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

Abstract

We study the parabolic Anderson problem, that is, the heat equation ${\partial }_{t}u=\Delta u+\xi u$ on $(0,\infty )\times {\mathbb{Z}}^d$ with independent identically distributed random potential $\{\xi (z):z\in {\mathbb{Z}}^d\}$ and localized initial condition $u(0,x)=1_0(x)$. Our interest is in the long-term behavior of the random total mass $U(t)={\sum }_{z}u(t,z)$ of the unique nonnegative solution in the case that the distribution of $\xi (0)$ is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.

Published in at http://dx.doi.org/10.1214/08-AAP526 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Geometric characterization of intermittency in the parabolic Anderson model
• The universality classes in the parabolic Anderson model
• Transition from the annealed to the quenched asymptotics for a random walk on random obstacles

Cited by in corpus (17)

• A two cities theorem for the parabolic Anderson model
• Localisation and ageing in the parabolic Anderson model with Weibull potential
• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian
• Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models
• Eigenvalue order statistics for random Schrödinger operators with doubly-exponential tails
• Mass concentration and aging in the parabolic Anderson model with doubly-exponential tails
• Intermittency for branching random walk in Pareto environment
• Localisation in the Bouchaud-Anderson Model
• Ageing in the parabolic Anderson model
• Quenched tail estimate for the random walk in random scenery and in random layered conductance
• Complete localisation in the parabolic Anderson model with Pareto-distributed potential
• Localization for random walks among random obstacles in a single Euclidean ball
• Delocalising the parabolic Anderson model through partial duplication of the potential
• Slowdown estimates for one-dimensional random walks in random environment with holding times
• Localization of a one-dimensional simple random walk among power-law renewal obstacles
• Complete Localisation and Exponential Shape of the Parabolic Anderson Model with Weibull Potential Field
• Poly-logarithmic localization for random walks among random obstacles