Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Weak convergence approach for parabolic equations with large, highly oscillatory, random potential

Yu Gu and Guillaume Bal

Full-text: Open access

Abstract

This paper concerns the macroscopic behavior of solutions to parabolic equations with large, highly oscillatory, random potential. When the correlation function of the random potential satisfies a specific integrability condition, we show that the random solution converges, as the correlation length of the medium tends to zero, to the deterministic solution of a homogenized equation in dimension $d\geq3$. Our derivation is based on a Feynman–Kac probabilistic representation and the Kipnis–Varadhan method applied to weak convergence of Brownian motions in random sceneries. For sufficiently mixing coefficients, we also provide an optimal rate of convergence to the homogenized limit using a quantitative martingale central limit theorem. As soon as the above integrability condition fails, the solution is expected to remain stochastic in the limit of a vanishing correlation length. For a large class of potentials given as functionals of Gaussian fields, we show the convergence of solutions to stochastic partial differential equations (SPDE) with multiplicative noise. The Feynman–Kac representation and the corresponding weak convergence of Brownian motions in random sceneries allows us to explain the transition from deterministic to stochastic limits as a function of the correlation function of the random potential.

Résumé

Nous considérons le comportement macroscopique de solutions d’équations paraboliques présentant un terme potentiel aléatoire de grande intensité et oscillant rapidement. Lorsque la fonction de corrélation du potentiel aléatoire satisfait une condition précise d’intégrabilité, nous démontrons que la solution aléatoire converge vers la solution déterministe d’une équation homogénéisée quand la longueur de corrélation du milieu tend vers zéro en toute dimension $d\geq3$. Notre preuve s’appuie sur une représentation probabiliste de type Feynman–Kac et sur la methodologie introduite par Kipnis et Varadhan permettant de montrer la convergence de mouvements browniens dans des milieux aléatoires. Lorsque les coefficients aléatoires sont suffisamment mélangeants, nous présentons un taux optimal de convergence grâce à une approche quantitative du théorème de la limite centrale pour les martingales. Dès que la condition d’intégrabilité mentionée ci-dessus n’est plus satisfaite, nous pensons que la solution restera fortement stochastique pour toute longueur de corrélation du milieu. Nous montrons, pour une classe de potentiels décrits comme des fonctionnelles de champs gaussiens, que la solution converge vers celle d’une équation différentielle stochastique. La représentation de Feynman–Kac et la convergence faible de mouvements browniens nous permet d’obtenir une description précise de la transition d’une limite déterministe vers une limite stochastique en fonction des propriétés de la fonction de corrélation du potentiel aléatoire.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 261-285.

Dates
Received: 2 December 2013
Revised: 22 July 2014
Accepted: 4 August 2014
First available in Project Euclid: 6 January 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089269

Digital Object Identifier
doi:10.1214/14-AIHP637

Mathematical Reviews number (MathSciNet)
MR3449303

Zentralblatt MATH identifier
1343.35019

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35K05: Heat equation 60G44: Martingales with continuous parameter 60F05: Central limit and other weak theorems 60K37: Processes in random environments

Keywords
Stochastic homogenization Brownian motion in random scenery Feynman–Kac formula Weak convergence

Citation

Gu, Yu; Bal, Guillaume. Weak convergence approach for parabolic equations with large, highly oscillatory, random potential. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 261--285. doi:10.1214/14-AIHP637. https://projecteuclid.org/euclid.aihp/1452089269


