Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

Gärtner, Jürgen; Hollander, Frank den; Maillard, G.

doi:10.1007/978-3-642-23811-6_7

Cited by 11 publications

(30 citation statements)

References 25 publications

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“…1 t log u(0, t). It was shown in Gärtner, den Hollander and Maillard [8] 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) λ 0 (κ) < ∞ for all κ ∈ [0, ∞); (3) κ → λ 0 (κ) is continuous on [0, ∞) but not Lipschitz at 0.…”

mentioning

confidence: 86%

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

Erhard¹,

Hollander²,

Maillard³

2014

Ann. Inst. H. Poincaré Probab. Statist.

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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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confidence: 86%

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

Erhard¹,

Hollander²,

Maillard³

2014

Ann. Inst. H. Poincaré Probab. Statist.

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“…(1. 8) In [6] we showed that λ 0 (0) = E(ξ(0, 0)) and λ 0 (κ) > E(ξ(0, 0)) for κ ∈ (0, ∞) as soon as the limit in (1.8) exists. In [3] we proved the following:…”

Section: Parabolic Anderson Modelmentioning

confidence: 97%

“…(1. 6) The formal solution of (1.1) is given by the Feynman-Kac formula u(x, t) = E x exp t 0 ξ(X κ (s), t − s) ds u 0 (X κ (t)) , (1. 7) where X κ = (X κ (t)) t≥0 is the continuous-time simple random walk jumping at rate 2dκ (i.e., the Markov process with generator κ∆), and P x is the law of X κ when X κ (0) = x.…”

Section: Parabolic Anderson Modelmentioning

confidence: 99%

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

Erhard

Hollander

Maillard

2014

Probab. Theory Relat. Fields

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35 pages, 4 figures.International audienceWe continue our study of 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 $\kappa \in [0,\infty)$ is the diffusion constant, $\Delta$ is the discrete Laplacian, and $\xi$ plays the role of a \emph{dynamic random environment} that drives the equation. 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 $(-\xi) \vee 0$. 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.\ $\xi$. Our main object of interest is the quenched Lyapunov exponent $\lambda_0 (\kappa) = \lim_{t\to\infty} \frac{1}{t}\log u(0,t)$. In earlier work we showed that under certain mild space-time mixing assumptions the limit exists $\xi$-a.s., is finite and continuous on $[0,\infty)$, is globally Lipschitz on $(0,\infty)$, is not Lipschitz at 0, and satisfies $\lambda_0(0) = \E(\xi(0,0))$ and $\lambda_0(\kappa) > \E(\xi(0,0))$ for $\kappa \in (0,\infty)$.In the present paper we show that $\lim_{\kappa\to\infty} \lambda_0(\kappa) =\E(\xi(0,0))$ under an additional space-time mixing condition on $\xi$. This result shows that the parabolic Anderson model exhibits space-time ergodicity in the limit of large diffusivity. This fact is interesting because there are choices of $\xi$ that fulfill our assumption for which the annealed Lyapunov exponent $\lambda_1(\kappa) = \lim_{t\to\infty} \frac{1}{t}\log \E(u(0,t))$ is infinite on $[0,\infty)$, a situation that is referred to as strongly catalytic behavior

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“…(1.10) and define the sets E l = {η ∈ : η l < ∞} and L l = {f : 11) and η x,y is defined by where, for a configuration η, c(x, η) is the rate for the spin at x to flip, and…”

Section: Choices Of Dynamic Random Environments (I) Space-time White mentioning

confidence: 99%

“…Further note that, by Jensen's inequality, 11) where the interchange of the expectations is justified because…”

Section: Proof Of Theorem 13mentioning

confidence: 99%

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

Erhard¹,

Hollander²,

Maillard³

2016

Math Phys Anal Geom

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The parabolic Anderson model is defined as the partial differential equation ∂u(x, t)/∂t = κ u(x, t) + ξ (x, t)u(x, t), x ∈ Z d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u 0 (x), x ∈ Z d , is typically 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 ξ ∨ 0, and die at rate (−ξ) ∨ 0. In earlier work we looked at the Lyapunov exponentsFor the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ . In the present paper we investigate what happens when κ is replaced by K , where where, for a fixed realisation of K, Supp(K) is the set of values taken by the Kfield. We also show that for the associated quenched Lyapunov exponent λ 0 (K) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(K) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ , and for all K satisfying a certain clustering property, namely, there are arbitrarily large balls where K is almost constant and close to any value in Supp(K). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of K where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of K where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

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Quenched Lyapunov Exponent for the Parabolic Anderson Model in a Dynamic Random Environment

Cited by 11 publications

References 25 publications

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

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

The parabolic Anderson model in a dynamic random environment: space-time ergodicity for the quenched Lyapunov exponent

Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

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