The Laguerre process and generalized Hartman--Watson law

TL;DR

This paper investigates the eigenvalue dynamics of Laguerre processes, deriving stochastic equations, generators, and distributions, including the generalized Hartman--Watson law, with explicit results for 2x2 matrices.

Contribution

It provides new stochastic differential equations, generator formulas, and explicit density functions for the generalized Hartman--Watson law in Laguerre processes.

Findings

Derived stochastic differential equations for eigenvalues

Computed the infinitesimal generator and semi-group

Explicit density functions for the generalized Hartman--Watson law for 2x2 matrices

Abstract

In this paper, we study complex Wishart processes or the so-called Laguerre processes $(X_t)_{t\geq0}$. We are interested in the behaviour of the eigenvalue process; we derive some useful stochastic differential equations and compute both the infinitesimal generator and the semi-group. We also give absolute-continuity relations between different indices. Finally, we compute the density function of the so-called generalized Hartman--Watson law as well as the law of $T_0:=\inf\{t,\det(X_t)=0\}$ when the size of the matrix is 2.