Export citation

References

  • [1] S. N. Armstrong and C. K. Smart. Quantitative stochastic homogenization of elliptic equations in nondivergence form. Arch. Ration. Mech. Anal. 214 (2014) 867–911.
  • [2] G. Bal. Central limits and homogenization in random media. Multiscale Model. Simul. 7 (2008) 677–702.
  • [3] G. Bal. Convergence to spdes in Stratonovich form. Comm. Math. Phys. 292 (2009) 457–477.
  • [4] G. Bal. Homogenization with large spatial random potential. Multiscale Model. Simul. 8 (2010) 1484–1510.
  • [5] G. Bal. Convergence to homogenized or stochastic partial differential equations. Appl. Math. Res. Express. AMRX 2011 (2) (2011) 215–241.
  • [6] G. Bal, J. Garnier, Y. Gu and W. Jing. Corrector theory for elliptic equations with oscillatory and random potentials with long range correlations. Asymptot. Anal. 77 (2012) 123–145.
  • [7] G. Bal, J. Garnier, S. Motsch and V. Perrier. Random integrals and correctors in homogenization. Asymptot. Anal. 59 (2008) 1–26.
  • [8] G. Bal and W. Jing. Corrector theory for elliptic equations in random media with singular green’s function. Application to random boundaries. Commun. Math. Sci. 19 (2011) 383–411.
  • [9] E. Bolthausen. A central limit theorem for two-dimensional random walks in random sceneries. Ann. Appl. Probab. 17 (1989) 108–115.
  • [10] A. Bourgeat and A. Piatnitski. Estimates in probability of the residual between the random and the homogenized solutions of one-dimensional second-order operator. Asymptot. Anal. 21 (1999) 303–315.
  • [11] L. A. Caffarelli and P. E. Souganidis. Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math. 180 (2010) 301–360.
  • [12] J. G. Conlon and A. Naddaf. On homogenization of elliptic equations with random coefficients. Electron. J. Probab. 5 (9) (2000) 1–58 (electronic).
  • [13] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 2009.
  • [14] R. Figari, E. Orlandi and G. Papanicolaou. Mean field and Gaussian approximation for partial differential equations with random coefficients. SIAM J. Appl. Math. 42 (1982) 1069–1077.
  • [15] A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 39 (2011) 779–856.
  • [16] Y. Gu and G. Bal. Random homogenization and convergence to integrals with respect to the Rosenblatt process. J. Differential Equations 253 (2012) 1069–1087.
  • [17] Y. Gu and G. Bal. An invariance principle for Brownian motion in random scenery. Electron. J. Probab. 19 (2014) 1–19.
  • [18] M. Hairer, E. Pardoux and A. Piatnitski. Random homogenisation of a highly oscillatory singular potential. SPDEs: Anal. Comp. 1 (2013) 571–605.
  • [19] Y. Hu, D. Nualart and J. Song. Feynman–Kac formula for heat equation driven by fractional white noise. Ann. Appl. Probab. 39 (2011) 291–326.
  • [20] H. Kesten and F. Spitzer. A limit theorem related to a new class of self similar processes. Probab. Theory Related Fields 50 (1979) 5–25.
  • [21] C. Kipnis and S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1–19.
  • [22] T. Komorowski, C. Landim and S. Olla. Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 345. Springer, Heidelberg, 2012.
  • [23] T. Komorowski and E. Nieznaj. On the asymptotic behavior of solutions of the heat equation with a random, long-range correlated potential. Potential Anal. 33 (2010) 175–197.
  • [24] S. M. Kozlov. Averaging of random operators. Mat. Sb. 151 (1979) 188–202.
  • [25] A. Lejay. Homogenization of divergence-form operators with lower-order terms in random media. Probab. Theory Related Fields 120 (2001) 255–276.
  • [26] D. Marahrens and F. Otto. Annealed estimates on the Green’s function. Probab. Theory Related Fields 163 (2015) 527–573.
  • [27] J.-C. Mourrat. Kantorovich distance in the martingale clt and quantitative homogenization of parabolic equations with random coefficients. Probab. Theory Related Fields 160 (2012) 1–36.
  • [28] G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979) 835–873. Colloq. Math. Soc. János Bolyai 27. North Holland, Amsterdam–New York, 1981.
  • [29] E. Pardoux and A. Piatnitski. Homogenization of a singular random one dimensional PDE. In Multi Scale Problems and Asymptotic Analysis 291–303. GAKUTO Internat. Ser. Math. Sci. Appl. 24. Gakkōtosho, Tokyo, 2006.
  • [30] É. Pardoux and A. Piatnitski. Homogenization of a singular random one-dimensional PDE with time-varying coefficients. Ann. Appl. Probab. 40 (2012) 1316–1356.
  • [31] B. Rémillard and D. Dawson. A limit theorem for Brownian motion in a random scenery. Canad. Math. Bull. 34 (1991) 385–391.
  • [32] M. S. Taqqu. Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Related Fields 31 (1975) 287–302.
  • [33] M. E. Taylor. Partial Differential Equations. I. Basic Theory, 2nd edition. Applied Mathematical Sciences 115. Springer, New York, 2011.
  • [34] V. Yurinskii. Averaging of symmetric diffusion in random medium. Sib. Math. J. 27 (1986) 603–613.
  • [35] N. Zhang and G. Bal. Convergence to SPDE of the Schrödinger equation with large, random potential. Commun. Math. Sci. 12 (2014) 825–841.
  • [36] N. Zhang and G. Bal. Homogenization of a Schrödinger equation with large random, potential. Stoch. Dyn. 14 (2014) 1–29.