  We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p >
1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact,
locally compact length space. We show that rays in the Wasserstein space can be
represented as probability measures concentrated on the set of rays in the
ambient space. We show the existence of co-rays for any prescribed initial
probability measure. We introduce Busemann functions on the Wasserstein space
and show that co-rays are negative gradient lines in some sense.
