Heat Flow on Alexandrov Spaces

Communications on Pure and Applied Mathematics - Tập 66 Số 3 - Trang 307-331 - 2013
Nicola Gigli1, Kazumasa Kuwada2, Shin‐ichi Ohta3
1University of Nice, Department of Mathematics, Valrose 06108 Nice Cedex 02, FRANCE
2Ochanomizu University, Ohtsuka 2-1-1, Graduate School of Humanities and Sciences, Bunkyo-ku, Tokyo 112-8610 JAPAN
3Kyoto University, Department of Mathematics, Kyoto 606-8502, Japan

Tóm tắt

Abstract

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐space produces the same evolution as the gradient flow of the relative entropy in the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}L^2\end{align*} \end{document}‐Wasserstein space. This means that the heat flow is well‐defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry‐Émery gradient estimates and the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\Gamma_2\end{align*} \end{document}‐condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift. © 2012 Wiley Periodicals, Inc.

Từ khóa


Tài liệu tham khảo

10.1007/978-3-540-75914-0

Ambrosio L., 2004, Topics on analysis in metric spaces

Bakry D., 1997, On Sobolev and logarithmic Sobolev inequalities for Markov semigroups, 43

10.1007/s11118-011-9232-2

10.1016/j.aim.2008.06.008

10.1515/9783110858389

Burago D., 2001, A course in metric geometry, Graduate Studies in Mathematics, 10.1090/gsm/033

Burago Yu., 1992, Alexandrov spaces with curvatures bounded below, (Russian summary) Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222; translation in Russian Math. Surveys, 47, 1

10.1007/s000390050094

10.1007/s00440-009-0199-5

10.1016/j.jfa.2008.03.006

10.1515/9783110889741

10.1007/s00526-009-0303-9

Gigli N., First variation formula in Wasserstein spaces over compact Alexandrov spaces, Canad. Math. Bull

10.1137/S0036141096303359

Juillet N.Diffusion by optimal transport in Heisenberg groups. Preprint. Available at:http://www‐irma.u‐strasbg.fr/∼juillet/papers/grad_2011.pdf

Kawabi H., 2007, The Littlewood‐Paley‐Stein inequality for diffusion processes on general metric spaces, J. Math. Sci. Univ. Tokyo, 14, 1

Korevaar N. J., 1993, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom, 1, 561, 10.4310/CAG.1993.v1.n4.a4

10.1016/j.jfa.2010.01.010

10.1007/s002090100252

10.1023/A:1011218425271

Kuwae K., 2003, Sobolev spaces and Dirichlet spaces over maps between metric spaces, J. Reine Angew. Math., 555, 39

10.5802/afst.962

10.1016/j.matpur.2007.06.003

10.1016/j.jfa.2006.10.018

10.4007/annals.2009.169.903

10.1353/ajm.0.0048

10.1002/cpa.20273

10.1007/s00205-012-0493-8

Otsu Y., 1994, The Riemannian structure of Alexandrov spaces, J. Differential Geom., 39, 629, 10.4310/jdg/1214455075

10.1081/PDE-100002243

Perelman G.DC structure on Alexandrov space with curvature bounded below. Available at:http://www.math.psu.edu/petrunin/papers/alexandrov/Cstructure.pdf

10.1090/S1079-6762-03-00120-3

Petrunin A., 2011, Alexandrov meets Lott‐Villani‐Sturm, Münster J. Math., 4, 53

Qian Z., Sharp spectral gap and Li‐Yau's estimate on Alexandrov spaces, Math. Z.

10.1017/S000497270003817X

10.1016/j.crma.2007.06.018

Sturm K.‐T., 1996, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9), 75, 273

10.1007/s11511-006-0002-8

10.1007/s11511-006-0003-7

10.1007/978-3-540-71050-9

10.1002/cpa.20060

Zhang H.‐C., 2010, On a new definition of Ricci curvature on Alexandrov spaces, Acta. Math. Sci. Ser. B Engl. Ed., 30, 1949

10.4310/CAG.2010.v18.n3.a4

Zhang H.‐C., 2012, Yau's gradient estimate on Alexandrov spaces, J. Differential Geom., 91, 445

Thông tin
Thông tin xuất bản

Communications on Pure and Applied Mathematics

Tập 66 Số 3

307-331

Thông tin tác giả