Tightness for the Cover Time of the two dimensional sphere

TL;DR

This paper establishes the tightness of the centered and scaled cover time of a Wiener sausage on the 2D sphere, revealing precise asymptotic behavior as the radius shrinks.

Contribution

It proves the tightness of the cover time for Wiener sausages on the 2D sphere, providing new insights into their asymptotic distribution.

Findings

The centered and scaled cover time is tight as the radius approaches zero.

Asymptotic behavior involves a logarithmic correction term.

Results contribute to understanding stochastic processes on curved surfaces.

Abstract

Let $C^*_{\epsilon,S^2}$ denote the cover time of the two dimensional sphere by a Wiener sausage of radius $\epsilon$. We prove that $$\sqrt{C^{*}_{\epsilon,S^2} } -\sqrt{\frac{2A_{S^2}}{\pi}}(\log \epsilon^{-1}-\frac14\log\log \epsilon^{-1})$$ is tight, where $A_{S^2}=4\pi$ denotes the Riemannian area of $S^2$.