On White Noise Space and Levy's Brownian Motion on the Circle

TL;DR

This paper explores the nature of Brownian motion and white noise on the circle, revealing its relation to Euclidean Brownian motion and addressing the concept of white noise in circular domains.

Contribution

It formally defines white noise space and Brownian bridge on the circle, clarifying their properties and connections to classical Brownian motion.

Findings

Brownian motion on the circle is a regular Euclidean Brownian motion on half-circle

The Brownian motion is degenerated in Minlos's sense

White noise on the circle is formally defined

Abstract

In this article, we show that the Brownian motion on the circle constructed in Levy (1959) is a regular Euclidean Brownian motion on the half-circle with its own mirror image on the other half-circle, and is degenerated in the sense of Minlos (1959). This raises the question of what the white noise is on the circle. We then formally define the white noise space and its associated Brownian bridge.