W-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds Dedicated to Professor Zhiming Ma on His Seventieth Birthday
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摘要
In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.
In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.
In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman's seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K, m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a(K, m)-super Ricci flow, where K ∈ R and m ∈ [n, ∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds.Our result improves an important result due to Lott and Villani(2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.
引文
[1] Anderson G,Guionnet A,Zeitouni O.An Introduction to Random Matrices.Cambridge:Cambridge University Press,2010
2 Bakry D,Emery M.Diffusion hypercontractives.In:S′eminaire de probabilit′es,XIX.Lecture Notes in Mathematics,vol.1123.Berlin:Springer-Verlag,1985,177–206
3 Bakry D,Ledoux M.A logarithmic Sobolev form of the Li-Yau parabolic inequality.Rev Mat Iberoam,2006,22:683 –702
4 Baudoin F,Garofalo N.Perelman’s entropy and doubling property on Riemannian manifolds.J Geom Anal,2011,21:1119–1131
5 Benamou J-D,Brenier Y.A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numer Math,2000,84:375–393
6 Bismut J-M.The hypoelliptic Laplacian on the cotangent bundle.J Amer Math Soc,2005,18:379–476
7 Bismut J-M.Hypoelliptic Laplacian and orbital integrals.Annals of Mathematics Studies,vol.177.Princeton:Princeton University Press,2011
8 Boltzmann L.Weitere Studien ber das W¨armegleichgewicht unter Gasmolek¨ulen.Wiener Berichte,1872,66:275–370
9 Boltzmann L.ber die beziehung dem zweiten Haubtsatze der mechanischen W¨armetheorie und der Wahrscheinlichkeitsrechnung respektive den S¨atzen¨uber das W¨armegleichgewicht.Wiener Berichte,1877,76:373–435
10 Brenier Y.Polar factorization and monotone rearrangement of vector-valued functions.Comm Pure Appl Math,1991,44 :375–417
11 Clausius R.ber die W¨armeleitung gasfrmiger Krper.Ann Phys(8),1865,125:353–400
12 Ecker K.A formula relating entropy monotonicity to Harnack inequalities.Comm Anal Geom,2007,15:1025–1061
13 Evans L.Lecture notes on entropy and partial differential equations.Http://math.berkeley.edu/～evans/entropy.and.PDE.pdf
14 Gawedzki K.Lectures on Conformal Field Theory.Quantum Fields and Strings:A Course for Mathematicians,vol.2 .Providence:Amer Math Soc,1999
15 Hamilton R S.Three manifolds with positive Ricci curvature.J Differential Geom,1982,17:255–306
16 Hamilton R S.The formation of singularities in the Ricci flow.In:Surveys in Differential Geometry,vol.2.Boston:International Press,1995,7–136
17 Kotschwar B,Ni L.Gradient estimate for p-harmonic functions,1/H flow and an entropy formula.Ann Sci′Ec Norm Sup′er(4),2009,42:1–36
18 Li J,Xu X.Differential Harnack inequalities on Riemannian manifolds I:Linear heat equation.Adv Math,2011,226:4456–4491
19 Li P,Yau S-T.On the parabolic kernel of the Schrdinger operator.Acta Math,1986,156:153–201
20 Li S.W-entropy formulas on super Ricci flows and matrices Dirichlet processes.Ph D Thesis.Shanghai:Fudan University;Toulouse:Universit′e Paul Sabatier,2015
21 Li X-D.Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds.J Math Pures Appl(9),2005,84:1295–1361
22 Li X-D.On Perelman’s W-entropy functional on Riemannian manifolds with weighted volume measure.Included in Th`ese d’Habilitation`a Diriger des Rechercher.Toulouse:Universit′e Paul Sabatier,2007
23 Li X-D.Perelman’s W-entropy for the Fokker-Planck equation over complete Riemannian manifolds.Bull Sci Math,2011,135:871–882
24 Li X-D.Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature.Math Ann,2012,353:403–437
25 Li X-D.From the Boltzmann H-theorem to Perelman’s W-entropy formula for the Ricci flow.In:Emerging Topics on Differential Equations and Their Applications.Nankai Series in Pure,Applied Mathematics and Theoretical Physics,vol.10.Hackensack:World Scientific,2013,68–84
26 Li X-D.Hamilton’s Harnack inequality and the W-entropy formula on complete Riemannian manifolds.Stochastic Process Appl,2016,126:1264–1283
