The scaling limit behaviour of periodic stable-like processes

Abstract

We prove a functional non-central limit theorem for scaled Markov processes generated by pseudo-differential operators of periodic variable order. Two different situations occur. If the measure of the set where the order function attains its minimum α_o is positive with respect to the invariant measure, the limit turns out to be an α_o -stable Lévy process. In the other case the scaled sequence converges in probability to the zero function. The large deviation for this convergence is typical of processes having heavy-tail increments. It turns out that only a finite number of large jumps can be recovered on large scales. We also apply the results in order to understand the recurrence and transience of periodic stable-like processes.