Super-Gaussian Decay of Exponentials: A Sufficient Condition
Benjamin Hinrichs, Daan Willem Janssen, Jobst Ziebell
TL;DR
The paper establishes Fernique-type integrability conditions for tails stronger than Gaussian of expressions of the form $e^{-p(x)^{2+\varepsilon} + \alpha q(x)^2}$ under centred Gaussian measures on locally convex spaces. By combining absolutely 2-summing map theory with finite-dimensional projections, it proves a general integrability theorem for continuous seminorms $p$ and $q$ under suitable factorisation conditions on the Cameron–Martin space, and shows that nuclear spaces yield integrability for all $\alpha>0$ and $\varepsilon>0$. It further proves uniform convergence of finite-dimensional approximations and extends the framework to μ-measurable seminorms via pushforward to Banach spaces, broadening the applicability to quantum field theory contexts. Collectively, the results provide a robust, Fernique-type tail-control toolkit for infinite-dimensional Gaussian measures with potential applications in quantum field theory and related areas.
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 $\exp({-p(x)^{2+\varepsilon}+αq(x)^2})$ is integrable for all $α,\varepsilon>0$ 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.