Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, and G. Maillard

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Abstract

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa\varDelta u(x,t)+\xi(x,t)u(x,t)$, $x\in\mathbb{Z}^{d}$, $t\geq0$, where the $u$-field and the $\xi$-field are $\mathbb{R}$-valued, $\kappa\in[0,\infty)$ is the diffusion constant, and $\varDelta $ is the discrete Laplacian. The $\xi$-field plays the role of a dynamic random environment that drives the equation. The initial condition $u(x,0)=u_{0}(x)$, $x\in\mathbb{Z}^{d}$, is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2d\kappa$, split into two at rate $\xi\vee0$, and die at rate $(-\xi)\vee0$. Our goal is to prove a number of basic properties of the solution $u$ under assumptions on $\xi$ that are as weak as possible. These properties will serve as a jump board for later refinements.

Throughout the paper we assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies $\mathbb{E}(|\xi(0,0)|)<\infty$, where $\mathbb{E}$ denotes expectation w.r.t. $\xi$. Under a mild assumption on the tails of the distribution of $\xi$, we show that the solution to the parabolic Anderson equation exists and is unique for all $\kappa\in[0,\infty)$. Our main object of interest is the quenched Lyapunov exponent $\lambda_{0}(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t)$. It was shown in Gärtner, den Hollander and Maillard (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) that this exponent exists and is constant $\xi$-a.s., satisfies $\lambda_{0}(0)=\mathbb{E}(\xi(0,0))$ and $\lambda_{0}(\kappa)>\mathbb{E}(\xi(0,0))$ for $\kappa\in(0,\infty)$, and is such that $\kappa\mapsto\lambda_{0}(\kappa)$ is globally Lipschitz on $(0,\infty)$ outside any neighborhood of $0$ where it is finite. Under certain weak space–time mixing assumptions on $\xi$, we show the following properties: (1) $\lambda_{0}(\kappa)$ does not depend on the initial condition $u_{0}$; (2) $\lambda_{0}(\kappa)<\infty$ for all $\kappa\in[0,\infty)$; (3) $\kappa\mapsto\lambda_{0}(\kappa)$ is continuous on $[0,\infty)$ but not Lipschitz at $0$. We further conjecture: (4) $\lim_{\kappa\to\infty}[\lambda_{p}(\kappa)-\lambda_{0}(\kappa)]=0$ for all $p\in\mathbb{N}$, where $\lambda_{p}(\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\mathbb{E}([u(0,t)]^{p})$ is the $p$th annealed Lyapunov exponent. (In (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) properties (1), (2) and (4) were not addressed, while property (3) was shown under much more restrictive assumptions on $\xi$.) Finally, we prove that our weak space–time mixing conditions on $\xi$ are satisfied for several classes of interacting particle systems.

Résumé

Dans cet article on étudie l’équation parabolique d’Anderson $\partial u(x,t)/\partial t=\kappa\varDelta u(x,t)+\xi(x,t)u(x,t)$, $x\in\mathbb{Z}^{d}$, $t\geq0$, où les champs $u$ et $\xi$ sont à valeurs dans $\mathbb{R}$, $\kappa\in[0,\infty)$ est la constante de diffusion, et $\varDelta $ est le laplacien discret. Le champ $\xi$ joue le rôle d’environnement aléatoire dynamique et dirige l’équation. La condition initiale $u(x,0)=u_{0}(x)$, $x\in\mathbb{Z}^{d}$, est choisie positive et bornée. La solution de l’équation parabolique d’Anderson décrit l’évolution d’un champ de particules effectuant des marches aléatoires simples avec un branchement binaire : les particules sautent au taux $2d\kappa$, se divisent en deux au taux $\xi\vee0$, et meurent au taux $(-\xi)\vee0$. Notre but est de prouver un certain nombre de propriétés basiques de la solution $u$ sous des conditions sur $\xi$ qui sont aussi faibles que possible. Ces propriétés vont servir d’impulsion pour de futur améliorations.

Tout au long de cet article nous supposons que $\xi$ est stationnaire et ergodique sous les translations en espace et en temps, n’est pas constant et satisfait $\mathbb{E}(|\xi(0,0)|)<\infty$, où $\mathbb{E}$ représente l’espérance par rapport à $\xi$. Sous une hypothèse très faible sur les queues de la distribution de $\xi$, nous montrons que la solution de l’équation parabolique d’Anderson existe et est unique pour tout $\kappa\in[0,\infty)$. Notre principal objet d’intérêt est l’exposant de Lyapunov quenched $\lambda_{0}(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t)$. Il a été prouvé dans Gärtner, den Hollander et Maillard (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) que cet exposant existe et est constant $\xi$-a.s., satisfait $\lambda_{0}(0)=\mathbb{E}(\xi(0,0))$ et $\lambda_{0}(\kappa)>\mathbb{E}(\xi(0,0))$ pour $\kappa\in(0,\infty)$, et est tel que $\kappa\mapsto\lambda_{0}(\kappa)$ est globalement lipschitzienne sur $(0,\infty)$ à l’extérieur de n’importe quel voisinage de $0$ où il est fini. Sous certaines conditions faibles de mélange en espace-temps sur $\xi$, nous montrons les propriétés suivantes : (1) $\lambda_{0}(\kappa)$ ne dépend pas de la condition initiale $u_{0}$; (2) $\lambda_{0}(\kappa)<\infty$ pour tout $\kappa\in[0,\infty)$; (3) $\kappa\mapsto\lambda_{0}(\kappa)$ est continue sur $[0,\infty)$ mais pas lipschitzienne en $0$. Nous conjecturons en outre : (4) $\lim_{\kappa\to\infty}[\lambda_{p}(\kappa)-\lambda_{0}(\kappa)]=0$ pour tout $p\in\mathbb{N}$, où $\lambda_{p}(\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\mathbb{E}([u(0,t)]^{p})$ est le $p$-ième exposant de Lyapunov annealed. (Dans (In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner (2012) 159–193 Springer) les propriétés (1), (2) et (4) n’ont pas été abordées, tandis que la propriété (3) a été prouvée sous des hypothèses beaucoup plus restrictives sur $\xi$.) Finalement, nous prouvons que nos conditions faibles de mélange en espace-temps sur $\xi$ sont satisfaites par plusieurs systèmes de particules en interaction.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 4 (2014), 1231-1275.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1413555499

Digital Object Identifier
doi:10.1214/13-AIHP558

Mathematical Reviews number (MathSciNet)
MR3269993

Zentralblatt MATH identifier
1314.60155

Subjects
Primary: 60H25: Random operators and equations [See also 47B80] 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60F10: Large deviations 35B40: Asymptotic behavior of solutions

Keywords
Parabolic Anderson equation Percolation Quenched Lyapunov exponent Large deviations Interacting particle systems

Citation

Erhard, D.; den Hollander, F.; Maillard, G. The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 4, 1231--1275. doi:10.1214/13-AIHP558. https://projecteuclid.org/euclid.aihp/1413555499


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