Stationary parabolic Anderson model and intermittency

Abstract

This paper is devoted to the analysis of the large time behavior of the solutions of the Anderson parabolic problem: $$\frac{{\partial u}}{{\partial t}} = \kappa \Delta u\xi (x)u$$ when the potential ξ(x) is a homogeneous ergodic random field on ℝ d . Our goal is to prove the asymptotic spatial intermittency of the solution and for this reason, we analyze the large time properties of all the moments of the positive solutions. This provides an extension to the continuous space ℝ d of the work done originally by Gartner and Molchanov in the case of the lattice ℤ d . In the process of our moment analysis, we show that it is possible to exhibit new asymptotic regimes by considering a special class of generalized Gaussian fields, interpolating continuously between the exponent 2 which is found in the case of bona fide continuous Gaussian fields ξ(x) and the exponent 3/2 appearing in the case of a one dimensional white noise. Finally, we also determine the precise almost sure large time asymptotics of the positive solutions.