  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.
