A Remark on Zeros of Brownian Motion

TL;DR

This paper investigates the behavior of zeros of standard Brownian motion, providing almost sure lower bounds for the minimum absolute value of the process over intervals where it has no zeros, within a specific covering family.

Contribution

It establishes almost sure lower bounds for the minimum absolute value of Brownian motion on zero-free intervals from a countable covering, advancing understanding of zero distribution.

Findings

Provides explicit lower bounds for Brownian motion on zero-free intervals.

Shows the bounds hold almost surely for a countable family of intervals.

Enhances understanding of the zero set structure of Brownian motion.

Abstract

Let $ \{W(t), t\ge 0\}$ be a standard Brownian motion. If $I$ is a bounded interval on which $W $ has no zero, an almost sure lower bound to $\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is taken from a given countable family of intervals covering the positive half-line.