Intermittency on catalysts

Gaertner, J.; Hollander, Frank den; Maillard, G.

doi:10.1017/cbo9781139107020.011

Cited by 17 publications

(9 citation statements)

References 18 publications

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“…We are now ready to formulate our main result (which was already announced in Gärtner, den Hollander and Maillard [4]). Note that the expression in the r.h.s.…”

Section: Lyapunov Exponentsmentioning

confidence: 79%

Intermittency on catalysts: three-dimensional simple symmetric exclusion

Gärtner

Hollander

Maillard³

2009

Electron. J. Probab.

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We continue our study of intermittency for the parabolic Anderson model ∂u/∂t = κ∆u + ξu in a space-time random medium ξ, where κ is a positive diffusion constant, ∆ is the lattice Laplacian on Z d , d ≥ 1, and ξ is a simple symmetric exclusion process on Z d in Bernoulli equilibrium. This model describes the evolution of a reactant u under the influence of a catalyst ξ.In [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as t → ∞ of the successive moments of the solution u. This led to an almost complete picture of intermittency as a function of d and κ. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as κ → ∞ in the critical dimension d = 3, which was left open in [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in d ≥ 4, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for κ.MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.

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“…We are now ready to formulate our main result (which was already announced in Gärtner, den Hollander and Maillard [4]). Note that the expression in the r.h.s.…”

Section: Lyapunov Exponentsmentioning

confidence: 79%

Intermittency on catalysts: three-dimensional simple symmetric exclusion

Gärtner

Hollander

Maillard³

2009

Electron. J. Probab.

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“…The annealed results found by that team for the models (i)-(iii) in a series of papers are summarized in Examples 8.3, 8.5, and 8.6, see also the survey article [GärHolMai09b]. The annealed results found by that team for the models (i)-(iii) in a series of papers are summarized in Examples 8.3, 8.5, and 8.6, see also the survey article [GärHolMai09b].…”

Section: Potentials Built Out Of Particlesmentioning

confidence: 96%

The Parabolic Anderson Model

König

2016

Pathways in Mathematics

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Each "Pathways in Mathematics" book offers a roadmap to a currently well developing mathematical research field and is a first-hand information and inspiration for further study, aimed both at students and researchers. It is written in an educational style, i.e., in a way that is accessible for advanced undergraduate and graduate students. It also serves as an introduction to and survey of the field for researchers who want to be quickly informed about the state of the art. The point of departure is typically a bachelor/masters level background, from which the reader is expeditiously guided to the frontiers. This is achieved by focusing on ideas and concepts underlying the development of the subject while keeping technicalities to a minimum. Each volume contains an extensive annotated bibliography as well as a discussion of open problems and future research directions as recommendations for starting new projects More information about this series at

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“…Interesting recent mathematical progress not discussed here can be found, for example, in [HKM06], [GKM07], [BMR07], [GHM07] and [D08], two survey articles emphasising recent work on a range of potentials are [GK05] and [GHM08]. Note that in some of these references the potential field is allowed to have a nontrivial time-dependence, a feature which we shall exclude from the discussions of the present paper.…”

Section: The Parabolic Anderson Problemmentioning

confidence: 99%

The parabolic Anderson model with heavy-tailed potential

Mörters

2011

EMS Series of Congress Reports

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Abstract. The parabolic Anderson model is the Cauchy problem for the heat equation with random potential. It offers a case study for the effects that a random, or irregular, environment can have on a diffusion process. The main focus in the present survey is on phenomena that are due to a highly irregular potential, which we model by a spatially independent, identically distributed random field with heavy tails. Among the effects we discuss are random fluctuations in the growth of the total mass, localisation in the weak and almost sure sense, and ageing.2010 Mathematics Subject Classification. Primary 60K37 Secondary 82C44.Keywords. Anderson Hamiltonian, parabolic problem, intermittency, localization, aging, out of equilibrium, random disorder, random medium, heavy tail, polynomial tail, Pareto distribution, Poisson point process, scaling limit. The parabolic Anderson problemWe consider the heat equation with random potential on the integer lattice Z d and study the Cauchy problem with localised initial datum,whereis the discrete Laplacian, and the potential (ξ(z) :

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Intermittency on catalysts

Cited by 17 publications

References 18 publications

Intermittency on catalysts: three-dimensional simple symmetric exclusion

Intermittency on catalysts: three-dimensional simple symmetric exclusion

The Parabolic Anderson Model

The parabolic Anderson model with heavy-tailed potential

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