A two cities theorem for the parabolic Anderson model

Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation ${\partial }_{t}u(t,z)=\Delta u(t,z)+\xi (z)u(t,z)$ on $(0,\infty )\times {\mathbb{Z}}^d$ with random potential $(\xi (z):z\in {\mathbb{Z}}^d)$. We consider independent and identically distributed potentials, such that the distribution function of $\xi (z)$ converges polynomially at infinity. If $u$ is initially localized in the origin, that is, if $u(0,{z})={\mathbh1}_0({z})$, we show that, as time goes to infinity, the solution is completely localized in two points almost surely and in one point with high probability. We also identify the asymptotic behavior of the concentration sites in terms of a weak limit theorem.

Published in at http://dx.doi.org/10.1214/08-AOP405 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Geometric characterization of intermittency in the parabolic Anderson model
• The universality classes in the parabolic Anderson model
• Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials
• Intermittency in a catalytic random medium
• Complete localisation in the parabolic Anderson model with Pareto-distributed potential

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