 Skorokhod reflection map on the half-line

For a continuous driving path with $g(0)\ge 0$, the running supremum supplies the minimal nondecreasing regulator keeping the path nonnegative. It increases only at a new negative running minimum, where the reflected path is zero. Any regulator satisfying the Skorokhod problem constraints must coincide with it. On a finite time interval, $\Vert \Gamma (g)-\Gamma (h){\Vert }_{\infty }\le 2\Vert g-h{\Vert }_{\infty }$, so the map is continuous and transfers a functional central limit theorem to reflected workloads.