We give decay estimates of the solution to the linear Schrödinger equation in dimension \(d \ge 3\) 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.

中文翻译：

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具有较小乘性噪声的$$ d \ ge 3 $$d≥3中的随机线性Schrödinger方程的衰减

我们给出了线性Schrödinger方程在维\（d \ ge 3 \）中的解决方案的衰减估计，该噪声具有小的噪声，该噪声在时间上是白色的，在空间上是彩色的。结果，我们还获得了溶液的某些渐近行为。该证明依赖于Journé-Soffer-Sogge使用的自举参数来确定性Schrödinger算子的衰减。