The Donsker-Varadhan $I$-Function which measures the asymptotic rate of decay of the distribution of the occupation measure of a Markov process is evaluated for diffusion processes with boundaries.
Ross Pinsky. "The $I$-Function for Diffusion Processes with Boundaries." Ann. Probab. 13 (3) 676 - 692, August, 1985. https://doi.org/10.1214/aop/1176992902