PARABOLIC ANDERSON PROBLEM AND INTERMITTENCY

We consider the stochastic partial differential equation partial deriv ative u/partial derivative t = kappaDELTAu + xi(t)(x)u. The potential xi(t)(x) is assumed to be a mean zero homogeneous Gaussian field. We p ay special attention to the white noise case. In order to minimize the technical difficulties we consider only the case the discrete Laplaci an DELTA on the lattice Z(d). We prove existence and uniqueness (for a lmost every realization of the random potential) for nonnegative initi al conditions. These results are proved by means of the Feynman-Kac re presentation of the minimal solutions. Infinite dimensional Ito and St ratonovich equations are needed to study the white noise case. We then prove that the solutions have moments of all orders. In the case of a white noise potential we derive a family of closed equations for thes e moments. We then prove the existence of the moment Lyapunov exponent s and we study their dependence upon the diffusion constant kappa. As a consequence, we show that there is full intermittency of the solutio n when the dimension d is not greater than 2 while the same intermitte ncy only holds for large values of the diffusion constant in higher di mensions. The fundamental equation can be viewed as a parabolic Anders on model and this phase transition is natural from the point of view o f localization theory. Finally, the last chapter is devoted to the stu dy of the almost sure Lyapunov exponent. We prove its existence and we derive their asymptotic behavior for small kappa.