Complete localisation in the parabolic Anderson model with Pareto-distributed potential

الأصل · EN

The parabolic Anderson problem is the Cauchy problem for the heat equation ∂ₜ u(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 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)=₀(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 ᵈ such that ∞ u(t,Zₜ)/∑z∈ᵈ u(t,z) =1, in probability. We also identify the asymptotic behaviour of Zₜ in terms of a weak limit theorem.