Source: Ann. Probab.
Volume 37, Number 1
The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z)=Δu(t, z)+ξ(z)u(t, z) on (0, ∞)×ℤd with random potential (ξ(z):z∈ℤd). We consider independent and identically distributed potentials, such that the distribution function of ξ(z) converges polynomially at infinity. If u is initially localized in the origin, that is, if , 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.
 Anderson, P. W. (1958). Absence of diffusion in certain random lattices. Phys. Rev. 109 1492–1505.
 Ben Arous, G., Molchanov, S. and Ramírez, A. F. (2007). Transition asymptotics for reaction-diffusion in random media. In Probability and Mathematical Physics. CRM Proc. Lecture Notes 42 1–40. Amer. Math. Soc., Providence, RI.
 Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125.
 Fleischmann, K. and Greven, A. (1992). Localization and selection in a mean field branching random walk in a random environment. Ann. Probab. 20 2141–2163.
 Fleischmann, K. and Molchanov, S. A. (1990). Exact asymptotics in a mean field model with random potential. Probab. Theory Related Fields 86 239–251.
 Gärtner, J. and den Hollander, F. (2006). Intermittency in a catalytic random medium. Ann. Probab. 34 2219–2287.
 Gärtner, J. and König, W. (2005). The parabolic Anderson model. In Interacting Stochastic Systems 153–179. Springer, Berlin.
 Gärtner, J., König, W. and Molchanov, S. (2007). Geometric characterization of intermittency in the parabolic Anderson model. Ann. Probab. 35 439–499.
 Gärtner, J. and Molchanov, S. A. (1990). Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 613–655.
 van der Hofstad, R., König, W. and Mörters, P. (2006). The universality classes in the parabolic Anderson model. Comm. Math. Phys. 267 307–353.
 van der Hofstad, R., Mörters, P. and Sidorova, N. (2008). Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials. Ann. Appl. Probab. 18 2450–2494.
 König, W., Mörters, P. and Sidorova, N. (2006). Complete localisation in the parabolic Anderson model with Pareto-distributed potential. Unpublished. Available at arXiv:math.PR/0608544.
 Molchanov, S. (1994). Lectures on random media. In Lectures on Probability Theory (Saint-Flour, 1992). 1581 242–411. Springer, Berlin.
 Sznitman, A.-S. (1998). Brownian Motion, Obstacles and Random Media. Springer, Berlin.
 Zel’dovich, Y. B., Molchanov, S. A., Ruzmaǐkin, A. A. and Sokolov, D. D. (1987). Intermittency in random media. Uspekhi Fiz. Nauk 152 3–32.