27 Li S,Li X-D.W-entropy formula for the Witten Laplacian on manifolds with time-dependent metrics and potentials.Pacific J Math,2015,278:173–199
28 Li S,Li X-D.Harnack inequalities and W-entropy formula for Witten Laplacian on Riemannian manifolds with K-super Perelman Ricci flow.Ar Xiv:1412.7034,2014
29 Li S,Li X-D.W-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds.Ar Xiv:1604.02596,2016
30 Li S,Li X-D.On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows.Asian J Math,2018,in press
31 Li S,Li X-D.W-entropy,super Perelman Ricci flows and(K,m)-Ricci solitons.Ar Xiv:1706.07040,2017
32 Li S,Li X-D.Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds.J Funct Anal,2018,274:3263–3290
33 Li S,Li X-D,Xie Y-X.Generalized Dyson Brownian motion,Mc Kean-Vlasov equation and eigenvalues of random matrices.Ar Xiv:1303.1240,2013
34 Li S,Li X-D,Xie Y-X.On the law of large numbers for empirical measure process of generalized Dyson Brownian motion.Ar Xiv:1407.7234,2014
35 Lott J.Some geometric calculations on Wasserstein space.Comm Math Phys,2008,277:423–437
36 Lott J.Optimal transport and Perelman’s reduced volume.Calc Var Partial Differential Equations,2009,36:49–84
37 Lott J,Villani C.Ricci curvature for metric-measure spaces via optimal transport.Ann of Math(2),2009,169:903 –991
38 Lu P,Ni L,Vazquez J-J,et al.Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds.J Math Pures Appl(9),2009,91:1–19
39 Mc Cann R.Polar factorization of maps on Riemannian manifolds.Geom Funct Anal,2001,11:589–608
40 Mc Cann R,Topping P.Ricci flow,entropy and optimal transpotation.Amer J Math,2010,132:711–730
41 Nash J.Continuity of solutions of parabolic and elliptic equations.Amer J Math,1958,80:931–954
42 Ni L.The entropy formula for linear equation.J Geom Anal,2004,14:87–100
43 Ni L.Addenda to“The entropy formula for linear equation”.J Geom Anal,2004,14:329–334
44 Otto F.The geometry of dissipative evolution equations:The porous medium equation.Comm Partial Differential Equations,2001,26:101–174
45 Otto F,Villani C.Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality.J Funct Anal,2000,173:361–400
46 Perelman G.The entropy formula for the Ricci flow and its geometric applications.Ar Xiv:math/0211159,2002
47 Shannon C.A mathematical theory of communication.Bell Syst Tech J,1948,27:379–423
48 Sturm K-T.Convex functionals of probability measures and nonlinear diffusions on manifolds.J Math Pures Appl(9),2005,84:149–168
49 Sturm K-T.On the geometry of metric measure spaces,I.Acta Math,2006,196:65–131
50 Sturm K-T.On the geometry of metric measure spaces,II.Acta Math,2006,196:133–177
51 Sturm K-T,von Renesse M-K.Transport inequalities,gradient estimates,entropy,and Ricci curvature.Comm Pure Appl Math,2005,58:923–940
52 Topping P.Lectures on the Ricci Flow.London Mathematical Society Lecture Note Series,vol.325.Cambridge:Cambridge University Presse,2006
53 Topping P.L-optimal transportation for Ricci flow.J Reine Angew Math,2009,636:93–122
54 Villani C.Topics in Optimal Transportation.Providence:Amer Math Soc,2003
55 Villani C.Optimal Transport:Old and New.Berlin:Springer,2008
56 Villani C.Entropy production and convergence to equilibrium.In:Entropy Methods for the Boltzmann Equation.Lecture Notes in Mathematics,vol.1916.Berlin:Springer,2008,1–70
57 Villani C.H-theorem and beyond:Boltzmann’s entropy in today’s mathematics.In:Boltzmann’s Legacy.ESI Lectures in Mathematics and Physics.Z¨urich:Eur Math Soc,2008,129–143
58 Villani C.(Ir)reversibility and entropy.Progress in Mathematical Physics,vol.63.Basel:Birkh¨auser/Springer,2013,19 –79
59 Wang Y Z.W-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds.Comm Pure Appl Math,2018,in press
1)Here,we say that(M,g)satisfies the bounded geometry condition if the Riemannian curvature tensor Riem and its covariant derivatives?kRiem are uniformly bounded on M,k=1,2,3.
2)Li S,Li X-D.On the convexity of the Boltzmann-Shannon type entropy on compact Riemannian manifolds equipped with Perelman’s Ricci flow.Preprint,2013
3)Following[54,55],we call Hm(u,t)the relative entropy even though it is slightly different from the classical definition of the relative entropy in probability theory.
