Last zero time or Maximum time of the winding number of Brownian motions

TL;DR

This paper investigates the asymptotic behavior of the maximum time and last zero time of the winding number of planar Brownian motion, revealing their limit laws and growth rates.

Contribution

It introduces new results on the limit law of the maximum time process and establishes the equivalence in law between the maximum time and last zero time processes.

Findings

Limit law of the logarithm of maximum time with normalization

Upper growth rate of the maximum time process

Equivalence in law of maximum time and last zero time processes

Abstract

In this paper we consider the winding number, $\theta(s)$, of planar Brownian motion and study asymptotic behavior of the process of the maximum time, the time when $\theta(s)$ attains the maximum in the interval $0\le s \le t$. We find the limit law of its logarithm with a suitable normalization factor and the upper growth rate of the maximum time process itself. We also show that the process of the last zero time of $\theta(s)$ in $[0,t]$ has the same law as the maximum time process.