Brownian Motion, "Diverse and Undulating"

TL;DR

This paper reviews the history, mathematical significance, and recent advances in Brownian motion, highlighting its applications in physics and biophysics, and discussing recent progress in the geometry of Brownian paths.

Contribution

It provides a comprehensive overview of Brownian motion's historical development, mathematical importance, and recent geometric research, connecting theory with modern applications.

Findings

Brownian motion's role in biophysical experiments, such as DNA force measurement

Elementary explanation of Newtonian potential representation via Brownian motion

Recent progress in understanding the geometry of planar Brownian curves, including conformal invariance and multifractality

Abstract

We describe in detail the history of Brownian motion, as well as the contributions of Einstein, Sutherland, Smoluchowski, Bachelier, Perrin and Langevin to its theory. The always topical importance in physics of the theory of Brownian motion is illustrated by recent biophysical experiments, where it serves, for instance, for the measurement of the pulling force on a single DNA molecule. In a second part, we stress the mathematical importance of the theory of Brownian motion, illustrated by two chosen examples. The by-now classic representation of the Newtonian potential by Brownian motion is explained in an elementary way. We conclude with the description of recent progress seen in the geometry of the planar Brownian curve. At its heart lie the concepts of conformal invariance and multifractality, associated with the potential theory of the Brownian curve itself.