Multiscale functional inequalities in probability: Constructive approach

Duerinckx, Mitia; Gloria, Antoine

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

Soit $A$ un champ aléatoire ergodique et stationnaire sur ${ℝ}^d$. Afin d’établir des propriétés de concentration pour des fonctions non linéaires $Z\left(A\right)$, il est courant de faire appel à des inégalités fonctionnelles de type Poincaré ou Sobolev logarithmique dans l’espace de probabilité. Ces inégalités ne sont cependant valables que pour une classe restreinte de lois (mesure produit, mesure gaussienne avec covariance intégrable, ou plus généralement mesure de Gibbs avec Hamiltionien spécifique). Dans cette contribution nous introduisons des variantes de ces inégalités que nous appelons inégalités fonctionnelles multiéchelles et qui jouissent de propriétés de concentration non linéaires comme leur version standard. Nous développons ensuite une approche constructive de ces inégalités. Nous considérons à cet effet des champs aléatoires qui peuvent s’écrire comme des transformations de structure produit, pour lesquelles la question revient à établir une règle de dérivation composée pour des changements de variables aléatoires et non linéaires. Cette approche s’applique à la plupart des exemples de champs aléatoires utilisés en modélisation des matériaux aléatoires dans les sciences appliquées, comprenant notamment les champs gaussiens avec covariance arbitraire, processus d’inclusions de Poisson avec rayons aléatoires (non bornés), la mesure de parking aléatoires et les processus de Matérn, ou encore les pavages de l’espace basés sur le processus de Poisson. Ces inégalités fonctionnelles multiéchelles, que nous développons ici principalement en vue de leur utilisation en homogénéisation stochastique quantitative, ont un intérêt propre.

Consider an ergodic stationary random field $A$ on the ambient space ${ℝ}^d$. In order to establish concentration properties for nonlinear functions $Z\left(A\right)$, 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.

Classification : 78A48, 28C15, 60E15
Mots clés : random media, functional inequalities, multiscale, concentration of measure

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | Zbl

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

[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 | DOI | MR | Zbl

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

[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 | Zbl

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

[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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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 | DOI | Numdam | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

[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

[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

[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 | Zbl

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

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

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

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

[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 | DOI | MR | Zbl

[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 | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

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

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

[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 | Zbl