On the expected diameter of planar Brownian motion

James McRedmond; Chang Xu

arXiv:1707.03725·math.PR·September 17, 2018

On the expected diameter of planar Brownian motion

James McRedmond, Chang Xu

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TL;DR

This paper refines bounds on the expected diameter of planar Brownian motion's trace, narrowing the range and providing simulation-based estimates for its average size.

Contribution

It improves existing bounds on the expected diameter of planar Brownian motion and offers simulation estimates for its value.

Findings

01

Refined bounds: 1.601 to 2.355 for expected diameter

02

Simulation suggests expected diameter around 1.99

03

Enhanced understanding of Brownian motion trace geometry

Abstract

Known results show that the diameter $d_{1}$ of the trace of planar Brownian motion run for unit time satisfies $1.595 \leq E d_{1} \leq 2.507$ . This note improves these bounds to $1.601 \leq E d_{1} \leq 2.355$ . Simulations suggest that $E d_{1} \approx 1.99$ .

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