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

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014), p. 1231-1275 / Harvested from Numdam

Résumé

Dans cet article on étudie l’équation parabolique d’Anderson $\partial u(x,t)/\partial t=\kappa \varDelta u(x,t)+\xi (x,t)u(x,t)$, $x\in \mathbb {Z}^{d}$, $t\ge 0$, 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 $\xi \vee 0$, et meurent au taux $(-\xi )\vee 0$. 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 $\lambda _{0}(\kappa )=\lim _{t\rightarrow \infty }\frac{1}{t}\log u(0,t)$. 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) $\lim _{\kappa \rightarrow \infty }[\lambda _{p}(\kappa )-\lambda _{0}(\kappa )]=0$ pour tout $p\in \mathbb {N}$, où $\lambda _{p}(\kappa )=\lim _{t\rightarrow \infty }\frac{1}{pt}\log \mathbb {E}([u(0,t)]^{p})$ 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.

Abstract

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa \varDelta u(x,t)+\xi (x,t)u(x,t)$, $x\in \mathbb {Z}^{d}$, $t\ge 0$, 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 $\xi \vee 0$, and die at rate $(-\xi )\vee 0$. 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 $\lambda _{0}(\kappa )=\lim _{t\rightarrow \infty }\frac{1}{t}\log u(0,t)$. 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) $\lim _{\kappa \rightarrow \infty }[\lambda _{p}(\kappa )-\lambda _{0}(\kappa )]=0$ for all $p\in \mathbb {N}$, where $\lambda _{p}(\kappa )=\lim _{t\rightarrow \infty }\frac{1}{pt}\log \mathbb {E}([u(0,t)]^{p})$ 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.

Détail

Publié le : 2014-01-01

MR 3269993 | Zbl 06377553

DOI : https://doi.org/10.1214/13-AIHP558
Classification:  60H25,  82C44,  60F10,  35B40

Bibliographie

[1] E. D. Andjel. Invariant measure for the zero range process. Ann. Probab. 10 (1982) 525-547. | MR 659526 | Zbl 0492.60096

[2] R. A. Carmona and S. A. Molchanov. Parabolic Anderson Problem and Intermittency. AMS Memoirs 518. American Mathematical Society, Providence, RI, 1994. | MR 1185878 | Zbl 0925.35074

[3] J. T. Cox and D. Griffeath. Large deviations for Poisson systems of independent random walks. Z. Wahrsch. Verw. Gebiete 66 (1984) 543-558. | MR 753813 | Zbl 0551.60028

[4] M. Cranston, T. S. Mountford and T. Shiga. Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. (N.S.) 71 (2002) 163-188. | MR 1980378 | Zbl 1046.60057

[5] A. Drewitz, J. Gärtner, A. F. Ramirez and R. Sun. Survival probability of a random walk among a Poisson system of moving traps. In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner 119-158. J.-D. Deuschel, B. Gentz, W. König, M.-K. van Renesse, M. Scheutzow and U. Schmock (Eds). Springer Proceedings in Mathematics 11. Springer, Berlin, 2012. | Zbl 1267.60109

[6] P. Erdös. On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. 43 (1942) 437-450. | MR 6749 | Zbl 0061.07905

[7] J. Gärtner and F. Den Hollander. Intermittency in a catalytic random medium. Ann. Probab. 34 (2006) 2219-2287. | MR 2294981 | Zbl 1117.60065

[8] J. Gärtner, F. Den Hollander and G. Maillard. Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment. In Probability in Complex Physical Systems. In Honour of Erwin Bolthausen and Jürgen Gärtner 159-193. J.-D. Deuschel, B. Gentz, W. König, M.-K. van Renesse, M. Scheutzow and U. Schmock (Eds). Springer Proceedings in Mathematics 11. Springer, Berlin, 2012. | Zbl 1252.39027

[9] J. Gärtner and S. A. Molchanov. Parabolic problems for the Anderson model. Comm. Math. Phys. 132 (1990) 613-655. | MR 1069840 | Zbl 0711.60055

[10] G. H. Hardy and S. A. Ramanujan. Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. (3) 17 (1918) 75-115. | JFM 46.0198.04 | MR 1575586

[11] C. Kipnis and C. Landim. Scaling Limits of Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 320. Springer, Berlin, 1999. | MR 1707314 | Zbl 0927.60002

[12] H. Kesten and V. Sidoravicius. Branching random walks with catalysts. Electron. J. Probab. 8 (2003) 1-51. | MR 1961167 | Zbl 1064.60196

[13] C. Landim. Occupation time large deviations for the symmetric simple exclusion process. Ann. Probab. 20 (1992) 206-231. | MR 1143419 | Zbl 0751.60098

[14] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften 276. Springer, New York, 1985. | MR 776231 | Zbl 0559.60078

[15] F. Redig and F. Völlering. Concentration of additive functionals for Markov processes. Preprint, 2011. Available at http://arXiv.org/abs/1003.0006v2.

[16] L. Wu. Feynman-Kac semigroups, ground state diffusions, and large deviations. J. Funct. Anal. 123 (1994) 202-231. | MR 1279300 | Zbl 0798.60067

[17] L. Wu. A deviation inequality for non-reversible Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 4 (2000) 435-445. | Numdam | MR 1785390 | Zbl 0972.60003