  We develop a theory of optimal transport for stationary random measures with
a focus on stationary point processes and construct a family of distances on
the set of stationary random measures. These induce a natural notion of
interpolation between two stationary random measures along a shortest curve
connecting them. In the setting of stationary point processes we leverage this
transport distance to give a geometric interpretation for the evolution of
infinite particle systems with stationary distribution. Namely, we characterise
the evolution of infinitely many Brownian motions as the gradient flow of the
specific relative entropy w.r.t.~the Poisson point process. Further, we
establish displacement convexity of the specific relative entropy along optimal
interpolations of point processes and establish an stationary analogue of the
HWI inequality, relating specific entropy, transport distance, and a specific
relative Fisher information.
