Linear formulations for the occupation measures of a diffusion process with singular boundary behavior and an arbitrary but fixed control process are motivated and derived. Under specific assumptions on the control process, the existence and uniqueness of solutions to these linear formulations for both long-term average or discounted cost criteria in infinite time are shown. In particular, absolute continuity for the occupation measure of the nonsingular behavior is proven. Two examples of simple Brownian diffusion illustrate the applicability of the results on specific control problems, and their relevance for ongoing research in stochastic control is discussed.