Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise

Jingyu Huang, Khoa Lê, and David Nualart

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Abstract

In this paper we study the linear stochastic heat equation, also known as parabolic Anderson model, in multidimension driven by a Gaussian noise which is white in time and it has a correlated spatial covariance. Examples of such covariance include the Riesz kernel in any dimension and the covariance of the fractional Brownian motion with Hurst parameter $H\in(\frac{1}{4},\frac{1}{2}]$ in dimension one. First we establish the existence of a unique mild solution and we derive a Feynman–Kac formula for its moments using a family of independent Brownian bridges and assuming a general integrability condition on the initial data. In the second part of the paper we compute Lyapunov exponents, lower and upper exponential growth indices in terms of a variational quantity. The last part of the paper is devoted to study the phase transition property of the Anderson model.

Résumé

Dans cet article nous étudions l’équation de la chaleur linéaire stochastique multidimensionnelle, connue aussi comme model d’Anderson parabolique, perturbée par un bruit gaussien qui est blanc en temps et qui a une covariance corrélée en espace. Le noyau de Riesz en dimension quelconque et la covariance du mouvement Brownien fractionnaire avec paramètre de Hurst $H\in(\frac{1}{4},\frac{1}{2}]$ en une dimension, sont des examples d’une telle covariance. D’abord, on établit l’existence d’une solution d’evolution unique et on obtient une formule de Feynman–Kac pour les moments de la solution, en utilisant une famille de ponts browniens indépendants et en supposant une condition générale d’intégrabilité sur la condition initiale. Dans la deuxième partie du travail nous calculons les exposants de Lyapunov et les exposants supérieur et inférieur de croissance exponentielle en fonction d’une quantité variationnelle. La dernière partie du travail est consacré à l’etude de la transition de phase pour le model d’Anderson.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 3 (2017), 1305-1340.

Dates
Received: 9 September 2015
Revised: 11 March 2016
Accepted: 30 March 2016
First available in Project Euclid: 21 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1500624043

Digital Object Identifier
doi:10.1214/16-AIHP756

Mathematical Reviews number (MathSciNet)
MR3689969

Zentralblatt MATH identifier
1372.60092

Subjects
Primary: 60G15: Gaussian processes 60H07: Stochastic calculus of variations and the Malliavin calculus 60H15: Stochastic partial differential equations [See also 35R60] 60F10: Large deviations 65C30: Stochastic differential and integral equations

Keywords
Stochastic heat equation Brownian bridge Feynman–Kac formula Exponential growth index Phase transition

Citation

Huang, Jingyu; Lê, Khoa; Nualart, David. Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 3, 1305--1340. doi:10.1214/16-AIHP756. https://projecteuclid.org/euclid.aihp/1500624043


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