• Bernoulli
  • Volume 12, Number 3 (2006), 551-570.

The scaling limit behaviour of periodic stable-like processes

Rice Franke

Full-text: Open access


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 αo is positive with respect to the invariant measure, the limit turns out to be an αo -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.

Article information

Bernoulli, Volume 12, Number 3 (2006), 551-570.

First available in Project Euclid: 28 June 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

functional non-central limit theorem homogenization stable-like process large deviations heavy-tail increment


Franke, Rice. The scaling limit behaviour of periodic stable-like processes. Bernoulli 12 (2006), no. 3, 551--570. doi:10.3150/bj/1151525136.

Export citation


  • [1] Applebaum, D. (2004) Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press.
  • [2] Bhattacharya, R. (1985) A central limit theorem for diffusions with periodic coefficients. Ann. Probab., 13, 385-396.
  • [3] Bensoussan, A., Lions, J.L. and Papanicolaou, G. (1978) Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland.
  • [4] Doob, J.L. (1953) Stochastic Processes. New York: Wiley.
  • [5] Jacob, N. and Leopold, H.G. (1993) Pseudodifferential operators with variable order of differentiation generating Feller semigroups. Integral Equations Operator Theory, 17, 544-553.
  • [6] Jacod, J. and Shiryaev, A. (1987) Limit Theorems for Stochastic Processes. Berlin: Springer-Verlag.
  • [7] Kikuchi, K. and Negoro, A. (1997) On Markov process generated by pseudodifferential operator of variable order. Osaka J. Math., 34, 319-335.
  • [8] Kolokoltsov, V.N. (2000) Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. (3), 80, 725-768.
  • [9] Kolokoltsov, V.N., Schilling, R.L. and Tyukov, A.E. (2002) Transience and non-explosion of certain stochastic Newtonian systems. Electron. J. Probab., 7, no. 19.
  • [10] Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press.
  • [11] Schilling, R.L. (1998) Growth and Hölder conditions for the sample paths of Feller processes, Probab. Theory Related Fields, 112, 565-611.
  • [12] Wentzell, A.D. (1990) Limit Theorems on Large Deviations for Markov Stochastic Processes. Dordrecht: Kluwer Academic Press.