  We give decay estimates of the solution to the linear Schr\"odinger equation
in dimension $d \geq 3$ with a small noise which is white in time and colored
in space. As a consequence, we also obtain certain asymptotic behaviour of the
solution. The proof relies on the bootstrapping argument used by
Journ\'e-Soffer-Sogge for decay of deterministic Schr\"odinger operators.
