Annales Henri Lebesgue

Duerinckx, Mitia; Gloria, Antoine

Multiscale functional inequalities in probability: Constructive approach

Annales Henri Lebesgue, Volume 3 (2020) , pp. 825-872.

Metadata

Keywordsrandom media, functional inequalities, multiscale, concentration of measure

Abstract

Consider an ergodic stationary random field $A$ on the ambient space $ℝ^{d}$ . In order to establish concentration properties for nonlinear functions $Z (A)$ , it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscale functional inequalities and which still imply fine concentration properties, and we develop a constructive approach to such inequalities. We consider random fields that can be viewed as transformations of a product structure, for which the question is reduced to devising approximate chain rules for nonlinear random changes of variables. This approach allows us to cover most examples of random fields arising in the modelling of heterogeneous materials in the applied sciences, including Gaussian fields with arbitrary covariance function, Poisson random inclusions with (unbounded) random radii, random parking and Matérn-type processes, as well as Poisson random tessellations. The obtained multiscale functional inequalities, which we primarily develop here in view of their application to concentration and to quantitative stochastic homogenization, are of independent interest.

References

[AKM16] Armstrong, Scott N.; Kuusi, Tuomo; Mourrat, Jean-Christophe Mesoscopic higher regularity and subadditivity in elliptic homogenization., Comm. Math. Phys, Volume 347 (2016) no. 2, pp. 315-361 | Article | MR 3545509 | Zbl 1357.35025

[AKM17] Armstrong, Scott N.; Kuusi, Tuomo; Mourrat, Jean-Christophe The additive structure of elliptic homogenization, Invent. Math., Volume 208 (2017) no. 3, pp. 999-1154 | Article | MR 3648977 | Zbl 1377.35014

[AKM19] Armstrong, Scott N.; Kuusi, Tuomo; Mourrat, Jean-Christophe Quantitative stochastic homogenization and large-scale regularity, Grundlehren der Mathematischen Wissenschaften, Volume 352, Springer, 2019 | MR 3932093 | Zbl 07053909

[AM16] Armstrong, Scott N.; Mourrat, Jean-Christophe Lipschitz regularity for elliptic equations with random coefficients, Arch. Ration. Mech. Anal., Volume 219 (2016) no. 1, pp. 255-348 | Article | MR 3437852 | Zbl 1344.35048

[AS16] Armstrong, Scott N.; Smart, C. K. Quantitative stochastic homogenization of convex integral functionals, Ann. Sci. Éc. Norm. Supér. (4), Volume 49 (2016) no. 2, pp. 423-481 | MR 3481355 | Zbl 1344.49014

[BCJ03] Barron, Emmanuel Nicholas; Cardaliaguet, Pierre; Jensen, Reed Conditional essential suprema with applications, Appl. Math. Optim., Volume 48 (2003) no. 3, pp. 229-253 | Article | MR 2004284 | Zbl 1049.60003

[BGM93] Burton, Robert M.; Goulet, Marc; Meester, Ronald On $1$ -dependent processes and $k$ -block factors, Ann. Probab., Volume 21 (1993) no. 4, pp. 2157-2168 | Article | MR 1245304 | Zbl 0788.60049

[BLM03] Boucheron, Stéphane; Lugosi, Gábor; Massart, Pascal Concentration inequalities using the entropy method, Ann. Probab., Volume 31 (2003) no. 3, pp. 1583-1614 | MR 1989444 | Zbl 1051.60020

[BP16] Bachmann, Sascha; Peccati, Giovanni Concentration bounds for geometric poisson functionals: Logarithmic sobolev inequalities revisited, Electron. J. Probab., Volume 21 (2016) no. 6, pp. 1-44 | MR 3485348 | Zbl 1337.60011

[Bra94] Bradley, Richard C. On regularity conditions for random fields, Proc. Amer. Math. Soc., Volume 121 (1994) no. 2, pp. 593-598 | Article | MR 1219721 | Zbl 0802.60047

[DG18a] Duerinckx, Mitia; Gloria, Antoine Multiscale functional inequalities in probability: Concentration properties (2018) (https://arxiv.org/abs/1711.03148, in press, to appear in ALEA. Latin American Journal of Probability and Mathematical Statistics) | Zbl 07202874

[DG18b] Duerinckx, Mitia; Gloria, Antoine Multiscale second-order Poincaré inequalities in probability (2018) (https://arxiv.org/abs/1711.03158)

[DGO18] Duerinckx, Mitia; Gloria, Antoine; Otto, Felix Robustness of the pathwise structure of fluctuations in stochastic homogenization (2018) (https://arxiv.org/abs/1807.11781, in press, to appear in Probability Theory and Related Fields)

[DGO20] Duerinckx, Mitia; Gloria, Antoine; Otto, Felix The structure of fluctuations in stochastic homogenization, Commun. Math. Phys., Volume 377 (2020) no. 1, pp. 259-306 | Article | MR 4107930 | Zbl 07209661

[ES81] Efron, Bradley; Stein, Charles M. The jackknife estimate of variance, Ann. Stat., Volume 9 (1981) no. 3, pp. 586-596 | Article | MR 615434 | Zbl 0481.62035

[FO16] Fischer, Julian; Otto, Felix A higher-order large-scale regularity theory for random elliptic operators, Comm. Part. Diff. Equa., Volume 41 (2016) no. 7, pp. 1108-1148 | Article | MR 3528529 | Zbl 1349.35440

