Matrix valued Brownian motion and a paper by Polya

TL;DR

This paper explores the geometric behavior of eigenvalues in matrix-valued Brownian motion and links Pólya's function, related to the Riemann hypothesis, to Brownian motion on symmetric spaces.

Contribution

It provides a geometric description of eigenvalue dynamics in matrix Brownian motion and connects Pólya's function to Brownian motion on symmetric spaces, offering new insights.

Findings

Eigenvalues of matrix Brownian motion have a geometric description.

Pólya's function related to the Riemann hypothesis can be modeled by Brownian motion.

Both functions are connected to Brownian motion on symmetric spaces.

Abstract

We give a geometric description of the motion of eigenvalues of a Brownian motion with values in some matrix spaces. In the second part we consider a paper by Polya where he introduced a function close to the Riemann zeta function, which satisfies Riemann hypothesis. We show that each of these two functions can be related to Brownian motion on a symmetric space.