Abstract
We consider the α-stable Ornstein–Uhlenbeck process in \({\mathbb{R}}^d\) with the generator \(L = \Delta^{\alpha/2} - \lambda x \cdot\nabla_x\) . We show that if 2 > α ≥ 1 or α < 1 = d the Harnack inequality holds. For α < 1 < d we construct a counterexample that shows that the Harnack inequality does not hold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bass R.F., Levin D.A. (2002). Harnack inequalities for jump processes. Potential Anal. 17(4): 375–388
Blumenthal, R.M., Getoor, R.K.: Markov processes and potential theory. vol. 29, Pure and Applied Mathematics, Academic, New York (1968)
Bogdan K., Byczkowski T. (1999). Probabilistic proof of boundary Harnack principle for α-harmonic functions. Potential Anal. 11(2): 135–156
Bogdan K., Stós A., Sztonyk P. (2003). Harnack inequality for stable processes on d-sets. Studia Math. 158(2): 163–198
Chen Z.-Q., Fitzsimmons P.J., Takeda M., Ying J., Zhang T.-S. (2004). Absolute continuity of symmetric Markov processes. Ann. Probab. 32(3A): 2067–2098
Chen Z.-Q., Kumagai T. (2003). Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108(1): 27–62
Chen Z.-Q., Song R. (2003). Drift transforms and Green function estimates for discontinuous processes. J. Funct. Anal. 201(1): 262–281
Chung K.L., Zhao Z.X. (1995). From Brownian motion to Schrödinger’s equation, vol. 312, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin
Hebisch W., Saloff-Coste L. (2001). On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier (Grenoble) 51(5): 1437–1481
Ikeda N., Watanabe S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2: 79–95
Jakubowski, T.: The estimates of the mean first exit time from a ball for the α-stable Ornstein–Uhlenbeck processes. Stoch. Process. Appl. (to appear)
Janicki A., Weron A. (1994). Simulation and chaotic behavior of α-stable stochastic processes, vol. 178, Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, New York
Samorodnitsky, G., Taqqu, M.S.: Stable non-Gaussian random processes (Stochastic models with infinite variance). Stochastic Modeling. Chapman & Hall, New York (1994)
Song R., Vondraček Z. (2004). Harnack inequality for some classes of Markov processes. Math. Z. 246(1–2): 177–202
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by KBN and MEN.
Rights and permissions
About this article
Cite this article
Jakubowski, T. On Harnack inequality for α-stable Ornstein–Uhlenbeck processes. Math. Z. 258, 609–628 (2008). https://doi.org/10.1007/s00209-007-0188-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-007-0188-2