As a generalization of scale functions of spectrally negative Lévy processes, we define generalized scale functions of general standard processes with no positive jumps. For this purpose, we utilize the excursion theory. Using the generalized scale functions, we study Laplace transforms of hitting times, potential measures and duality.
"Generalized scale functions of standard processes with no positive jumps." Electron. Commun. Probab. 25 1 - 12, 2020. https://doi.org/10.1214/20-ECP289