Small Noise Asymptotic Expansion for a Infinite Dimensional Stochastic Reaction-Diffusion Forced Van Der Pol Equation

Starting from the classical Van der Pol equation, after suitable changes of variables, we derive a reaction-diffusion type forcedVan der Pol equation with values in a suitable infinite dimensionalHilbert space. In particular we will perturb previous equation with a small additive Brownian noise. After some preliminary results concerning the non trivial existence and uniqueness of a solution, due to the presence of anon-Lipschitz non-linearity, we provide a rigorous asymptoticexpansions in term of the small parameter ε of the related solution up to order 3. We will then explicitly write the first threeorder of the rigorous expansion and provide as well an upperbound for the remainder.