An application of the Gaussian correlation inequality to the small deviations for a Kolmogorov diffusion

TL;DR

This paper applies the Gaussian correlation inequality to analyze small deviations and Chung's laws of iterated logarithm for an iterated Kolmogorov diffusion, advancing understanding of its probabilistic behavior.

Contribution

It introduces a novel application of the Gaussian correlation inequality to small deviations and Chung's laws for Kolmogorov diffusions.

Findings

Solved the small ball problem for the diffusion using Gaussian correlation inequality

Proved Chung's laws of iterated logarithm at zero and infinity for the process

Enhanced understanding of the probabilistic properties of Kolmogorov diffusions

Abstract

We consider an iterated Kolmogorov diffusion $X_{t}$ of step $n$. The small ball problem for $X_{t}$ is solved by means of the Gaussian correlation inequality. We also prove Chung's laws of iterated logarithm for $X_{t}$ both at time zero and infinity.