A storage model in which the growth rate is proportional to the level of the dam plus a factor, both proportionality constant and factor depending on a finite Markov chain, is shown to be a Hunt process. Hitting distributions are shown to satisfy certain integral equations, communicating and recurrence classes are studied, and stationary measures are shown to exist when the dam is finite and the level has at least one recurrent linear growth phase, and in some other cases as well.
R. V. Erickson. "Stationary Measures for a Class of Storage Models Driven by a Markov Chain." Ann. Math. Statist. 43 (3) 997 - 1007, June, 1972. https://doi.org/10.1214/aoms/1177692563