Precise Laplace asymptotics for singular stochastic PDEs: The case of 2D gPAM

TL;DR

This paper develops a Laplace asymptotic method for the renormalised solutions of the 2D generalized Parabolic Anderson Model driven by small noise, leveraging Hairer's regularity structures to handle singularities.

Contribution

It introduces a novel Laplace method for singular SPDEs using regularity structures, providing precise asymptotic bounds and extending classical path and rough path techniques.

Findings

Established a Taylor expansion of the solution in noise intensity.

Derived bounds for expansion terms and remainders.

Outlined adaptation to other models.

Abstract

We implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. Our work rests upon Hairer's theory of regularity structures which allows to generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter: We prove precise bounds for its terms and the remainder and use them to estimate asymptotically irrevelant terms to arbitrary order. While most of our arguments are not specific to gPAM, we also outline how to adapt those that are.