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We develop a theory of optimal transport for stationary random measures witha particular focus on stationary point processes. This provides us with anotion of geodesic distance between distributions of stationary random measuresand induces a natural displacement interpolation between them. In the settingof stationary point processes we leverage this transport distance to give ageometric interpretation for the evolution of infinite particle systems withstationary distribution. Namely, we characterise the evolution of infinitelymany Brownian motions as the gradient flow of the specific relative entropyw.r.t.~the Poisson process. Further, we establish displacement convexity of thespecific relative entropy along optimal interpolations of point processes.

Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy