×

Hunt’s hypothesis (H) for Markov processes: survey and beyond. (English) Zbl 1480.60226

Summary: The goal of this paper is threefold. First, we survey the existing results on Hunt’s hypothesis (H) for Markov processes and Getoor’s conjecture for Lévy processes. Second, we investigate (H) for multidimensional Lévy processes from the viewpoint of projections. Third, we present a few open questions for further study.

MSC:

60J45 Probabilistic potential theory
60G51 Processes with independent increments; Lévy processes

References:

[1] Bertoin, J., Lévy Processes (1996), Cambridge: Cambridge University Press, Cambridge · Zbl 0861.60003
[2] Bertoin, J., Regenerative embeddings of Markov sets, Probab. Th. Rel. Fields, 108, 559-571 (1997) · Zbl 0895.60011 · doi:10.1007/s004400050121
[3] Blumenthal, R. M.; Getoor, R. K., Markov Processes and Potential Theory (1968), New York and London: Academic Press, New York and London · Zbl 0169.49204
[4] Blumenthal, R. M., Getoor, R. K.: Dual processes and potential theory. In: Proc. 12th Biennial Seminar of the Canadian Math. Congress on Time Series and Stochastic Processes, 137-156, Canad. Math. Congr., Montreal, 1970 · Zbl 0238.60055
[5] Bochner, S., Harmonic Analysis and the Theory of Probability (1955), Berkeley and Los Angeles: Univ. California Press, Berkeley and Los Angeles · Zbl 0068.11702 · doi:10.1525/9780520345294
[6] Bogdan, K., Byczkowski, T., Kulczycki, T., et al.: Potential analysis of stable processes and its extensions (Edited by Piotr G., Andrzej S.). Lecture Notes in Mathematics 1980, Springer-Verlag, Berlin, 2009
[7] Bretagnolle, J., Résults de Kesten sur les processus à accroissements indépendants, Séminare de Probabilités V, 21-36 (1971), Berlin: Springer-Verlag, Berlin
[8] Fitzsimmons, P. J., On the equivalence of three potential principles for right Markov processes, Probab. Th. Rel. Fields, 84, 251-265 (1990) · Zbl 0668.60065 · doi:10.1007/BF01197847
[9] Fitzsimmons, P. J., On the quasi-regularity of semi-Dirichlet forms, Potential Anal., 15, 151-185 (2001) · Zbl 0993.60075 · doi:10.1023/A:1011249920221
[10] Fitzsimmons, P. J., Gross’s Bwownian motion fails to satisfy the polarity principle, Rev. Roumaine Math. Pures Appl., 59, 87-91 (2014) · Zbl 1389.60094
[11] Fitzsimmons, P. J.; Kanda, M., On Choquet’s dichotomy of capacity for Markov processes, Ann. Probab., 20, 342-349 (1992) · Zbl 0748.60065 · doi:10.1214/aop/1176989930
[12] Forst, G., The definition of energy in non-symmetric translation invariant Dirichlet spaces, Math. Ann., 216, 165-172 (1975) · Zbl 0293.31018 · doi:10.1007/BF01432544
[13] Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet Forms and Symmetric Markov Processes (2011), Berlin: De Gruyter, Berlin · Zbl 1227.31001
[14] Glover, J., Energy and the maximum principle for nonsymmetric Hunt processes, Probability Theory and Its Applications, 26, 4, 757-768 (1981) · Zbl 0475.60058
[15] Glover, J., Topics in energy and potential theory, 195-202 (1983), Boston: Birkhäuser, Boston · Zbl 0529.60080
[16] Glover, J.; Rao, M., Hunt’s hypothesis (H) and Getoor’s conjecture, Ann. Probab., 14, 1085-1087 (1986) · Zbl 0602.60063 · doi:10.1214/aop/1176992463
