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 $\partial_tu=\Delta u+\xi u$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{\xi(z):z\in{\mathbb{Z}}^d\}$ and localized initial condition $u(0,x)={\mathbf{1}}_0(x)$. Our interest is in the longterm behavior of the random total mass $U(t)=\sum_zu(t,z)$ 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606527
 Bibcode:
 2006math......6527V
 Keywords:

 Mathematics  Probability;
 60H25;
 82C44 (Primary);
 60F05;
 60F15;
 60G70 (Secondary)
 EPrint:
 Published in at http://dx.doi.org/10.1214/08AAP526 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)