On the long time behavior of the stochastic heat equation

We consider the stochastic heat equation in one space dimension and compute - for a particular choice of the initial datum - the exact long time asymp totic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent, Equivalently, by interpreting the solution as the partition func tion of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is b ased on a representation of the solution in terms of the weakly asymmetric exclusion process.