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

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