Super-Gaussian Decay of Exponentials: A Sufficient Condition

TL;DR

This paper establishes a sufficient condition for exponential functions to exhibit tail decay faster than Gaussian, with implications for integrability in Gaussian measures and applications in quantum field theory.

Contribution

It provides a new criterion for super-Gaussian decay of exponentials on locally convex spaces, extending Fernique's theorem.

Findings

Exponential decay stronger than Gaussian under certain conditions

Integrability of specific exponential functions with super-Gaussian decay

Application to quantum field theory contexts

Abstract

In this article, we present a sufficient condition for the exponential $\exp({-f})$ to have a tail decay stronger than any Gaussian, where $f$ is defined on a locally convex space $X$ and grows faster than a squared seminorm on $X$. In particular, our result proves that $\exp({-p(x)^{2+\varepsilon}+\alpha q(x)^2})$ is integrable for all $\alpha,\varepsilon>0$ w.r.t. a Radon Gaussian measure on a nuclear space $X$, if $p$ and $q$ are continuous seminorms on $X$ with compatible kernels. This can be viewed as an adaptation of Fernique's theorem and, for example, has applications in quantum field theory.