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, (). 

[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.

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.

[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, (). 

[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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