4)Li S,Li X-D.On the Li-Yau-Hamilton-Perelman Harnack inequality and W-entropy formula for super Ricci flows.In preparation,2018
5)Kuwada K,Li X-D.Monotonicity and rigidity of the W-entropy on RCD(0,N)spaces.In preparation,2018
6)Karp L,Li P.The heat equation on complete Riemannian manifolds.Unpublished manuscript
2 Bakry D,Emery M.Diffusion hypercontractives.In:S′eminaire de probabilit′es,XIX.Lecture Notes in Mathematics,vol.1123.Berlin:Springer-Verlag,1985,177–206
3 Bakry D,Ledoux M.A logarithmic Sobolev form of the Li-Yau parabolic inequality.Rev Mat Iberoam,2006,22:683 –702
4 Baudoin F,Garofalo N.Perelman’s entropy and doubling property on Riemannian manifolds.J Geom Anal,2011,21:1119–1131
5 Benamou J-D,Brenier Y.A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numer Math,2000,84:375–393
6 Bismut J-M.The hypoelliptic Laplacian on the cotangent bundle.J Amer Math Soc,2005,18:379–476
7 Bismut J-M.Hypoelliptic Laplacian and orbital integrals.Annals of Mathematics Studies,vol.177.Princeton:Princeton University Press,2011
8 Boltzmann L.Weitere Studien ber das W¨armegleichgewicht unter Gasmolek¨ulen.Wiener Berichte,1872,66:275–370
9 Boltzmann L.ber die beziehung dem zweiten Haubtsatze der mechanischen W¨armetheorie und der Wahrscheinlichkeitsrechnung respektive den S¨atzen¨uber das W¨armegleichgewicht.Wiener Berichte,1877,76:373–435
10 Brenier Y.Polar factorization and monotone rearrangement of vector-valued functions.Comm Pure Appl Math,1991,44 :375–417
11 Clausius R.ber die W¨armeleitung gasfrmiger Krper.Ann Phys(8),1865,125:353–400
12 Ecker K.A formula relating entropy monotonicity to Harnack inequalities.Comm Anal Geom,2007,15:1025–1061
13 Evans L.Lecture notes on entropy and partial differential equations.Http://math.berkeley.edu/～evans/entropy.and.PDE.pdf
14 Gawedzki K.Lectures on Conformal Field Theory.Quantum Fields and Strings:A Course for Mathematicians,vol.2 .Providence:Amer Math Soc,1999
15 Hamilton R S.Three manifolds with positive Ricci curvature.J Differential Geom,1982,17:255–306
16 Hamilton R S.The formation of singularities in the Ricci flow.In:Surveys in Differential Geometry,vol.2.Boston:International Press,1995,7–136
17 Kotschwar B,Ni L.Gradient estimate for p-harmonic functions,1/H flow and an entropy formula.Ann Sci′Ec Norm Sup′er(4),2009,42:1–36
18 Li J,Xu X.Differential Harnack inequalities on Riemannian manifolds I:Linear heat equation.Adv Math,2011,226:4456–4491
19 Li P,Yau S-T.On the parabolic kernel of the Schrdinger operator.Acta Math,1986,156:153–201
20 Li S.W-entropy formulas on super Ricci flows and matrices Dirichlet processes.Ph D Thesis.Shanghai:Fudan University;Toulouse:Universit′e Paul Sabatier,2015
21 Li X-D.Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds.J Math Pures Appl(9),2005,84:1295–1361
22 Li X-D.On Perelman’s W-entropy functional on Riemannian manifolds with weighted volume measure.Included in Th`ese d’Habilitation`a Diriger des Rechercher.Toulouse:Universit′e Paul Sabatier,2007
23 Li X-D.Perelman’s W-entropy for the Fokker-Planck equation over complete Riemannian manifolds.Bull Sci Math,2011,135:871–882
24 Li X-D.Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature.Math Ann,2012,353:403–437
25 Li X-D.From the Boltzmann H-theorem to Perelman’s W-entropy formula for the Ricci flow.In:Emerging Topics on Differential Equations and Their Applications.Nankai Series in Pure,Applied Mathematics and Theoretical Physics,vol.10.Hackensack:World Scientific,2013,68–84
26 Li X-D.Hamilton’s Harnack inequality and the W-entropy formula on complete Riemannian manifolds.Stochastic Process Appl,2016,126:1264–1283
27 Li S,Li X-D.W-entropy formula for the Witten Laplacian on manifolds with time-dependent metrics and potentials.Pacific J Math,2015,278:173–199
28 Li S,Li X-D.Harnack inequalities and W-entropy formula for Witten Laplacian on Riemannian manifolds with K-super Perelman Ricci flow.Ar Xiv:1412.7034,2014
29 Li S,Li X-D.W-entropy formulas and Langevin deformation of flows on Wasserstein space over Riemannian manifolds.Ar Xiv:1604.02596,2016
