A multiscale guide to Brownian motion

Author: 

Belyaev, D
Grebenkov, D
Jones, P

Publication Date: 

1 January 2015

Journal: 

Journal of Physics A: Mathematical and Theoretical

Last Updated: 

2020-06-27T08:46:12.693+01:00

Issue: 

4

Volume: 

49

DOI: 

10.1088/1751-8113/49/4/043001

page: 

043001-043001

abstract: 

We revise the Levy's construction of Brownian motion as a simple though 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 Brownian bridge, 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.

Symplectic id: 

573427

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article