### A two cities theorem for the parabolic Anderson model

Sidorova, N *(UCL)*

Thursday 24 July 2008, 16:00-16:45

Seminar Room 1, Newton Institute

#### Abstract

The parabolic Anderson problem is the Cauchy problem for the heat equation u_t(t,z)=Delta u(t,z)+xi(z)u(t,z) on the d-dimensional integer lattice with random potential. We consider independent and identically distributed potentials, such that the corresponding distribution function converges polynomially at infinity. If the solution is initially localised in the origin we show that, as time goes to infinity, it will be completely localised in two points almost surely and in one point with high probability. We also identify the asymptotic behaviour of the concentration sites in terms of a weak limit theorem.

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