30 Li S,Li X-D.On Harnack inequalities for Witten Laplacian on Riemannian manifolds with super Ricci flows.Asian J Math,2018,in press
31 Li S,Li X-D.W-entropy,super Perelman Ricci flows and(K,m)-Ricci solitons.Ar Xiv:1706.07040,2017
32 Li S,Li X-D.Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds.J Funct Anal,2018,274:3263–3290
33 Li S,Li X-D,Xie Y-X.Generalized Dyson Brownian motion,Mc Kean-Vlasov equation and eigenvalues of random matrices.Ar Xiv:1303.1240,2013
34 Li S,Li X-D,Xie Y-X.On the law of large numbers for empirical measure process of generalized Dyson Brownian motion.Ar Xiv:1407.7234,2014
35 Lott J.Some geometric calculations on Wasserstein space.Comm Math Phys,2008,277:423–437
36 Lott J.Optimal transport and Perelman’s reduced volume.Calc Var Partial Differential Equations,2009,36:49–84
37 Lott J,Villani C.Ricci curvature for metric-measure spaces via optimal transport.Ann of Math(2),2009,169:903 –991
38 Lu P,Ni L,Vazquez J-J,et al.Local Aronson-Benilan estimates and entropy formulae for porous medium and fast diffusion equations on manifolds.J Math Pures Appl(9),2009,91:1–19
39 Mc Cann R.Polar factorization of maps on Riemannian manifolds.Geom Funct Anal,2001,11:589–608
40 Mc Cann R,Topping P.Ricci flow,entropy and optimal transpotation.Amer J Math,2010,132:711–730
41 Nash J.Continuity of solutions of parabolic and elliptic equations.Amer J Math,1958,80:931–954
42 Ni L.The entropy formula for linear equation.J Geom Anal,2004,14:87–100
43 Ni L.Addenda to“The entropy formula for linear equation”.J Geom Anal,2004,14:329–334
44 Otto F.The geometry of dissipative evolution equations:The porous medium equation.Comm Partial Differential Equations,2001,26:101–174
45 Otto F,Villani C.Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality.J Funct Anal,2000,173:361–400
46 Perelman G.The entropy formula for the Ricci flow and its geometric applications.Ar Xiv:math/0211159,2002
47 Shannon C.A mathematical theory of communication.Bell Syst Tech J,1948,27:379–423
48 Sturm K-T.Convex functionals of probability measures and nonlinear diffusions on manifolds.J Math Pures Appl(9),2005,84:149–168
49 Sturm K-T.On the geometry of metric measure spaces,I.Acta Math,2006,196:65–131
50 Sturm K-T.On the geometry of metric measure spaces,II.Acta Math,2006,196:133–177
51 Sturm K-T,von Renesse M-K.Transport inequalities,gradient estimates,entropy,and Ricci curvature.Comm Pure Appl Math,2005,58:923–940
52 Topping P.Lectures on the Ricci Flow.London Mathematical Society Lecture Note Series,vol.325.Cambridge:Cambridge University Presse,2006
53 Topping P.L-optimal transportation for Ricci flow.J Reine Angew Math,2009,636:93–122
54 Villani C.Topics in Optimal Transportation.Providence:Amer Math Soc,2003
55 Villani C.Optimal Transport:Old and New.Berlin:Springer,2008
56 Villani C.Entropy production and convergence to equilibrium.In:Entropy Methods for the Boltzmann Equation.Lecture Notes in Mathematics,vol.1916.Berlin:Springer,2008,1–70
57 Villani C.H-theorem and beyond:Boltzmann’s entropy in today’s mathematics.In:Boltzmann’s Legacy.ESI Lectures in Mathematics and Physics.Z¨urich:Eur Math Soc,2008,129–143
58 Villani C.(Ir)reversibility and entropy.Progress in Mathematical Physics,vol.63.Basel:Birkh¨auser/Springer,2013,19 –79
59 Wang Y Z.W-entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds.Comm Pure Appl Math,2018,in press
1)Here,we say that(M,g)satisfies the bounded geometry condition if the Riemannian curvature tensor Riem and its covariant derivatives?kRiem are uniformly bounded on M,k=1,2,3.
2)Li S,Li X-D.On the convexity of the Boltzmann-Shannon type entropy on compact Riemannian manifolds equipped with Perelman’s Ricci flow.Preprint,2013
3)Following[54,55],we call Hm(u,t)the relative entropy even though it is slightly different from the classical definition of the relative entropy in probability theory.
4)Li S,Li X-D.On the Li-Yau-Hamilton-Perelman Harnack inequality and W-entropy formula for super Ricci flows.In preparation,2018
5)Kuwada K,Li X-D.Monotonicity and rigidity of the W-entropy on RCD(0,N)spaces.In preparation,2018
6)Karp L,Li P.The heat equation on complete Riemannian manifolds.Unpublished manuscript