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

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.

Electron. Commun. Probab. 2014