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We put forward a general framework for the study of a pathwise central limittheorem (CLT) and a moderate deviation principle (MDP) for stochastic partialdifferential equations perturbed with a small multiplicative linear noise bymeans of the theory of rough paths. The CLT can be interpreted as theconvergence to a pathwise derivative of the It\^o-Lyons map. The result followsby applying a pathwise Malliavin-like calculus for rough paths and fromcompactness methods. The convergence in the CLT is quantified by an optimalspeed of convergence. From the exponential equivalence principle and theknowledge of the speed of convergence, we can derive easily a MDP. Inparticular, we do not apply the weak convergence approach usually employed inthis framework. We derive a pathwise CLT and a MDP for the stochasticLandau-Lifschitz-Gilbert equation in one dimension, for the heat equation andfor a stochastic reaction-diffusion equation. As a further application, wederive a pathwise convergence to the CLT limit and a corresponding MDP forequations driven by linear It\^o noise.

Pathwise central limit theorem and moderate deviations via rough paths for SPDEs with multiplicative noise