Brownian Functionals in Physics and Computer Science

TL;DR

This paper reviews Brownian functionals in one dimension, highlighting their derivations, properties, and diverse applications in physics and computer science, emphasizing the Feynman-Kac formula and first-passage functionals.

Contribution

It provides a pedagogical overview of path integral methods and the Feynman-Kac formula, illustrating their use in analyzing Brownian functionals with applications in multiple fields.

Findings

Derivation of diffusion equation from Einstein's work

Application of Feynman-Kac formula to Brownian functionals

Analysis of first-passage Brownian functionals in various contexts

Abstract

This is a brief review on Brownian functionals in one dimension and their various applications, a contribution to the special issue ``The Legacy of Albert Einstein" of Current Science. After a brief description of Einstein's original derivation of the diffusion equation, this article provides a pedagogical introduction to the path integral methods leading to the derivation of the celebrated Feynman-Kac formula. The usefulness of this technique in calculating the statistical properties of Brownian functionals is illustrated with several examples in physics and probability theory, with particular emphasis on applications in computer science. The statistical properties of "first-passage Brownian functionals" and their applications are also discussed.