  Probabilistic frames are a generalization of finite frames into the
Wasserstein space of probability measures with finite second moment. We
introduce new probabilistic definitions of duality, analysis, and synthesis and
investigate their properties. In particular, we formulate a theory of transport
duals for probabilistic frames and prove certain properties of this class. We
also investigate paths of probabilistic frames, identifying conditions under
which geodesic paths between two such measures are themselves probabilistic
frames. In the discrete case, this is related to ranks of convex combinations
of matrices, while, in the continuous case, this is related to the continuity
of the optimal transport plan.
