  Large deviation for Markov processes can be studied by Hamilton--Jacobi
equation techniques. The method of proof involves three steps: First, we apply
a nonlinear transform to generators of the Markov processes, and verify that
limit of the transformed generators exists. Such limit induces a
Hamilton--Jacobi equation. Second, we show that a strong form of uniqueness
(the comparison principle) holds for the limit equation. Finally, we verify an
exponential compact containment estimate. The large deviation principle then
follows from the above three verifications. This paper illustrates such a
method applied to a class of Hilbert-space-valued small diffusion processes.
The examples include stochastically perturbed Allen--Cahn, Cahn--Hilliard PDEs
and a one-dimensional quasilinear PDE with a viscosity term. We prove the
comparison principle using a variant of the Tataru method. We also discuss
different notions of viscosity solution in infinite dimensions in such context.
