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

Decay of the stochastic linear Schr\"odinger equation in $d \geq 3$ with small multiplicative noise