Brownian functionals in physics and computer science

This is a brief review on Brownian functionals in one dimension and their various applications. 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.