Weak and Almost Sure Limits for the Parabolic Anderson Model with Heavy Tailed Potentials

Weak and Almost Sure Limits for the Parabolic Anderson Model with Heavy Tailed Potentials

We study the parabolic Anderson problem, that is, the heat equation $\partial _{t}u=\Delta u+\xi u$ on $(0,\infty)\times {\Bbb Z}^{d}$ with independent identically distributed random potential $\{\xi (z)\colon z\in {\Bbb Z}^{d}\}$ and localized initial condition u(0, x) = 1₀(x). Our interest is in t...

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Veröffentlicht in:The Annals of Applied Probability. - Institute of Mathematical Statistics. - 18(2008), 6, Seite 2450-2494
1. Verfasser: van der Hofstad, Remco (VerfasserIn)
Weitere Verfasser: Mörters, Peter, Sidorova, Nadia
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 2008
Zugriff auf das übergeordnete Werk:The Annals of Applied Probability
Schlagworte:Anderson Hamiltonian Parabolic Anderson problem Long term behavior Intermittency Localization Random environment Random potential Partial differential equations with random coefficients Heavy tails Extreme value theory mehr... Pareto distribution Weibull distribution Weak limit theorem Law of the iterated logarithm Mathematics Physical sciences Philosophy
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520 |a We study the parabolic Anderson problem, that is, the heat equation $\partial _{t}u=\Delta u+\xi u$ on $(0,\infty)\times {\Bbb Z}^{d}$ with independent identically distributed random potential $\{\xi (z)\colon z\in {\Bbb Z}^{d}\}$ and localized initial condition u(0, x) = 1₀(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 ξ(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. 
540 |a Copyright 2008 The Institute of Mathematical Statistics 
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