# Ergodicity of affine processes on the cone of symmetric positive semidefinite matrices

## TL;DR

This paper studies the long-term behavior of affine processes on the cone of positive semidefinite matrices, establishing conditions for the existence of a unique limit distribution and analyzing convergence rates in specific metrics.

## Contribution

It provides new results on ergodicity and convergence rates for affine processes on matrix cones, under moment and diffusion conditions.

## Key findings

- Finite log-moment of jump measure ensures unique limit distribution.
- Convergence in Laplace transform metric is established.
- Wasserstein convergence rate derived under moment and diffusion assumptions.

## Abstract

This article investigates the long-time behavior of conservative affine processes on the cone of symmetric positive semidefinite $d\times d$-matrices. In particular, for conservative and subcritical affine processes on this cone we show that a finite $\log$-moment of the state-independent jump measure is sufficient for the existence of a unique limit distribution. Moreover, we study the convergence rate of the underlying transition kernel to the limit distribution: firstly, in a specific metric induced by the Laplace transform and secondly, in the Wasserstein distance under a first moment assumption imposed on the state-independent jump measure and an additional condition on the diffusion parameter.