American Institute of Mathematical Sciences

Advanced
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

Alexander V. Kolesnikov 1,

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 $M$.
Keywords: metric-measure space, convex geometry., Optimal transportation, Bakry--Émery tensor, Sobolev spaces, Hessian manifolds, Monge--Ampère equation, log-concave measures.
Mathematics Subject Classification: Primary: 35J60; Secondary: 46E3.
Citation: Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems, 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, Berlin, 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 "Séminaire de Probabilités, XIX, 1983/84," Lecture Notes in Math., 1123, Springer, Berlin, (1985), 130-174. doi: 10.1007/BFb0075843.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in "Séminaire de Probabilités, XIX, 1983/84," Lecture Notes in Math., 1123, Springer, Berlin, (1985), 177-206. 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), 547-572. 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), 135-150. doi: 10.2307/1971510.  Google Scholar

[7]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI, 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), 369-402. 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), 105-126. 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 "Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3" (Beijing, 1980), Science Press, Beijing, (1982), 339-370.  Google Scholar

[11]

R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture, in "Concentration, Functional Inequalities and Isoperimetry," Contermporary Mathematics, 545, Amer. Math. Soc., Providence, RI, (2011), 55-68. 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), 347-385. 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

M. Gromov, Convex sets and Kähler manifolds, in "Advances in Differential Geometry and Topology," World Sci. Publ., Teaneck, NJ, (1990), 1-38.  Google Scholar

[15]

C. E. Gutièrrez, "The Monge-Ampère Equation," Progress in Nonlinear Differential Equations and Their Applications, 44, Birkhäuser Boston, Inc., Boston, MA, 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), 1-41. doi: 10.5802/afst.1366.  Google Scholar

[17]

A. V. Kolesnikov, Global Hölder estimates for optimal transportation, Mat. Zametki, 88 (2010), 708-728. doi: 10.1134/S0001434610110076.  Google Scholar

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2012), 243-264. 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), 1373-1404. doi: 10.1016/j.matpur.2004.03.005.  Google Scholar

[20]

A. V. Kolesnikov, Mass transportation and contractions, MIPT Proc., 2 (2010), 90-99. Google Scholar

[21]

N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569-595.  Google Scholar

[22]

M. Ledoux, Concentration of measure and logarithmic Sobolev inequality, in "Séminaire de Probabilités, XXXIII," Lecture Notes in Math., 1709, Springer, Berlin, (1999), 120-216. doi: 10.1007/BFb0096511.  Google Scholar

[23]

E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound, Duke Math. J., 154 (2010), 207-239. 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), 1-43. 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), 286-292. doi: 10.1007/BF00252910.  Google Scholar

[26]

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

[27]

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

[28]

N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications in "Handbook of Geometric Analysis," No. 1, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, (2008), 467-524.  Google Scholar

[29]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 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, Berlin, 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 "Séminaire de Probabilités, XIX, 1983/84," Lecture Notes in Math., 1123, Springer, Berlin, (1985), 130-174. doi: 10.1007/BFb0075843.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in "Séminaire de Probabilités, XIX, 1983/84," Lecture Notes in Math., 1123, Springer, Berlin, (1985), 177-206. 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), 547-572. 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), 135-150. doi: 10.2307/1971510.  Google Scholar

[7]

L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI, 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), 369-402. 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), 105-126. 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 "Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3" (Beijing, 1980), Science Press, Beijing, (1982), 339-370.  Google Scholar

[11]

R. Eldan and B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture, in "Concentration, Functional Inequalities and Isoperimetry," Contermporary Mathematics, 545, Amer. Math. Soc., Providence, RI, (2011), 55-68. 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), 347-385. 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, Classics in Mathematics, Springer-Verlag, Berlin, 2001.  Google Scholar

[14]

M. Gromov, Convex sets and Kähler manifolds, in "Advances in Differential Geometry and Topology," World Sci. Publ., Teaneck, NJ, (1990), 1-38.  Google Scholar

[15]

C. E. Gutièrrez, "The Monge-Ampère Equation," Progress in Nonlinear Differential Equations and Their Applications, 44, Birkhäuser Boston, Inc., Boston, MA, 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), 1-41. doi: 10.5802/afst.1366.  Google Scholar

[17]

A. V. Kolesnikov, Global Hölder estimates for optimal transportation, Mat. Zametki, 88 (2010), 708-728. doi: 10.1134/S0001434610110076.  Google Scholar

[18]

A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2012), 243-264. 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), 1373-1404. doi: 10.1016/j.matpur.2004.03.005.  Google Scholar

[20]

A. V. Kolesnikov, Mass transportation and contractions, MIPT Proc., 2 (2010), 90-99. Google Scholar

[21]

N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569-595.  Google Scholar

[22]

M. Ledoux, Concentration of measure and logarithmic Sobolev inequality, in "Séminaire de Probabilités, XXXIII," Lecture Notes in Math., 1709, Springer, Berlin, (1999), 120-216. doi: 10.1007/BFb0096511.  Google Scholar

[23]

E. Milman, Isoperimetric and concentration inequalities: Equivalence under curvature lower bound, Duke Math. J., 154 (2010), 207-239. 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), 1-43. 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), 286-292. doi: 10.1007/BF00252910.  Google Scholar

[26]

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

[27]

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

[28]

N. S. Trudinger and X.-L. Wang, The Monge-Ampère equation and its geometric applications in "Handbook of Geometric Analysis," No. 1, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, (2008), 467-524.  Google Scholar

[29]

C. Villani, "Optimal Transport. Old and New," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 338, Springer-Verlag, Berlin, 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, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447
[2]

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, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641
[3]

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
[4]

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
[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, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069
[7]

Zhijun Zhang. Optimal global asymptotic behavior of the solution to a singular monge-ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1129-1145. doi: 10.3934/cpaa.2020053
[8]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559
[9]

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
[10]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705
[11]

Ziwei Zhou, Jiguang Bao, Bo Wang. A Liouville theorem of parabolic Monge-AmpÈre equations in half-space. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1561-1578. doi: 10.3934/dcds.2020331
[12]

Shuyu Gong, Ziwei Zhou, Jiguang Bao. Existence and uniqueness of viscosity solutions to the exterior problem of a parabolic Monge-Ampère equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4921-4936. doi: 10.3934/cpaa.2020218
[13]

Diego Maldonado. On interior $C^2$-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058
[14]

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
[15]

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
[16]

Yahui Niu. Monotonicity of solutions for a class of nonlocal Monge-Ampère problem. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5269-5283. doi: 10.3934/cpaa.2020237
[17]

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
[18]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121
[19]

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
[20]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (2) : 915-931. doi: 10.3934/cpaa.2020297

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy