Parabolic Anderson model with voter catalysts: dichotomy in the behavior of Lyapunov exponents

We consider the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ with $u\colon\, \Z^d\times R^+\to \R^+$, where $\kappa\in\R^+$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in\R^+$ is the coupling constant, and $\xi\colon\,\Z^d\times \R^+\to\{0,1\}$ is the voter model starting from Bernoulli product measure $\nu_{\rho}$ with density $\rho\in (0,1)$. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$. In G\"artner, den Hollander and Maillard 2010 the behavior of the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In the present paper we address some questions left open in G\"artner, den Hollander and Maillard 2010 by considering specifically when the Lyapunov exponents are the a priori maximal value in terms of strong transience of the Markov process underlying the voter model.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 33 pages. Updated version following the referee's comment

Quenched Lyapunov exponent for the parabolic Anderson model in a dynamic random environment

We continue our study of the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ for the space-time field $u\colon\,\Z^d\times [0,\infty)\to\R$, where $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in (0,\infty)$ is the coupling constant, and $\xi\colon\,\Z^d\times [0,\infty)\to\R$ is a space-time random environment that drives the equation. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$, both living on $\Z^d$. In earlier work we considered three choices for $\xi$: independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$, and showed that these exponents display an interesting dependence on the diffusion constant $\kappa$, with qualitatively different behavior in different dimensions $d$. In the present paper we focus on the \emph{quenched} Lyapunov exponent, i.e., the exponential growth rate of $u$ conditional on $\xi$. We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general $\xi$ that is stationary and ergodic w.r.t.\ translations in $\Z^d$ and satisfies certain noisiness conditions. After that we focus on the three particular choices for $\xi$ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.Comment: In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 33 pages. Final revised versio

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

In this paper we study the parabolic Anderson equation \partial u(x,t)/\partial t=\kappa\Delta u(x,t)+\xi(x,t)u(x,t), x\in\Z^d, t\geq 0, where the u-field and the \xi-field are \R-valued, \kappa \in [0,\infty) is the diffusion constant, and $\Delta$ is the discrete Laplacian. The initial condition u(x,0)=u_0(x), x\in\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. Throughout the paper we assume that $\xi$ is stationary and ergodic under translations in space and time, is not constant and satisfies \E(|\xi(0,0)|)<\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(\kappa)=\lim_{t\to\infty}\frac{1}{t}\log u(0,t). Under certain weak space-time mixing conditions 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\N, where \lambda_p (\kappa)=\lim_{t\to\infty}\frac{1}{pt}\log\E([u(0,t)]^p) is the p-th annealed Lyapunov exponent. Finally, we prove that our weak space-time mixing conditions on \xi are satisfied for several classes of interacting particle systems.Comment: 50 pages. The comments of the referee are incorporated into the paper. A missing counting estimate was added in the proofs of Lemma 3.6 and Lemma 4.

Chaînes à liaisons complètes et mesures de Gibbs unidimensionnelles

On introduit un formalisme de mécanique statistique pour l'étude des processus stochastiques discrets(chaînes) pour lesquels on prouve : (i) des propriétés générales de chaînes extrémales, incluant la trivialité de la tribu queue, les corrélations à courtes portées, la réalisation via des limites à volumes infinis et l'ergodicité, (ii) deux nouvelles conditions pour l'unicité de la chaîne cohérente, (iii) des résultats de perte de mémoire et des propriétés de mélange pour des chaînes sous le régime de Dobrushin. On considère des systèmes à alphabet fini, pouvant avoir une grammaire. On établit des conditions pour qu'une chaîne définisse une mesure de Gibbs et vice-versa. On discute de l'équivalence des critères d'unicité pour les chaînes et les champs et on établit des bornes pour les taux de continuité des systèmes respectifs de probabilités conditionnelles. On prouve un théorème de (re)construction pour les spécifications en partant de conditionnement sur un site.ROUEN-BU Sciences (764512102) / SudocROUEN-BU Sciences Madrillet (765752101) / SudocTOULON-BU Centrale (830622101) / SudocROUEN-Bib.maths (764512206) / SudocSudocFranceF

Regular gg-measures are not always Gibbsian

9 pagesInternational audienceRegular $g$-measures are discrete-time processes determined by conditional expectations with respect to the past. One-dimensional Gibbs measures, on the other hand, are fields determined by simultaneous conditioning on past and future. For the Markovian and exponentially continuous cases both theories are known to be equivalent. Its equivalence for more general cases was an open problem. We present a simple example settling this issue in a negative way: there exist $g$-measures that are continuous and non-null but are not Gibbsian. Our example belongs, in fact, to a well-studied family of processes with rather nice attributes: It is a chain with variable-length memory, characterized by the absence of phase coexistence and the existence of a visible renewal scheme