# 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},
journal={Annals of Applied Probability},
year={2008},
volume={18},
pages={2450-2494}
}
• Published 21 June 2006
• Mathematics
• Annals of Applied Probability
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…

### A scaling limit theorem for the parabolic Anderson model with exponential potential

• Mathematics
• 2011
The parabolic Anderson problem is the Cauchy problem for the heat equation∂tu(t,z) = ∆u(t,z) + ξ (t,z)u(t,z) on (0,∞)×Zd with random potential (ξ (t,z) : z∈ Zd) and localized initial condition. In

### A two cities theorem for the parabolic Anderson model.

• Mathematics
• 2009
The parabolic Anderson problem is the Cauchy problem for the heat equation partial derivative(t)u(t, z) = Delta u(t,z) + xi(z)u(t,z) on (0,infinity) x Z(d) with random potential (xi(z): z is an

### Ageing in the parabolic Anderson model

• Mathematics
• 2009
The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on

### Delocalising the parabolic Anderson model through partial duplication of the potential

• Mathematics
• 2016
The parabolic Anderson model on $$\mathbb {Z}^d$$Zd with i.i.d. potential is known to completely localise if the distribution of the potential is sufficiently heavy-tailed at infinity. In this paper

### The Parabolic Anderson Model

• Mathematics
• 2016
This is a survey on the intermittent behavior of the parabolic Anderson model, which is the Cauchy problem for the heat equation with random potential on the lattice ℤd. We first introduce the model

### A Scaling Limit Theorem for the Parabolic Anderson Model with Exponential Potential

• Mathematics
• 2012
The parabolic Anderson problem is the Cauchy problem for the heat equation with random potential and localized initial condition. In this paper, we consider potentials which are constant in time and

• Mathematics
• 2006

## References

SHOWING 1-10 OF 14 REFERENCES

### Long-time tails in the parabolic Anderson model

• Mathematics
• 2000
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

• Mathematics
• 2007
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

• Mathematics
• 2006
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

• Mathematics
• 2005
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

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.

### Parabolic problems for the Anderson model

• Mathematics
• 1990
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

### 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 *