The Parabolic Anderson Model

Gärtner, Jürgen; König, Wolfgang

doi:10.1007/3-540-27110-4_8

Cited by 73 publications

(62 citation statements)

References 38 publications

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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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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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“…We refer the readers to [7,8,24] about the interpretations of S t, x and S − t, x . The connection between the asymptotics of N (t) and S t, x can be found in the literature for the case that {q + ξ q } q is replaced by an R d -stationary random field (see e.g.…”

Section: Asymptotics For Associated Wiener Integralsmentioning

confidence: 99%

Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

Fukushima¹,

Ueki²

2010

Ann. Henri Poincaré

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The asymptotic behavior of the integrated density of states for a randomly perturbed lattice at the infimum of the spectrum is investigated. The leading term is determined when the decay of the single site potential is slow. The leading term depends only on the classical effect from the scalar potential. To the contrary, the quantum effect appears when the decay of the single site potential is fast. The corresponding leading term is estimated and the leading order is determined. In the multidimensional cases, the leading order varies in different ways from the known results in the Poisson case. The same problem is considered for the negative potential. These estimates are applied to investigate the long time asymptotics of Wiener integrals associated with the random potentials.

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“…ξ. One says that the solution u is p-intermittent if 10) and intermittent if (1.10) holds for all p ∈ N \ {1}. Carmona and Molchanov [2] succeeded to investigate the annealed Lyapunov exponents, and to obtain the qualitative picture of intermittency (in terms of these exponents), for potentials of the form ξ(x, t) =Ẇ x (t), (1.11) where {W x (t) : x ∈ Z d , t ≥ 0} denotes a collection of independent Brownian motions.…”

Section: Intermittencymentioning

confidence: 99%

Intermittency on catalysts

Gaertner¹,

Hollander²,

Maillard³

2009

Trends in Stochastic Analysis

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The present paper provides an overview of results obtained in four recent papers by the authors. These papers address the problem of intermittency for the Parabolic Anderson Model in a time-dependent random medium, describing the evolution of a "reactant" in the presence of a "catalyst". Three examples of catalysts are considered: (1) independent simple random walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is on the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the reactant. It turns out that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant.MSC 2000. Primary 60H25, 82C44; Secondary 60F10, 35B40.

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The Parabolic Anderson Model

Cited by 73 publications

References 38 publications

The parabolic Anderson model with heavy-tailed potential

The parabolic Anderson model with heavy-tailed potential

Classical and Quantum Behavior of the Integrated Density of States for a Randomly Perturbed Lattice

Intermittency on catalysts

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