April  2014, 34(4): 1511-1532. doi: 10.3934/dcds.2014.34.1511

Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures

1. 

Higher School of Economics, Faculty of Mathematics, 117312, Vavilova 7, Moscow, Russian Federation

Received  November 2012 Revised  April 2013 Published  October 2013

We study the optimal transportation mapping $\nabla \Phi : \mathbb{R}^d \mapsto \mathbb{R}^d$ pushing forward a probability measure $\mu = e^{-V} \ dx$ onto another probability measure $\nu = e^{-W} \ dx$. Following a classical approach of E. Calabi we introduce the Riemannian metric $g = D^2 \Phi$ on $\mathbb{R}^d$ and study spectral properties of the metric-measure space $M=(\mathbb{R}^d, g, \mu)$. We prove, in particular, that $M$ admits a non-negative Bakry--Émery tensor provided both $V$ and $W$ are convex. If the target measure $\nu$ is the Lebesgue measure on a convex set $\Omega$ and $\mu$ is log-concave we prove that $M$ is a $CD(K,N)$ space. Applications of these results include some global dimension-free a priori estimates of $\| D^2 \Phi\|$. With the help of comparison techniques on Riemannian manifolds and probabilistic concentration arguments we proof some diameter estimates for $M$.
Citation: Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
References:
[1]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", Springer-Verlag, (1994). doi: 10.1007/978-3-642-69881-1. Google Scholar

[2]

D. Bakry, Transformation de Riesz pour les semi-groupes symétrique. I. Étude de la dimension $1$,, in, 1123 (1985), 130. doi: 10.1007/BFb0075843. Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, in, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar

[4]

V. I. Bogachev and A. V. Kolesnikov, On the Monge-Ampère equation in infinite dimensions,, Infin. Dimen. Anal. Quantum Probab. and Relat. Topics, 8 (2005), 547. doi: 10.1142/S0219025705002141. Google Scholar

[5]

V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space,, preprint, (). Google Scholar

[6]

L. A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510. Google Scholar

[7]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995). Google Scholar

[8]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear elliptic differential equations. I. Monge-Ampère equation,, CPAM, 37 (1984), 369. doi: 10.1002/cpa.3160370306. Google Scholar

[9]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,, Michigan Math. J., 5 (1958), 105. doi: 10.1307/mmj/1028998055. Google Scholar

[10]

S.-Y. Cheng and S.-T. Yau, The real Monge-Ampère equation and affine flat structures,, in, (1982), 339. Google Scholar

[11]

R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture,, in, 545 (2011), 55. doi: 10.1090/conm/545/10764. Google Scholar

[12]

D. Feyel and A. S. Üstünel, Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener space,, Prob. Theory and Related Fields, 128 (2004), 347. doi: 10.1007/s00440-003-0307-x. Google Scholar

[13]

D. Gilbarg and N. S. Trudinge, "Elliptic Partial Differential Equation of the Second Order,", Reprint of the 1998 edition, (1998). Google Scholar

[14]

M. Gromov, Convex sets and Kähler manifolds,, in, (1990), 1. Google Scholar

[15]

C. E. Gutièrrez, "The Monge-Ampère Equation,", Progress in Nonlinear Differential Equations and Their Applications, 44 (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar

[16]

B. Klartag, Poincaré inequalities and moment maps,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 22 (2013), 1. doi: 10.5802/afst.1366. Google Scholar

[17]

A. V. Kolesnikov, Global Hölder estimates for optimal transportation,, Mat. Zametki, 88 (2010), 708. doi: 10.1134/S0001434610110076. Google Scholar

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities,, Theory Probab. Appl., 57 (2012), 243. doi: 10.1137/S0040585X97985947. Google Scholar

[19]

A. V. Kolesnikov, Convexity inequalities and optimal transport of infinite-dimensional measures,, J. Math. Pures Appl. (9), 83 (2004), 1373. doi: 10.1016/j.matpur.2004.03.005. Google Scholar

[20]

A. V. Kolesnikov, Mass transportation and contractions,, MIPT Proc., 2 (2010), 90. Google Scholar

[21]

N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments,, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569. Google Scholar

[22]

M. Ledoux, Concentration of measure and logarithmic Sobolev inequality,, in, 1709 (1999), 120. doi: 10.1007/BFb0096511. Google Scholar

[23]

E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound,, Duke Math. J., 154 (2010), 207. doi: 10.1215/00127094-2010-038. Google Scholar

[24]

E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration,, Invent. Math., 177 (2009), 1. doi: 10.1007/s00222-009-0175-9. Google Scholar

[25]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar

[26]

{A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type,", Noordhoff, (1964). Google Scholar

[27]

H. Shima, "The Geometry of Hessian Structures,", World Scientific Publishing Co. Pte. Ltd., (2007). doi: 10.1142/9789812707536. Google Scholar

[28]

N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications, in, 7 (2008), 467. Google Scholar

[29]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

show all references

References:
[1]

I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations,", Springer-Verlag, (1994). doi: 10.1007/978-3-642-69881-1. Google Scholar

[2]

