Mathematics – Probability
Scientific paper
2006-06-21
Annals of Applied Probability 2008, Vol. 18, No. 6, 2450-2494
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/08-AAP526 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/08-AAP526
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 long-term 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.
der Hofstad Remco van
Mörters Peter
Sidorova\tsup Nadia
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