In this article, the asymptotic behavior of the solution to the following one dimensional Schrodinger equations with white noise dispersion idu + u(xx) o dW + vertical bar u vertical bar(p-1)udt = 0 is studied. Here the equation is written in the Stratonovich formulation, and W (t) is a standard real valued Brownian motion. After establishing the global well-posedness, theoretical proof and numerical investigations are provided showing that, for a deterministic small enough initial data in L-x(1) boolean AND H-x(1), the expectation of the L-x(infinity) norm of the solutions decay to zero at O(t(-1/4)) as t goes to +infinity, as soon as p > 7.