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

Erhard, Dirk; Hollander, Frank den; Maillard, G.

doi:10.48550/arxiv.1208.0330

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“…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: 83%

“…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. In [3] we proved the following:…”

Section: Parabolic Anderson Modelmentioning

confidence: 90%

“…The definition of Gärtner-hyper-mixing is given in Definitions 1.3-1.5 below. A weaker form of these definitions was introduced and exploited in [3]. Here are two examples of ξ-fields that are Gärtner-hyper-mixing.…”

Section: Main Theorem and Examplesmentioning

confidence: 99%

“…t log u(0, t). In earlier work [6], [3] we established a number of basic properties of κ → λ 0 (κ) under certain mild space-time mixing and noisiness assumptions on ξ. In particular, we showed that the limit exists ξ-a.s., is finite and continuous on [0, ∞), is globally Lipschitz on (0, ∞), is not Lipschitz at 0, and satisfies λ 0 (0) = E(ξ(0, 0)) and λ 0 (κ) > E(ξ(0, 0)) for κ ∈ (0, ∞).…”

mentioning

confidence: 99%

“…Our ultimate goal is to arrive at a full qualitative picture of the quenched Lyapunov exponent for general dynamic random environments subject to certain mild space-time mixing and noisiness assumptions. Section 1.1 defines the parabolic Anderson model and recalls the main results from [6,3]. Section 1.2 contains our main theorem, which states that the quenched Lyapunov exponent converges to the average value of the environment in the limit of large diffusivity.…”

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confidence: 99%

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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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“…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: 83%

Section: Parabolic Anderson Modelmentioning

confidence: 90%

Section: Main Theorem and Examplesmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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