Super-Gaussian Decay of Exponentials: A Sufficient Condition

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 $\exp (-p(x{)}^{2+\epsilon }+\alpha q(x{)}^2)$ is integrable for all $\alpha ,\epsilon >0$ 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.

19 pages, shortened and improved proofs, added examples, version accepted for publication in J. Math. Anal. Appl

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