The Wiener chaos order of the solution to the stochastic Schrödinger and heat equations is asymptotically Gaussian around a mean proportional to time, so noise sensitivity sets in at perturbation scale 1/t.
Gaussian fluctuations from random Schr¨ odinger equation, Communications in Partial Differential Equations 46.2 (2021): 201-232
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Noise sensitivity for stochastic heat and Schr\"odinger equation
The Wiener chaos order of the solution to the stochastic Schrödinger and heat equations is asymptotically Gaussian around a mean proportional to time, so noise sensitivity sets in at perturbation scale 1/t.