## The Annals of Probability

### Lazy random walks and optimal transport on graphs

Christian Léonard

#### Abstract

This paper is about the construction of displacement interpolations of probability distributions on a discrete metric graph. Our approach is based on the approximation of any optimal transport problem whose cost function is a distance on a discrete graph by a sequence of entropy minimization problems under marginal constraints, called Schrödinger problems, which are associated with random walks. Displacement interpolations are defined as the limit of the time-marginal flows of the solutions to the Schrödinger problems as the jump frequencies of the random walks tend down to zero. The main convergence results are based on $\Gamma$-convergence of entropy minimization problems.

As a by-product, we obtain new results about optimal transport on graphs.

#### Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 1864-1915.

Dates
Revised: February 2015
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410034

Digital Object Identifier
doi:10.1214/15-AOP1012

Mathematical Reviews number (MathSciNet)
MR3502596

Zentralblatt MATH identifier
06603564

#### Citation

Léonard, Christian. Lazy random walks and optimal transport on graphs. Ann. Probab. 44 (2016), no. 3, 1864--1915. doi:10.1214/15-AOP1012. https://projecteuclid.org/euclid.aop/1463410034

#### References

• Ambrosio, L. (2003). Lecture notes on optimal transport problems. In Mathematical Aspects of Evolving Interfaces (Funchal, 2000). Lecture Notes in Math. 1812 1–52. Springer, Berlin.
• Ambrosio, L., Kirchheim, B. and Pratelli, A. (2004). Existence of optimal transport maps for crystalline norms. Duke Math. J. 125 207–241.
• Ambrosio, L. and Pratelli, A. (2003). Existence and stability results in the $L^{1}$ theory of optimal transportation. In Optimal Transportation and Applications (Martina Franca, 2001). Lecture Notes in Math. 1813 123–160. Springer, Berlin.
• Anzellotti, G. and Baldo, S. (1993). Asymptotic development by $\Gamma$-convergence. Appl. Math. Optim. 27 105–123.
• Attouch, H. (1996). Viscosity solutions of minimization problems. SIAM J. Optim. 6 769–806.
• Benamou, J.-D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84 375–393.
• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Bonciocat, A.-I. and Sturm, K.-T. (2009). Mass transportation and rough curvature bounds for discrete spaces. J. Funct. Anal. 256 2944–2966.
• Caffarelli, L. A., Feldman, M. and McCann, R. J. (2002). Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15 1–26 (electronic).
• Champion, T. and De Pascale, L. (2011). The Monge problem in $\mathbb{R}^{d}$. Duke Math. J. 157 551–572.
• Cordero-Erausquin, D., McCann, R. J. and Schmuckenschläger, M. (2001). A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math. 146 219–257.
• Dal Maso, G. (1993). An Introduction to $\Gamma$-Convergence. Progress in Nonlinear Differential Equations and Their Applications 8. Birkhäuser, Boston, MA.
• Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics (New York) 38. Springer, New York.
• Erbar, M. and Maas, J. (2012). Ricci curvature of finite Markov chains via convexity of the entropy. Arch. Ration. Mech. Anal. 206 997–1038.
• Evans, L. C. and Gangbo, W. (1999). Differential equations methods for the Monge–Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 viii+66.
• Gozlan, N., Roberto, C., Samson, P. M. and Tetali, P. (2014). Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Related Fields 160 47–94.
• Hillion, E. (2012). Concavity of entropy along binomial convolutions. Electron. Commun. Probab. 17 no. 4, 9.
• Hillion, E. (2014a). $W_{1,+}$-interpolation of probability measures on graphs. Electron. J. Probab. 41 679–698.
• Hillion, E. (2014b). Entropy along ${W}_{1,+}$-geodesics on graphs. Electron. J. Probab. 19 1–29.
• Hillion, E. (2014c). Contraction of measures on graphs. Potential Anal. 41 679–698.
• Johnson, O. (2007). Log-concavity and the maximum entropy property of the Poisson distribution. Stochastic Process. Appl. 117 791–802.
• Léonard, C. (2012a). From the Schrödinger problem to the Monge–Kantorovich problem. J. Funct. Anal. 262 1879–1920.
• Léonard, C. (2012b). Girsanov theory under a finite entropy condition. In Séminaire de Probabilités XLIV. Lecture Notes in Math. 2046 429–465. Springer, Heidelberg.
• Léonard, C. (2012c). On the convexity of the entropy along entropic interpolations. Preprint. Available at arXiv:1310.1274.
• Léonard, C. (2014a). A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. 34 1533–1574.
• Léonard, C. (2014b). Some properties of path measures. In Séminaire de Probabilités de Strasbourg, Vol. 46. 207–230. Springer, Berlin.
• Léonard, C., Rœlly, S. and Zambrini, J.-C. (2014). Reciprocal processes. A measure-theoretical point of view. Probab. Surv. 11 237–269.
• Lott, J. and Villani, C. (2009). Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169 903–991.
• Maas, J. (2011). Gradient flows of the entropy for finite Markov chains. J. Funct. Anal. 261 2250–2292.
• McCann, R. J. (1994). A convexity theory for interacting gases and equilibrium crystals. Ph.D. thesis, Princeton Univ., Princeton, NJ.
• McCann, R. J. (1997). A convexity principle for interacting gases. Adv. Math. 128 153–179.
• Mielke, A. (2011). A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems. Nonlinearity 24 1329–1346.
• Nagasawa, M. (1993). Schrödinger Equations and Diffusion Theory. Monographs in Mathematics 86. Birkhäuser, Basel.
• Ollivier, Y. (2009). Ricci curvature of Markov chains on metric spaces. J. Funct. Anal. 256 810–864.
• Ollivier, Y. and Villani, C. (2012). A curved Brunn–Minkowski inequality on the discrete hypercube, or: What is the Ricci curvature of the discrete hypercube? SIAM J. Discrete Math. 26 983–996.
• Otto, F. and Villani, C. (2000). Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 361–400.
• Schrödinger, E. (1931). Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144 144–153.
• Schrödinger, E. (1932). Sur la théorie relativiste de l’électron et l’interprétation de la mécanique quantique. Ann. Inst. H. Poincaré 2 269–310.
• Sturm, K.-T. (2006a). On the geometry of metric measure spaces. I. Acta Math. 196 65–131.
• Sturm, K.-T. (2006b). On the geometry of metric measure spaces. II. Acta Math. 196 133–177.
• Sudakov, V. N. (1979). Geometric problems in the theory of infinite-dimensional probability distributions. Proc. Steklov Inst. Math. 2 i–v, 1–178. Cover to cover translation of Trudy Mat. Inst. Steklov 141 (1976).
• Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
• von Renesse, M.-K. and Sturm, K.-T. (2005). Transport inequalities, gradient estimates, entropy, and Ricci curvature. Comm. Pure Appl. Math. 58 923–940.