We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail σ-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. These results are complementary of those existing in the literature, and generalize the Markovian results based on the Dobrushin ergodic coefficient.
We discuss the relationship between discrete-time processes (chains) and one-dimensional Gibbs measures. We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions generalize well known equivalence results between ergodic Markov chains and fields, as well as the known Gibbsian character of processes with exponential continuity rate. Our arguments are purely probabilistic; they are based on the study of regular systems of conditional probabilities (specifications). Furthermore, we discuss the equivalence of uniqueness criteria for chains and fields and we establish bounds for the continuity rates of the respective systems of finite-volume conditional probabilities. As an auxiliary result we prove a (re)construction theorem for specifications starting from single-site conditioning, which applies in a more general setting (general spin space, specifications not necessarily 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
53 pagesInternational audienceWe continue our study of intermittency for the parabolic Anderson equation $\partial u/\partial t = \kappa\Delta u + \xi u$, where $u\colon \Z^d\times [0,\infty)\to\R$, $\kappa$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi\colon \Z^d\times [0,\infty)\to\R$ is a space-time random medium. The solution of the equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. In this paper we focus on the case where $\xi$ is exclusion with a symmetric random walk transition kernel, starting from equilibrium with density $\rho\in (0,1)$. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant $\kappa$ when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents for $\kappa\to\infty$, which is controlled by moderate deviations of $\xi$ requiring a delicate expansion argument. In G\"artner and den Hollander \cite{garhol04} the case where $\xi$ is a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role
We continue our study of the parabolic Anderson equation ∂ u/∂t = κ∆ u + γξ u for the space-time field u :is the diffusion constant, ∆ is the discrete Laplacian, γ ∈ (0, ∞) is the coupling constant, and ξ : Z d × [0, ∞) → 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" ξ , both living on Z d .In earlier work we considered three choices for ξ : independent simple random walks, the symmetric exclusion process, and the symmetric voter model, all in equilibrium at a given density. We analyzed the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ξ , and showed that these exponents display an interesting dependence on the diffusion constant κ, with qualitatively different behavior in different dimensions d. In the present paper we focus on the quenched Lyapunov exponent, i.e., the exponential growth rate of u conditional on ξ .We first prove existence and derive some qualitative properties of the quenched Lyapunov exponent for a general ξ 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 ξ mentioned above and derive some more detailed properties. We close by formulating a number of open problems.
In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=\kappa\Delta u+\gamma\xi u$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $\kappa\in[0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in(0,\infty)$ is the coupling constant, and $\xi:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $\xi$. We focus on the case where $\xi$ is the voter model with opinions 0 and 1 that are updated according to a random walk transition kernel, starting from either the Bernoulli measure $\nu_{\rho}$ or the equilibrium measure $\mu_{\rho}$, where $\rho\in(0,1)$ is the density of 1's. We consider the annealed Lyapunov exponents, that is, the exponential growth rates of the successive moments of $u$. We show that if the random walk transition kernel has zero mean and finite variance, then these exponents are trivial for $1\leq d\leq4$, but display an interesting dependence on the diffusion constant $\kappa$ for $d\geq 5$, with qualitatively different behavior in different dimensions. In earlier work we considered the case where $\xi$ is a field of independent simple random walks in a Poisson equilibrium, respectively, a symmetric exclusion process in a Bernoulli equilibrium, which are both reversible dynamics. In the present work a main obstacle is the nonreversibility of the voter model dynamics, since this precludes the application of spectral techniques. The duality with coalescing random walks is key to our analysis, and leads to a representation formula for the Lyapunov exponents that allows for the application of large deviation estimates.Comment: Published in at http://dx.doi.org/10.1214/10-AOP535 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org
Abstract. In this paper we study the parabolic Anderson equation ∂u(x, t)/∂t = κΔu(x, t) + ξ(x, t)u(x, t), x ∈ Z d, t ≥ 0, where the u-field and the ξ -field are R-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 ∈ Z d , is taken to be nonnegative 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 ξ ∨ 0, 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 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 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 κ → λ 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) 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. (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.MSC: Primary 60H25; 82C44; secondary 60F10; 35B40
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