Large deviations for the dynamic ${\Phi }_d^{2n}$ model

Abstract

We are dealing with the validity of a large deviation principle for a class of reaction-diffusion equations with polynomial nonlinearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\epsilon$ and $\delta (\epsilon )$, respectively, with $0<\epsilon ,\delta (\epsilon )<<1$. We prove that, under the assumption that $\epsilon$ and $\delta (\epsilon )$ satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension.