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) no. 4, pp. 1231-1275.

Dans cet article on étudie l’équation parabolique d’Anderson $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, où les champs $u$ et $\xi$ sont à valeurs dans $ℝ$, $\kappa \in [0,\infty )$ est la constante de diffusion, et $𝛥$ est le laplacien discret. Le champ $\xi$ joue le rôle d’environnement aléatoire dynamique et dirige l’équation. La condition initiale $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, 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 $𝔼\left(\right|\xi (0,0)\left|\right)<\infty$, où $𝔼$ 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 ${\lambda }_0\left(\kappa \right)={lim}_{t\to \infty }\frac{1}{t}logu(0,t)$. 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 ${\lambda }_0\left(0\right)=𝔼\left(\xi (0,0)\right)$ et ${\lambda }_0\left(\kappa \right)>𝔼\left(\xi (0,0)\right)$ pour $\kappa \in (0,\infty )$, et est tel que $\kappa \mapsto {\lambda }_0\left(\kappa \right)$ 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) ${\lambda }_0\left(\kappa \right)$ ne dépend pas de la condition initiale $u_0$; (2) ${\lambda }_0\left(\kappa \right)<\infty$ pour tout $\kappa \in [0,\infty )$; (3) $\kappa \mapsto {\lambda }_0\left(\kappa \right)$ est continue sur $[0,\infty )$ mais pas lipschitzienne en $0$. Nous conjecturons en outre : (4) ${lim}_{\kappa \to \infty }[{\lambda }_p\left(\kappa \right)-{\lambda }_0\left(\kappa \right)]=0$ pour tout $p\in \mathbb{N}$, où ${\lambda }_p\left(\kappa \right)={lim}_{t\to \infty }\frac{1}{pt}log𝔼\left({\left[u(0,t)\right]}^p\right)$ 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.

In this paper we study the parabolic Anderson equation $\partial u(x,t)/\partial t=\kappa 𝛥u(x,t)+\xi (x,t)u(x,t)$, $x\in {\mathbb{Z}}^d$, $t\ge 0$, where the $u$-field and the $\xi$-field are $ℝ$-valued, $\kappa \in [0,\infty )$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $\xi$-field plays the role of a dynamic random environment that drives the equation. The initial condition $u(x,0)=u_0\left(x\right)$, $x\in {\mathbb{Z}}^d$, 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 $𝔼\left(\right|\xi (0,0)\left|\right)<\infty$, where $𝔼$ 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 ${\lambda }_0\left(\kappa \right)={lim}_{t\to \infty }\frac{1}{t}logu(0,t)$. 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 ${\lambda }_0\left(0\right)=𝔼\left(\xi (0,0)\right)$ and ${\lambda }_0\left(\kappa \right)>𝔼\left(\xi (0,0)\right)$ for $\kappa \in (0,\infty )$, and is such that $\kappa \mapsto {\lambda }_0\left(\kappa \right)$ 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) ${\lambda }_0\left(\kappa \right)$ does not depend on the initial condition $u_0$; (2) ${\lambda }_0\left(\kappa \right)<\infty$ for all $\kappa \in [0,\infty )$; (3) $\kappa \mapsto {\lambda }_0\left(\kappa \right)$ is continuous on $[0,\infty )$ but not Lipschitz at $0$. We further conjecture: (4) ${lim}_{\kappa \to \infty }[{\lambda }_p\left(\kappa \right)-{\lambda }_0\left(\kappa \right)]=0$ for all $p\in \mathbb{N}$, where ${\lambda }_p\left(\kappa \right)={lim}_{t\to \infty }\frac{1}{pt}log𝔼\left({\left[u(0,t)\right]}^p\right)$ 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.

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