On Small Deviations of Gaussian Multiplicative Chaos with A Strictly Logarithmic Covariance on Euclidean Ball

In the elementary and self-contained study we prove that in any natural spatial dimension d the small deviations of a Gaussian multiplicative chaos (GMC) M gamma\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_\gamma $$\end{document} are of lognormal type. We place the small deviations in the context of the regime of positive definiteness for a strictly logarithmic covariance kernel and provide the explicit bounds on the associated constants. We also provide a new representation of the Laplace transform of GMC related to a strictly logarithmic covariance kernel.