  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 L\'evy 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, H\"ogele 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.

Probability