Matrix-valued Bessel processes

Abstract

This paper introduces a matrix analog of the Bessel processes, taking values in the closed set $E$ of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map $x\mapsto x^\top x$. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function $w(x)=|\det x|^\alpha$ is a weight function in the Muckenhoupt $A_p$ class for $-1 < \alpha\le 0$ ($p=1$) and $-1 < \alpha < p-1$ ($p > 1$). The set of matrices of co-rank at least two has zero capacity with respect to the measure $m(dx)=|\det x|^\alpha dx$ if $\alpha >-1$, and if $\alpha\ge 1$ this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) $E$ with Neumann boundary conditions. The highly non-convex, non-Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods.