erbar, M., Kuwada, K., Sturm, K.-T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)
Nicola Gigli. Nonsmooth differential geometry-An approach tailored for spaces with Ricci curvature bounded from below. arXiv:1407.0809 (2014)
Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236(1113), vi–91 (2015)
Hamilton, R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
Haslhofer, R., Naber, A.: Weak solutions for the Ricci flow I. arXiv:1504.00911 (2015)
Kopfer, E., Sturm, K.-T.: Heat flows on time-dependent metric measure spaces and super-Ricci Flows. Commun. Pure Appl. Math. arXiv:1611.02570 (2017)
Kuwada, K.: Duality on gradient estimates and Wasserstein controls. J. Funct. Anal. 258(11), 3758–3774 (2010)
Kuwada, K., Philipowski, R.: Coupling of Brownian motions and Perelman’s L-functional. J. Funct. Anal. 260(9), 2742–2766 (2011)
Lierl, J., Saloff-Coste, L.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. arXiv:1205.6493 (2012)
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(2), 903–991 (2009)
Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)
McCann, R.J., Topping, P.M.: Ricci flow, entropy and optimal transportation. Am. J. Math. 132(3), 711–730 (2010)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159 (2002)
Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245 (2003)
Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109 (2003)
Savaré, G.: Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\(\infty \)) metric measure spaces. Discr. Cont. Dyn. Syst. A 34(4), 1641–1661 (2014)
Sturm, K.-T.: On the geometry of metric measure spaces I and II. Acta Math. 169(1), 65–131 (2006)
Sturm, K.-T.: Metric measure spaces with variable Ricci bounds and couplings of Brownian motions. In Festschrift Masatoshi Fukushima, volume 17 of Interdisciplinary Mathematical Sciences, pp. 553–575. World Science Publinsher, Hackensack (2015)
Sturm, K.-T.: Super Ricci flows for metric measure spaces, I. J. Funct. Anal. 275(12), 3504–3569 (2018)
Topping, P.: \(\cal{L}\)-optimal transportation for Ricci flow. J. Reine Angew. Math. 636, 93–122 (2009)
Villani, C.: Optimal Transport, Old and New. Springer, Berlin (2009)
von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)
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Kopfer, E. Super-Ricci flows and improved gradient and transport estimates. Probab. Theory Relat. Fields 175, 897–936 (2019). https://doi.org/10.1007/s00440-019-00904-6
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DOI: https://doi.org/10.1007/s00440-019-00904-6