Moment measures and stability for Gaussian inequalities

Alexander V. Kolesnikov ; Egor D. Kosov

Teoriâ slučajnyh processov, Tome 22 (2017) no. 2, pp. 47-61.

Résumé

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^n$ and let $\mathcal{P}_{\gamma}$ be the space of probability measures that are absolutely continuous with respect to $\gamma$. We study lower bounds for the functional $\mathcal{F}_{\gamma}(\mu) = {\rm Ent}(\mu) - \frac{1}{2} W^2_2(\mu, \nu)$, where $\mu \in \mathcal{P}_{\gamma}, \nu \in \mathcal{P}_{\gamma}$, ${\rm Ent}(\mu) = \int \log\bigl( \frac{\mu}{\gamma}\bigr) d \mu$ is the relative Gaussian entropy, and $W_2$ is the quadratic Kantorovich distance. The minimizers of $\mathcal{F}_{\gamma}$ are solutions to a dimension-free Gaussian analog of the (real) Kähler–Einstein equation. We show that $\mathcal{F}_{\gamma}(\mu) $ is bounded from below under the assumption that the Gaussian Fisher information of $\nu$ is finite and prove a priori estimates for the minimizers. Our approach relies on certain stability estimates for the Gaussian log-Sobolev and Talagrand transportation inequalities.

Détail

Keywords: Gaussian inequalities, Kähler-Einstein equation, moment measure.
Mots-clés : optimal transportation

Bibliographie

[1] D. Bakry, I. Gentil, M. Ledoux, Analysis and geometry of Markov diffusion operators, Springer, 2014 | MR | Zbl

[2] R. J. Berman, B. Berndtsson, “Real Monge-Ampère equations and Kähler-Ricci solitons on toric log Fano varieties”, Ann. Fac. Sci. Toulouse Math., 22:4 (2013), 649–711 | DOI | MR | Zbl

[3] S. G. Bobkov, N. Gozlan, C. Roberto, P.-M. Samson, “Bounds on the deficit in the logarithmic Sobolev inequality”, J. Funct. Anal., 267 (2014), 4110–4138 | DOI | MR | Zbl

[4] V. I. Bogachev, Gaussian measures, Amer. Math. Soc., Rhode Island, Providence, 1998 | MR | Zbl

[5] V. I. Bogachev, A. V. Kolesnikov, “Integrability of absolutely continuous transformations of measures and applications to optimal mass transportation”, Theory Probab. Appl., 50:3 (2006), 367–385 | DOI | MR | Zbl

[6] V. I. Bogachev, A. V. Kolesnikov, “On the Monge–Ampère equation in infinite dimensions”, Infin. Dimen. Anal. Quantum Probab. Related Topics, 8:4 (2005), 547–572 | DOI | MR | Zbl

[7] V. I. Bogachev, A. V. Kolesnikov, “Sobolev regularity for the Monge–Ampere equation in the Wiener space”, Kyoto J. Math., 53:4 (2013), 713–738 | DOI | MR | Zbl

[8] V. I. Bogachev, A. V. Kolesnikov, “The Monge–Kantorovich problem: achievements, connections, and perspectives”, Russian Math. Surveys, 67:5 (2012), 785–890 | DOI | MR | Zbl

[9] E. A. Carlen, “Superadditivity of Fisher's information and logarithmic Sobolev inequalities”, J. Funct. Anal., 101 (1991), 194–211 | DOI | MR | Zbl

[10] F. Cavalletti, “The Monge problem in Wiener space”, Calcul. Var. PDE's, 45:1-2 (2012), 101–124 | DOI | MR | Zbl

[11] D. Cordero-Erausquin, B. Klartag, “Moment measures”, J. Funct. Anal., 268 (2015), 3834–3866 | DOI | MR | Zbl

[12] T. A. Courtade, M. Fathi, A. Papanjady, “Wasserstein stability of the entropy power inequality for log-concave densities”, arXiv: 1610.07969

[13] S. Fang, V. Nolot, “Sobolev estimates for optimal transport maps on Gaussian spaces”, J. Funct. Anal., 266 (2014), 5045–5084 | DOI | MR | Zbl

[14] M. Fathi, E. Indrei, M. Ledoux, “Quantitative logarithmic Sobolev inequalities and stability estimates”, arXiv: 1410.6922 | MR

[15] D. Feyel, A. S. Üstünel, “Monge–Kantorovich measure transportation and Monge–Ampère equation on Wiener space”, Probab. Theory Related Fields, 128 (2004), 347–385 | DOI | MR | Zbl

[16] A. Figalli, “Quantitative isoperimetric inequalities, with applications to the stability of liquid drops and crystals”, Concentration, functional inequalities and isoperimetry, Contemp. Math., 545, Amer. Math. Soc., Providence, Rhode Island, 2011 | MR | Zbl

[17] J. Fontbona, N. Gozlan, J.-F. Jabir, “A variational approach to some transport inequalities”, Ann. Inst. H. Poincaré Probab. Statist., 53:4 (2017), 1719–1746 | DOI | MR | Zbl

[18] E. Indrei, D. Marcon, “A quantitative log-Sobolev inequality for a two parameter family of functions”, Internat. Math. Research Notices, 2014:20 (2014), 5563–5580 | DOI | MR | Zbl

[19] B. Klartag, A. V. Kolesnikov, “Remarks on curvature in the transportation metric”, Analysis Math., 43:1 (2017), 67–88 | DOI | MR | Zbl

[20] A. V. Kolesnikov, “Convexity inequalities and optimal transport of infinite-dimensional measures”, J. Math. Pures Appl., 83:11 (2004), 1373–1404 | DOI | MR | Zbl

[21] A. V. Kolesnikov, “Mass transportation and contractions”, MIPT Proc., 2:4 (2010), 90–99

[22] A. V. Kolesnikov, “On Sobolev regularity of mass transport and transportation inequalities”, Theory Probab. Appl., 57:2 (2012), 243–264 | DOI | MR

[23] M. Ledoux, Concentration of measure phenomenon, Amer. Math. Soc., Rhode Island, Providence, 2001 | MR | Zbl

[24] M. Ledoux, I. Nourdin, G. Peccati, “A Stein deficit for the logarithmic Sobolev inequality”, Sci. China Math., 60:7 (2017), 1163–1180 | DOI | MR | Zbl

[25] E. Milman, “On the role of convexity in isoperimetry, spectral gap and concentration”, Invent. Math., 177:1 (2009), 1–43 | DOI | MR | Zbl

[26] E. Milman, “Isoperimetric and concentration inequalities: Equivalence under curvature lower bound”, Duke Math. J., 154:2 (2010), 207–239 | DOI | MR | Zbl

[27] F. Santambrogio, “Dealing with moment measures via entropy and optimal transport”, J. Funct. Anal., 271 (2016), 418–436 | DOI | MR | Zbl

[28] X.-J. Wang, X. Zhu, “Kähler–Ricci solitons on toric manifolds with positive first Chern class”, Advances in Math., 188 (2004), 87–103 | DOI | MR | Zbl