Random walks on a halfaxis. I. Boundary problems and ergodic theorem

We investigate the Markov right continuous homogeneous Feller processes with the state space $\{1,\dots,d\}\times[0,\infty)$. It is assumed that up to the moment of the first entrance in the set of states $\{(i,0)\colon i=1,\dots,d\}$ the process develops like homogeneous Markov process which is second-component homogeneous without negative jumps of the second component. In § 1 the existence theorem is proved and all such processes are described. In § 2 some functionals associated with the moment when the second component leaves an interval are studied. The results of § 2 are then used in the proof of the ergodic theorem.