STATIONARY PARABOLIC ANDERSON MODEL AND INTERMITTENCY

This paper is devoted to the analysis of the large time behavior of th e solutions of the Anderson parabolic problem: partial derivative u/pa rtial derivative t = k Delta u + xi(x)u when the potentia xi(x) is a h omogeneous ergodic random field on IR(d). Our goal is to prove the asy mptotic spatial intermittency of the solution and for this reason, we analyze the large time properties of all the moments of the positive s olutions. This provides an extension to the continuous space IR(d) of the work done originally by Gartner and Molchanov in the case of the l attice Z(d). In the process of our moment analysis, we show that it is possible to exhibit new asymptotic regimes by considering a special c lass of generalized Gaussian fields, interpolating continuously betwee n the exponent 2 which is found in the case of bona fide continuous Ga ussian fields xi(x) and the exponent 3/2 appearing in the case of a on e dimensional white noise. Finally, we also determine the precise almo st sure large time asymptotics of the positive solutions.