This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and Levy noises in the Wasserstein distance. For the additive noise case, we obtain analogous infinite dimensional results to the respective finite dimensional cases obtained recently by Barrera, Hogele and Pardo (JSP2021), that is, the (stronger) profile cutoff phenomenon for the stochastic heat equation and the (weaker) window cutoff phenomenon for the stochastic wave equation. For the multiplicative noise case, which is studied in this context for the first time, the stochastic heat equation also exhibits profile cutoff phenomenon, while for the stochastic wave equation the methods break down due to the lack of symmetry. The methods rely strongly on the explicit knowledge of the respective eigensystem of the stochastic heat and wave operator and the explicit representation of the stochastic solution flows in terms of stochastic exponentials.
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