Boundary conditions and an ergodic theorem for processes with independent increments

All homogeneous Markov processes with state space $[0,\infty)$ which behave themselves as processes with independent increments and positive jumps up to the first exit time from $(0,\infty)$ are described. Conditions are obtained under which general ergodic theorems can be applied to such processes. The form of the stationary distribution is found which can be quite simply expressed in terms of the distribution of the absolute maximum of the process in question and coefficients of the boundary condition at zero.