# Geometric mean flows and the Cartan barycenter on the Wasserstein space over positive definite matrices

## TL;DR

This paper introduces flows on the Wasserstein space of probability measures over positive definite matrices, analyzing their differentiability, fixed points, and related inequalities, advancing understanding of geometric and probabilistic structures in matrix spaces.

## Contribution

It develops a new class of flows on Wasserstein space over positive definite matrices, establishing differentiability, a Lie-Trotter formula, and fixed point results related to the Karcher equation.

## Key findings

- Established differentiability of Cartan barycentric trajectories.
- Derived a version of the Lie-Trotter formula for these flows.
- Proved a fixed point theorem related to the Karcher equation.

## Abstract

We introduce a class of flows on the Wasserstein space of probability measures with finite first moment on the Cartan-Hadamard Riemannian manifold of positive definite matrices, and consider the problem of differentiability of the corresponding Cartan barycentric trajectory. As a consequence we have a version of Lie-Trotter formula and a related unitarily invariant norm inequality. Furthermore, a fixed point theorem related to the Karcher equation and the Cartan barycentric trajectory is also presented as an application.