# American Institute of Mathematical Sciences

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 and 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. [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. [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. [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. [5] V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space, preprint, arXiv:1110.1822. [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. [7] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI, 1995. [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. [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. [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. [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. [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. [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. [14] M. Gromov, Convex sets and Kähler manifolds, in "Advances in Differential Geometry and Topology," World Sci. Publ., Teaneck, NJ, (1990), 1-38. [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. [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. [17] A. V. Kolesnikov, Global Hölder estimates for optimal transportation, Mat. Zametki, 88 (2010), 708-728. doi: 10.1134/S0001434610110076. [18] A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2012), 243-264. doi: 10.1137/S0040585X97985947. [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. [20] A. V. Kolesnikov, Mass transportation and contractions, MIPT Proc., 2 (2010), 90-99. [21] N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569-595. [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. [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. [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. [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. [26] {A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type," Noordhoff, Ltd., Groningen, 1964. [27] H. Shima, "The Geometry of Hessian Structures," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812707536. [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. [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.

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##### References:
 [1] I. J. Bakelman, "Convex Analysis and Nonlinear Geometric Elliptic Equations," Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-69881-1. [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. [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. [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. [5] V. I. Bogachev and A. V. Kolesnikov, Sobolev regularity for the Monge-Ampère equation in the Wiener space, preprint, arXiv:1110.1822. [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. [7] L. A. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society Colloquium Publications, 43, Amer. Math. Soc., Providence, RI, 1995. [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. [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. [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. [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. [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. [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. [14] M. Gromov, Convex sets and Kähler manifolds, in "Advances in Differential Geometry and Topology," World Sci. Publ., Teaneck, NJ, (1990), 1-38. [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. [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. [17] A. V. Kolesnikov, Global Hölder estimates for optimal transportation, Mat. Zametki, 88 (2010), 708-728. doi: 10.1134/S0001434610110076. [18] A. V. Kolesnikov, On Sobolev regularity of mass transport and transportation inequalities, Theory Probab. Appl., 57 (2012), 243-264. doi: 10.1137/S0040585X97985947. [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. [20] A. V. Kolesnikov, Mass transportation and contractions, MIPT Proc., 2 (2010), 90-99. [21] N. V. Krylov, Fully nonlinear second order elliptic equations: Recent developments, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 25 (1997), 569-595. [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. [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. [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. [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. [26] {A. V. Pogorelov}, "Monge-Ampère Equations of Elliptic Type," Noordhoff, Ltd., Groningen, 1964. [27] H. Shima, "The Geometry of Hessian Structures," World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812707536. [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. [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.
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