Markov Processes with Asymptotically Zero Drifts

The researches of various Markov processes, in particular, of random walks in a quarter plane, showed the importance of investigating one-dimensional processes with “almost zero” mean drifts. For the first time this problem was formulated by Harris and investigated more deeply by Lamperti. The present work is devoted to the asymptotical analysis of these processes. The research method is the construction of Lyapunov functions. The obtained results essentially generate Lamperti's results.