The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, p. 1231-1275

In this paper we study the parabolic Anderson equation u(x,t)/t=κ𝛥u(x,t)+ξ(x,t)u(x,t), x d , t0, where the u-field and the ξ-field are -valued, κ[0,) is the diffusion constant, and 𝛥 is the discrete Laplacian. The ξ-field plays the role of a dynamic random environment that drives the equation. The initial condition u(x,0)=u 0 (x), x 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 2dκ, split into two at rate ξ0, and die at rate (-ξ)0. Our goal is to prove a number of basic properties of the solution u under assumptions on ξ that are as weak as possible. These properties will serve as a jump board for later refinements. Throughout the paper we assume that ξ is stationary and ergodic under translations in space and time, is not constant and satisfies 𝔼(|ξ(0,0)|)<, where 𝔼 denotes expectation w.r.t. ξ. Under a mild assumption on the tails of the distribution of ξ, we show that the solution to the parabolic Anderson equation exists and is unique for all κ[0,). Our main object of interest is the quenched Lyapunov exponent λ 0 (κ)=lim t 1 tlogu(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 ξ-a.s., satisfies λ 0 (0)=𝔼(ξ(0,0)) and λ 0 (κ)>𝔼(ξ(0,0)) for κ(0,), and is such that κλ 0 (κ) is globally Lipschitz on (0,) outside any neighborhood of 0 where it is finite. Under certain weak space-time mixing assumptions on ξ, we show the following properties: (1) λ 0 (κ) does not depend on the initial condition u 0 ; (2) λ 0 (κ)< for all κ[0,); (3) κλ 0 (κ) is continuous on [0,) but not Lipschitz at 0. We further conjecture: (4) lim κ [λ p (κ)-λ 0 (κ)]=0 for all p, where λ p (κ)=lim t 1 ptlog𝔼([u(0,t)] p ) is the pth 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 ξ.) Finally, we prove that our weak space-time mixing conditions on ξ are satisfied for several classes of interacting particle systems.

Dans cet article on étudie l’équation parabolique d’Anderson u(x,t)/t=κ𝛥u(x,t)+ξ(x,t)u(x,t), x d , t0, où les champs u et ξ sont à valeurs dans , κ[0,) est la constante de diffusion, et 𝛥 est le laplacien discret. Le champ ξ joue le rôle d’environnement aléatoire dynamique et dirige l’équation. La condition initiale u(x,0)=u 0 (x), x 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 2dκ, se divisent en deux au taux ξ0, et meurent au taux (-ξ)0. Notre but est de prouver un certain nombre de propriétés basiques de la solution u sous des conditions sur ξ 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 ξ est stationnaire et ergodique sous les translations en espace et en temps, n’est pas constant et satisfait 𝔼(|ξ(0,0)|)<, où 𝔼 représente l’espérance par rapport à ξ. Sous une hypothèse très faible sur les queues de la distribution de ξ, nous montrons que la solution de l’équation parabolique d’Anderson existe et est unique pour tout κ[0,). Notre principal objet d’intérêt est l’exposant de Lyapunov quenched λ 0 (κ)=lim t 1 tlogu(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 ξ-a.s., satisfait λ 0 (0)=𝔼(ξ(0,0)) et λ 0 (κ)>𝔼(ξ(0,0)) pour κ(0,), et est tel que κλ 0 (κ) est globalement lipschitzienne sur (0,) à l’extérieur de n’importe quel voisinage de 0 où il est fini. Sous certaines conditions faibles de mélange en espace-temps sur ξ, nous montrons les propriétés suivantes : (1) λ 0 (κ) ne dépend pas de la condition initiale u 0 ; (2) λ 0 (κ)< pour tout κ[0,); (3) κλ 0 (κ) est continue sur [0,) mais pas lipschitzienne en 0. Nous conjecturons en outre : (4) lim κ [λ p (κ)-λ 0 (κ)]=0 pour tout p, où λ p (κ)=lim t 1 ptlog𝔼([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 ξ.) Finalement, nous prouvons que nos conditions faibles de mélange en espace-temps sur ξ sont satisfaites par plusieurs systèmes de particules en interaction.

DOI : https://doi.org/10.1214/13-AIHP558
Classification:  60H25,  82C44,  60F10,  35B40
Keywords: parabolic Anderson equation, percolation, quenched Lyapunov exponent, large deviations, interacting particle systems
@article{AIHPB_2014__50_4_1231_0,
     author = {Erhard, D. and den Hollander, Frank and Maillard, G.},
     title = {The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {4},
     year = {2014},
     pages = {1231-1275},
     doi = {10.1214/13-AIHP558},
     zbl = {06377553},
     mrnumber = {3269993},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_4_1231_0}
}
Erhard, D.; den Hollander, F.; Maillard, G. The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 4, pp. 1231-1275. doi : 10.1214/13-AIHP558. http://www.numdam.org/item/AIHPB_2014__50_4_1231_0/

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