I am currently studying the geodesics in the space of probability measures over a Polish space (complete and separable metric space). In the "Optimal Transport - Old and new" book of C. Villani, there's the definition 7.19 (page 126) of "dynamical coupling":
"Definition: Let $(\mathcal{X},d)$ be a Polish space. A dynamical transference plan $\Pi$ is a probability measure on the space $C([0,1],\mathcal{X})$. A dynamical coupling of two probability measures $\mu_0,\mu_1\in\mathscr{P}(\mathcal{X})$ is a random curve $\gamma:[0,1]\rightarrow \mathcal{X}$ such that $\mathrm{law}\,(\gamma_0)=\mu_0$, $\mathrm{law}\,(\gamma_1)=\mu_1$."
What I don't understand is the concept of $\textit{random curve}$: why can we talk about "laws" of the curve? To me it seems that $\gamma$ is just an usual curve over a metric space!
Thank you for your replies in advance!