A Multiscale Guide to Brownian Motion

TL;DR

This paper introduces a wavelet-based multiscale representation of Brownian motion that offers intuitive insights and simplifies the understanding of its properties, also extending to related Gaussian processes.

Contribution

It presents a novel wavelet construction of Brownian motion, providing a clear, explicit formula and extending the approach to other Gaussian processes and fields.

Findings

Wavelet representation elucidates classical properties of Brownian motion.

Explicit formula maps unit interval to Brownian paths.

Applicable to simulation of Gaussian processes in various domains.

Abstract

We revise the Levy's construction of Brownian motion as a simple though still rigorous approach to operate with various Gaussian processes. A Brownian path is explicitly constructed as a linear combination of wavelet-based "geometrical features" at multiple length scales with random weights. Such a wavelet representation gives a closed formula mapping of the unit interval onto the functional space of Brownian paths. This formula elucidates many classical results about Brownian motion (e.g., non-differentiability of its path), providing intuitive feeling for non-mathematicians. The illustrative character of the wavelet representation, along with the simple structure of the underlying probability space, is different from the usual presentation of most classical textbooks. Similar concepts are discussed for fractional Brownian motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional Gaussian fields. Wavelet representations and dyadic decompositions form the basis of many highly efficient numerical methods to simulate Gaussian processes and fields, including Brownian motion and other diffusive processes in confining domains.