Hunt’s Hypothesis (H) for the Sum of Two Independent Lévy Processes

Abstract

Which Lévy processes satisfy Hunt’s hypothesis (H) is a long-standing open problem in probabilistic potential theory. The study of this problem for one-dimensional Lévy processes suggests us to consider (H) from the point of view of the sum of Lévy processes. In this paper, we present theorems and examples on the validity of (H) for the sum of two independent Lévy processes. We also give a novel condition on the Lévy measure which implies (H) for a large class of one-dimensional Lévy processes.

Keywords

Hunt’s hypothesis (H) / Getoor’s conjecture / Lévy process

References

[1] Bertoin J. Lévy Processes. 1996 Cambridge: Cambridge University Press

[2] Blumenthal RM, Getoor RK. Sample functions of stochastic processes with stationary independent increments. J. Math. Mech.. 1961, 10 493-516

[3] Bretagnolle, J.: Résults de Kesten sur les processus à accroissements indépendants. In: Séminare de Probabilités V, Lecture Notes in Mathematics, vol. 191, pp. 21–36. Springer, Berlin (1971)

[4] Fitzsimmons PJ. Gross’s Bwownian motion fails to satisfy the polarity principle. Rev. Roum. Math. Pures Appl.. 2014, 59 87-91

[5] Forst G. The definition of energy in non-symmetric translation invariant Dirichlet spaces. Math. Ann.. 1975, 216 165-172

[6] Hansen W, Netuka I. Hunt’s hypothesis (H) and triangle property of the Green function. Expo. Math.. 2016, 34 95-100

[7] Hartman P, Wintner A. On the infinitesimal generators of integral convolutions. Am. J. Math.. 1942, 64 273-298

[8] Hu Z-C, Sun W. Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes. Stoch. Process. Appl.. 2012, 122 2319-2328

[9] Hu Z-C, Sun W. Further study on Hunt’s hypothesis (H) for Lévy processes. Sci. China Math.. 2016, 59 2205-2226

[10] Hu Z-C, Sun W, Zhang J. New results on Hunt’s hypothesis (H) for Lévy processes. Potential Anal.. 2015, 42 585-605

[11] Kanda M. Two theorems on capacity for Markov processes with stationary independent increments. Z. Wahrsch. Verw. Gebiete. 1976, 35 159-165

[12] Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence (1969)

[13] Rao M. Hunt’s hypothesis for Lévy processes. Proc. Am. Math. Soc.. 1988, 104 621-624

[14] Sato K. Lévy Processes and Infinitely Divisible Distributions. 1999 Cambridge: Cambridge University Press

Funding

NNSFC(11371191)

NSERC(311945-2013)

Fundamental Research Funds for the Central Universities of China