[17] Glover, J.; Rao, M., Nonsymmetric Markov processes and hypothesis (H), J. Theor. Probab., 1, 371-380 (1988) · Zbl 0655.60062 · doi:10.1007/BF01048726
[18] Han, X. F.; Ma, Z. M.; Sun, W., hĥ-transforms of positivity preserving semigroups and associated Markov processes, Acta Math. Sinica, English Series, 27, 369-376 (2011) · Zbl 1219.31002 · doi:10.1007/s10114-011-0597-3
[19] Hansen, W.; Netuka, I., Hunt’s hypothesis (H) and triangle property of the Green function, Expo. Math., 34, 95-100 (2016) · Zbl 1338.31018 · doi:10.1016/j.exmath.2014.12.009
[20] Hartman, P.; Wintner, A., On the infinitesimal generators of integral convolutions, Amer. J. Math., 64, 273-298 (1942) · Zbl 0063.01951 · doi:10.2307/2371683
[21] Hawkes, J., Potential theory of Lévy processes, Proc. London Math. Soc., 3, 335-352 (1979) · Zbl 0401.60069 · doi:10.1112/plms/s3-38.2.335
[22] Hu, Z. C.; Sun, W., Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes, Stoch. Proc. Appl., 122, 2319-2328 (2012) · Zbl 1247.60117 · doi:10.1016/j.spa.2012.03.013
[23] Hu, Z. C.; Sun, W., Further study on Hunt’s hypothesis (H) for Lévy processes, Sci. China Math., 59, 2205-2226 (2016) · Zbl 1367.60053 · doi:10.1007/s11425-015-0600-2
[24] Hu, Z. C.; Sun, W., Hunt’s Hypothesis (H) for the sum of two independent Lévy processes, Commun. Math. Stat., 6, 227-247 (2018) · Zbl 1393.60081 · doi:10.1007/s40304-018-0136-y
[25] Hu, Z. C., Sun, W., Wang, L. F.: Two theorems on Hunt’s hypothesis (H) for Markov processes. To appear in Potential Anal., DOI: doi:10.1007/s11118-020-09848-2 · Zbl 1474.60183
[26] Hu, Z. C.; Sun, W.; Zhang, J., New results on Hunt’s hypothesis (H) for Lévy processes, Potential Anal., 42, 585-605 (2015) · Zbl 1308.60057 · doi:10.1007/s11118-014-9446-1
[27] Kanda, M., Two theorems on capacity for Markov processes with stationary independent increments, Z. Wahrsch. verw. Gebiete, 35, 159-165 (1976) · Zbl 0316.60048 · doi:10.1007/BF00533321
[28] Kanda, M., Characterization of semipolar sets for processes with stationary independent increments, Z. Wahrsch. verw. Gebiete, 42, 141-154 (1978) · Zbl 0362.60078 · doi:10.1007/BF00536050
[29] Kesten, H., Hitting probabilities of single points for processes with stationary independent increments (1969), Providence, R.I.: American Mathematical Society, Providence, R.I. · Zbl 0201.19002
[30] Port, S. C.; Stone, C. J., The asymmetric Cauchy process on the line, Ann. Math. Statist., 40, 137-143 (1969) · Zbl 0211.21102 · doi:10.1214/aoms/1177697810
[31] Rao, M., On a result of M. Kanda, Z. Wahrsch. verw. Gebiete, 41, 35-37 (1977) · Zbl 0373.60099 · doi:10.1007/BF00535012
[32] Rao, M., On polar sets for Lévy processes, J. London Math. Soc., 35, 569-576 (1987) · Zbl 0587.60072
[33] Rao, M., Hunt’s hypothesis for Lévy processes, Proc. Amer. Math. Soc., 104, 621-624 (1988) · Zbl 0693.60063
[34] Silverstein, M. L., The sector condition implies that semipolar sets are quasi-polar, Z. Wahrsch. verw. Gebiete, 41, 13-33 (1977) · Zbl 0379.60075 · doi:10.1007/BF00535011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.