D. Bakry, Transformation de Riesz pour les semi-groupes symétrique. I. Étude de la dimension $1$,, in, 1123 (1985), 130. doi: 10.1007/BFb0075843. Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives,, in, 1123 (1985), 177. doi: 10.1007/BFb0075847. Google Scholar

[4]

V. I. Bogachev and A. V. Kolesnikov, On the Monge-Ampère equation in infinite dimensions,, Infin. Dimen. Anal. Quantum Probab. and Relat. Topics, 8 (2005), 547. doi: 10.1142/S0219025705002141. Google Scholar

[5]

V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space,, preprint, (). Google Scholar

[6]

L. A. Caffarelli, Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation,, Ann. of Math. (2), 131 (1990), 135. doi: 10.2307/1971510. Google Scholar

[7]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations,", American Mathematical Society Colloquium Publications, 43 (1995). Google Scholar

[8]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear elliptic differential equations. I. Monge-Ampère equation,, CPAM, 37 (1984), 369. doi: 10.1002/cpa.3160370306. Google Scholar

[9]

E. Calabi, Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens,, Michigan Math. J., 5 (1958), 105. doi: 10.1307/mmj/1028998055. Google Scholar

[10]

S.-Y. Cheng and S.-T. Yau, The real Monge-Ampère equation and affine flat structures,, in, (1982), 339. Google Scholar

[11]

R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture,, in, 545 (2011), 55. doi: 10.1090/conm/545/10764. Google Scholar

[12]

D. Feyel and A. S. Üstünel, Monge-Kantorovich measure transportation and Monge-Ampère equation on Wiener space,, Prob. Theory and Related Fields, 128 (2004), 347. doi: 10.1007/s00440-003-0307-x. Google Scholar

[13]

D. Gilbarg and N. S. Trudinge, "Elliptic Partial Differential Equation of the Second Order,", Reprint of the 1998 edition, (1998). Google Scholar

[14]

M. Gromov, Convex sets and Kähler manifolds,, in, (1990), 1. Google Scholar

[15]

C. E. Gutièrrez, "The Monge-Ampère Equation,", Progress in Nonlinear Differential Equations and Their Applications, 44 (2001). doi: 10.1007/978-1-4612-0195-3. Google Scholar

[16]

B. Klartag, Poincaré inequalities and moment maps,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 22 (2013), 1. doi: 10.5802/afst.1366. Google Scholar

[17]

A. V. Kolesnikov, Global Hölder estimates for optimal transportation,, Mat. Zametki, 88 (2010), 708. doi: 10.1134/S0001434610110076. Google Scholar

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities,, Theory Probab. Appl., 57 (2012), 243. doi: 10.1137/S0040585X97985947. Google Scholar

[19]

A. V. Kolesnikov, Convexity inequalities and optimal transport of infinite-dimensional measures,, J. Math. Pures Appl. (9), 83 (2004), 1373. doi: 10.1016/j.matpur.2004.03.005. Google Scholar

[20]

A. V. Kolesnikov, Mass transportation and contractions,, MIPT Proc., 2 (2010), 90. Google Scholar

[21]

N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments,, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569. Google Scholar

[22]

M. Ledoux, Concentration of measure and logarithmic Sobolev inequality,, in, 1709 (1999), 120. doi: 10.1007/BFb0096511. Google Scholar

[23]

E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound,, Duke Math. J., 154 (2010), 207. doi: 10.1215/00127094-2010-038. Google Scholar

[24]

E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration,, Invent. Math., 177 (2009), 1. doi: 10.1007/s00222-009-0175-9. Google Scholar

[25]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains,, Arch. Rational Mech. Anal., 5 (1960), 286. doi: 10.1007/BF00252910. Google Scholar

[26]

{A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type,", Noordhoff, (1964). Google Scholar

[27]

H. Shima, "The Geometry of Hessian Structures,", World Scientific Publishing Co. Pte. Ltd., (2007). doi: 10.1142/9789812707536. Google Scholar

[28]

N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications, in, 7 (2008), 467. Google Scholar

[29]

C. Villani, "Optimal Transport. Old and New,", Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338 (2009). doi: 10.1007/978-3-540-71050-9. Google Scholar

[1]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447

[2]

Luca Codenotti, Marta Lewicka. Visualization of the convex integration solutions to the Monge-Ampère equation. Evolution Equations & Control Theory, 2019, 8 (2) : 273-300. doi: 10.3934/eect.2019015

[3]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[4]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641

[5]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[6]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[7]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[8]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[9]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[10]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[11]

Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697

[12]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[13]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[14]

Fan Cui, Huaiyu Jian. Symmetry of solutions to a class of Monge-Ampère equations. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1247-1259. doi: 10.3934/cpaa.2019060

[15]

Limei Dai, Hongyu Li. Entire subsolutions of Monge-Ampère type equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 19-30. doi: 10.3934/cpaa.2020002

[16]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

[17]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[18]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[19]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076

[20]

Bang-Xian Han. New characterizations of Ricci curvature on RCD metric measure spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4915-4927. doi: 10.3934/dcds.2018214

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]