We study rays and co-rays in the Wasserstein space Pp(X) (p>1) whose ambient space X is a complete, separable, non-compact, locally compact length space.
Rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space.
Co-rays exist for any prescribed initial probability measure in the Wasserstein space.
Busemann functions in the Wasserstein space show that co-rays are negative gradient lines.