Abstract: The parabolic Anderson model is the Cauchy problem for the heat equation with a random potential. We consider this model in a setting which is continuous in time and discrete in space, and focus on time-constant, independent and identically distributed potentials with polynomial tails at infinity. We are concerned with the long-term temporal dynamics of this system. Our main result is that the periods, in which the profile of the solutions remains nearly constant, are increasing linearly over time, a phenomenon known as ageing. We describe this phenomenon in the weak sense, by looking at the asymptotic probability of a change in a given time window, and in the strong sense, by identifying the almost sure upper envelope for the process of the time remaining until the next change of profile. We also prove functional scaling limit theorems for profile and growth rate of the solution of the parabolic Anderson model.

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 d with random potential (ξ(z): z ∈ Z d). We consider independent and identically distributed potentials, such that the distribution function of ξ(z) converges polynomially at infinity. If u is initially localised in the origin, i.e., if u(0, x) = 1l0(x), we show that, as time goes to infinity, the solution is 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.

A scaling limit theorem for the parabolic Anderson model with exponential potential