Complete Localisation in the Parabolic Anderson Model with Pareto-distributed Potential

The parabolic Anderson problem is the Cauchy problem for the heat equation ∂tu(t, z) = ∆u(t, z) + ξ(z)u(t, z) on (0,∞) × Z with random potential (ξ(z) : z ∈ Z). We consider independent and identically distributed potential variables, such that Prob(ξ(z) > x) decays polynomially as x ↑ ∞. If u is initially localised in the origin, i.e. if u(0, x) = 1l0(x), we show that, at any large time t, the solution is completely localised in a single point with high probability. More precisely, we find a random process (Zt : t ≥ 0) with values in Z such that limt↑∞ u(t, Zt)/ ∑ z∈Z u(t, z) = 1, in probability. We also identify the asymptotic behaviour of Zt in terms of a weak limit theorem.