[GNO14] Gloria, Antoine; Neukamm, Stefan; Otto, Felix An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations, ESAIM, Math. Model. Numer. Anal., Volume 48 (2014) no. 2, pp. 325-346 | Article | Numdam | MR 3177848 | Zbl 1307.35029

[GNO15] Gloria, Antoine; Neukamm, Stefan; Otto, Felix Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Invent. Math., Volume 199 (2015) no. 2, pp. 455-515 | Article | MR 3302119 | Zbl 1314.39020

[GNO17] Gloria, Antoine; Neukamm, Stefan; Otto, Felix Quantitative estimates in stochastic homogenization for correlated fields (2017) (https://arxiv.org/abs/1409.2678)

[GNO20] Gloria, Antoine; Neukamm, Stefan; Otto, Felix A regularity theory for random elliptic operators, Milan J. Math., Volume 88 (2020) no. 1, pp. 99-170 | Article | MR 4103433 | Zbl 07216255

[GO11] Gloria, Antoine; Otto, Felix An optimal variance estimate in stochastic homogenization of discrete elliptic equations, Ann. Probab., Volume 39 (2011) no. 3, pp. 779-856 | Article | MR 2789576 | Zbl 1215.35025

[GO12] Gloria, Antoine; Otto, Felix An optimal error estimate in stochastic homogenization of discrete elliptic equations, Ann. Appl. Probab., Volume 22 (2012) no. 1, pp. 1-28 | Article | MR 2932541 | Zbl 1387.35031

[GO15] Gloria, Antoine; Otto, Felix The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations (2015) (https://arxiv.org/abs/1510.08290)

[GP13] Gloria, Antoine; Penrose, Mathew D. Random parking, Euclidean functionals, and rubber elasticity, Comm. Math. Phys., Volume 321 (2013) no. 1, pp. 1-31 | Article | MR 3089662 | Zbl 1277.60024

[Gro75] Gross, Leonard Logarithmic Sobolev inequalities, Am. J. Math., Volume 97 (1975) no. 4, pp. 1061-1083 | Article | MR 420249

[HPA95] Houdré, Christian; Pérez-Abreu, Víctor Covariance identities and inequalities for functionals on Wiener and Poisson spaces, Ann. Probab., Volume 23 (1995) no. 1, pp. 400-419 | Article | MR 1330776 | Zbl 0831.60029

[Lee97] Lee, Sungchum The central limit theorem for Euclidean minimal spanning trees. I, Ann. Appl. Probab., Volume 7 (1997) no. 4, pp. 996-1020 | MR 1484795

[Lee99] Lee, Sungchum The central limit theorem for Euclidean minimal spanning trees. II, Adv. Appl. Probab., Volume 31 (1999) no. 4, pp. 969-984 | MR 1747451

[LY93] Lu, Sheng Lin; Yau, Horng-Tzer Spectral gap and logarithmic Sobolev inequality for Kawasaki and Glauber dynamics, Comm. Math. Phys., Volume 156 (1993) no. 2, pp. 399-433 | MR 1233852 | Zbl 0779.60078

[MO15] Marahrens, Daniel; Otto, Felix Annealed estimates on the Green function, Probab. Theory Relat. Fields, Volume 163 (2015) no. 3-4, pp. 527-573 | Article | MR 3418749 | Zbl 1342.60101

[NP12] Nourdin, Ivan; Peccati, Giovanni Normal approximations with Malliavin calculus. From Stein’s method to universality, Cambridge Tracts in Mathematics, Volume 192, Cambridge University Press, 2012 | Zbl 1266.60001

[NS98] Naddaf, Ali; Spencer, Thomas Estimates on the variance of some homogenization problems, 1998 (Preprint) | Zbl 0871.35010

[Pen01] Penrose, Mathew D. Random parking, sequential adsorption, and the jamming limit, Comm. Math. Phys., Volume 218 (2001) no. 1, pp. 153-176 | Article | MR 1824203 | Zbl 0980.60020

[Pen05] Penrose, Mathew D. Multivariate spatial central limit theorems with applications to percolation and spatial graphs, Ann. Probab., Volume 33 (2005) no. 5, pp. 1945-1991 | Article | MR 2165584 | Zbl 1087.60022

[PY02] Penrose, Mathew D.; Yukich, Joseph E. Limit theory for random sequential packing and deposition, Ann. Appl. Probab., Volume 12 (2002) no. 1, pp. 272-301 | MR 1890065 | Zbl 1018.60023

[PY05] Penrose, Mathew D.; Yukich, Joseph E. Normal approximation in geometric probability, Stein’s method and applications (Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore) Volume 5, World Scientific; Singapore University Press, 2005, pp. 37-58 | Article | MR 2201885

[SPY07] Schreiber, Tomasz; Penrose, Mathew D.; Yukich, Joseph E. Gaussian limits for multidimensional random sequential packing at saturation, Comm. Math. Phys., Volume 272 (2007) no. 1, pp. 167-183 | Article | MR 2291806 | Zbl 1145.60017

[Ste86] Steele, John Michael An Efron–Stein inequality for nonsymmetric statistics, Ann. Statist., Volume 14 (1986) no. 2, pp. 753-758 | Article | MR 840528 | Zbl 0604.62017

[Tor02] Torquato, Salvatore Random heterogeneous materials. Microstructure and macroscopic properties, Interdisciplinary Applied Mathematics, Volume 16, Springer, 2002 | Zbl 0988.74001

[Wu00] Wu, Liming A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, Probab. Theory Related Fields, Volume 118 (2000) no. 3, pp. 427-438 | MR 1800540 | Zbl 0970.60093