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This article generalizes the small noise cutoff phenomenon to the strongsolutions of the stochastic heat equation and the damped stochastic waveequation over a bounded domain subject to additive and multiplicative Wienerand L\'evy noises in the Wasserstein distance. For the additive noise case, weobtain analogous infinite dimensional results to the respective finitedimensional cases obtained recently by Barrera, H\"ogele and Pardo (JSP2021),that is, the (stronger) profile cutoff phenomenon for the stochastic heatequation and the (weaker) window cutoff phenomenon for the stochastic waveequation. For the multiplicative noise case, which is studied in this contextfor the first time, the stochastic heat equation also exhibits profile cutoffphenomenon, while for the stochastic wave equation the methods break down dueto the lack of symmetry. The methods rely strongly on the explicit knowledge ofthe respective eigensystem of the stochastic heat and wave operator and theexplicit representation of the stochastic solution flows in terms of stochasticexponentials.

The cutoff phenomenon for the stochastic heat and the wave equation subject to small L\'evy noise