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We give decay estimates of the solution to the linear Schrödinger equation in dimension $$d \ge 3$$ d ≥ 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 Journé–Soffer–Sogge for decay of deterministic Schrödinger operators.

Decay of the stochastic linear Schrödinger equation in $$d \ge 3$$ with small multiplicative noise