The constant coefficient in precise Laplace asymptotics for gPAM

TL;DR

This paper derives an explicit formula for the constant coefficient in precise Laplace asymptotics for the gPAM, combining Gaussian analysis and regularity structures to advance understanding of asymptotic behavior.

Contribution

It provides a new explicit formula for the constant coefficient in Laplace asymptotics of gPAM using traces and Carleman-Fredholm determinants.

Findings

Explicit formula for the constant coefficient in asymptotics

Improved regularity of the phase functional minimiser

Integration of Gaussian analysis with regularity structures

Abstract

This article resumes the analysis of precise Laplace asymptotics for the generalised Parabolic Anderson Model (gPAM) initiated by Peter Friz and the author. More precisely, we provide an explicit formula for the constant coefficient in the asymptotic expansion in terms of traces and Carleman-Fredholm determinants of certain explicit operators. The proof combines classical Gaussian analysis in abstract Wiener spaces with arguments from the theory of regularity structures. As an ingredient, we prove that the minimiser in the (extended) phase functional of gPAM has better than just Cameron-Martin regularity.