Abstract
In this paper, we present new results on Hunt’s hypothesis (H) for Lévy processes. We start with a comparison result on Lévy processes which implies that big jumps have no effect on the validity of (H). Based on this result and the Kanda-Forst-Rao theorem, we give examples of subordinators satisfying (H). Afterwards we give a new necessary and sufficient condition for (H) and obtain an extended Kanda-Forst-Rao theorem. By virtue of this theorem, we give a new class of Lévy processes satisfying (H). Finally, we construct a type of subordinators that does not satisfy Rao’s condition.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech 10, 493–516 (1961)
Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York and London (1968)
Blumenthal, R.M., Getoor, R.K.: Dual processes and potential theory. In: Proceedings 12th Biennial Seminar of the Canadian Math. Congress, pp. 137–156 (1970)
Bretagnolle, J.: Résults de Kesten sur les processus à accroissements indépendants. Séminare de Probabilités V, Lect. Notes in Math., vol. 191, pp. 21–36. Springer-Verlag, Berlin (1971)
Fitzsimmons, P.J.: On the equivalence of three potential principles for right Markov processes. Probab. Th. Rel. Fields 84, 251–265 (1990)
Fitzsimmons, P.J.: On the quasi-regularity of semi-Dirichlet forms. Potential Anal 15, 151–185 (2001)
Fitzsimmons, P.J., Kanda, M.: On Choquet’s dichotomy of capacity for Markov processes. Ann. Probab 20, 342–349 (1992)
Forst, G.: The definition of energy in non-symmetric translation invariant Dirichlet spaces. Math. Ann 216, 165–172 (1975)
Glover, J.: Energy and the maximum principle for nonsymmetric Hunt processes. Probab. Theory Appl. XXVI(4), 757–768 (1981)
Glover, J.: Topics in energy and potential theory. Seminar on Stochastic Processes, 1982, pp. 195–202. Birkhäuser (1983)
Glover, J., Rao, M.: Hunt’s hypothesis (H) and Getoor’s conjecture. Ann. Probab 14, 1085–1087 (1986)
Han, X.-F., Ma, Z.-M., Sun, W.: hĥ-transforms of positivity preserving semigroups and associated Makov processes. Acta Math. Sinica English Series 27, 369–376 (2011)
Hartman, P., Wintner, A.: On the infinitesimal generators of integral convolutions. Amer. J. Math 64, 273–298 (1942)
Hawkes, J.: Potential theory of Lévy processes. Proc. London Math. Soc 3, 335–352 (1979)
Hu, Z.-C., Sun, W.: Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes. Stoch. Proc. Appl 122, 2319–2328 (2012)
Kanda, M.: Two theorems on capacity for Markov processes with stationary independent increments. Z. Wahrsch. verw Gebiete 35, 159–165 (1976)
Kanda, M.: Characterisation of semipolar sets for processes with stationary independent increments. Z. Wahrsch. verw Gebiete 42, 141–154 (1978)
Kesten, H.: Hitting probabilities of single points for processes with stationary independent increments. Memoirs of the American Mathematical Society, vol. 93. American Mathematical Society, Providence (1969)
Lukacs, E.: Characteristic Functions, 2nd edn. Griffin, London (1970)
Port, S.C., Stone, C.J.: The asymmetric Cauchy process on the line. Ann. Math. Statist 40, 137–143 (1969)
Rao, M.: On a result of M. Kanda. Z. Wahrsch. verw Gebiete 41, 35–37 (1977)
Rao, M.: Hunt’s hypothesis for Lévy processes. Proc. Amer. Math. Soc 104, 621–624 (1988)
Silverstein, M.L.: The sector condition implies that semipolar sets are quasi-polar. Z. Wahrsch. verw Gebiete 41, 13–33 (1977)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hu, ZC., Sun, W. & Zhang, J. New Results on Hunt’s Hypothesis (H) for Lévy Processes. Potential Anal 42, 585–605 (2015). https://doi.org/10.1007/s11118-014-9446-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-014-9446-1