For a single quantum particle, random position measurements and random noise give identical evolution of linear observables, but the nonlinear observable <x>^2 grows linearly in time under measurements and as time cubed under noise at short times.
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Randomly measured quantum particles and thermal noise
For a single quantum particle, random position measurements and random noise give identical evolution of linear observables, but the nonlinear observable <x>^2 grows linearly in time under measurements and as time cubed under noise at short times.