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

## TL;DR

This paper establishes a large deviation principle for a class of reaction-diffusion equations with polynomial nonlinearities under Gaussian noise, in a regime where noise strength and correlation vanish, applicable across various polynomial degrees and dimensions.

## Contribution

It proves the validity of a large deviation principle for the dynamic $\

## Key findings

- Large deviation principle holds in the space of continuous trajectories.
- Valid for any polynomial degree and space dimension.
- Applicable when noise strength and correlation vanish under suitable scaling.

## 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.