Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials

@article{Hofstad2008WeakAA,
  title={Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials},
  author={Remco van der Hofstad and Peter Morters and Nadia Sidorovatsup},
  journal={Annals of Applied Probability},
  year={2008},
  volume={18},
  pages={2450-2494}
}
We study the parabolic Anderson problem, that is, the heat equation partial derivative(t)u = Delta u + xi u on (0,infinity) x Z(d) with independent identically distributed random potential {xi(Z): Z is an element of Z(d)) and localized initial condition u(0, x) = 1(0)(x). Our interest is in the long-term behavior of the random total mass U(t) = Sigma(z) u (t, z) of the unique nonnegative solution in the case that the distribution of xi(0) is heavy tailed. For this, we study two paradigm cases… 

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References

SHOWING 1-10 OF 14 REFERENCES

Long-time tails in the parabolic Anderson model

We consider the parabolic Anderson problem ∂ t u = κΔu + ξu on (0, ∞) × Z d with random i.i.d. potential ξ = (ξ(z)) z ∈ Zd and the initial condition u(0,.) ≡ 1. Our main assumption is that esssup

Geometric characterization of intermittency in the parabolic Anderson model

We consider the parabolic Anderson problem @tu = u + (x)u on R+ Z d with localized initial condition u(0;x) = 0(x) and random i.i.d. potential . Under the assumption that the distribution of (0) has

The Universality Classes in the Parabolic Anderson Model

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on$$\mathbb{Z}^{d}$$. We consider general i.i.d. potentials and show

Transition from the annealed to the quenched asymptotics for a random walk on random obstacles

In this work we study a natural transition mechanism describing the passage from a quenched (almost sure) regime to an annealed (in average) one, for a symmetric simple random walk on random

Enlargement of Obstacles for the Simple Random Walk

We consider a continuous time simple random walk moving among obstacles, which are sites (resp., bonds) of the lattice Z d . We derive in this context a version of the technique of enlargement of

Applied Probability

TLDR
The second edition of Applied Probability adds two new chapters on asymptotic and numerical methods and an appendix that separates some of the more delicate mathematical theory from the steady flow of examples in the main text.

Modelling Extremal Events

Parabolic problems for the Anderson model

Summary. This is a continuation of our previous work [6] on the investigation of intermittency for the parabolic equation (∂/∂t)u=Hu on ℝ+×ℤd associated with the Anderson Hamiltonian H=κΔ+ξ(·) for

Parabolic Anderson Problem and Intermittency

Extreme Values, Regular Variation, and Point Processes

Contents: Preface * Preliminaries * Domains of Attraction and Norming Constants * Quality of Convergence * Point Processes * Records and Extremal Processes * Multivariate Extremes * References *