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

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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.


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

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

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

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Zentralblatt MATH identifier

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

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


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.

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