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Two Theorems on Hunt’s Hypothesis (H) for Markov Processes

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Abstract

Hunt’s hypothesis (H) and the related Getoor’s conjecture is one of the most important problems in the basic theory of Markov processes. In this paper, we investigate the invariance of Hunt’s hypothesis (H) for Markov processes under two classes of transformations, which are change of measure and subordination. Our first theorem shows that for two standard processes (Xt) and (Yt), if (Xt) satisfies (H) and (Yt) is locally absolutely continuous with respect to (Xt), then (Yt) satisfies (H). Our second theorem shows that a standard process (Xt) satisfies (H) if and only if \((X_{\tau _{t}})\) satisfies (H) for some (and hence any) subordinator (τt) which is independent of (Xt) and has a positive drift coefficient. Applications of the two theorems are given.

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References

  1. Aikawa, H., Essén, M.: Potential Theory - Selected Topics. Springer, Berlin (1996)

    MATH  Google Scholar 

  2. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  3. Beznea, L., Boboc, N.: Potential Theory and Right Processes. Springer, Berlin (2004)

    MATH  Google Scholar 

  4. Beznea, L., Cornea, A., Röckner, M.: Potential theory of infinite dimensional Lévy processes. J. Func. Anal. 261, 2845–2876 (2011)

    MATH  Google Scholar 

  5. Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York (1968)

    MATH  Google Scholar 

  6. Blumenthal, R.M., Getoor, R.K.: Dual processes and potential theory. Proc. 12th Biennial Seminar of the Canadian Math. Congress, pp 137–156 (1970)

  7. Bochner, S.: Harmonic Analysis and the Theory of Probability. Univ. California Press, Berkeley and Los Angeles (1955)

    MATH  Google Scholar 

  8. 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, Berlin (1971)

    Google Scholar 

  9. Chen, Z.-Q., Fukushima, M.: Symmetric Markov Processes, Time Change, and Boundary Theory. Princeton University Press, Princeton (2012)

    MATH  Google Scholar 

  10. Cheridito, P., Filipović, D., Yor, M.: Equivalent and absolutely continuous measure changes for jump-diffusion processes. Ann. Appl. Probab. 15, 1713–1732 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Dawson, D.: Equivalence of Markov processes. Trans. Amer. Math. Soc. 131, 1–31 (1968)

    MathSciNet  MATH  Google Scholar 

  12. Doob, J.L.: Semimartingales and subharmonic functions. Trans. Amer. Math. Soc. 77, 86–121 (1954)

    MathSciNet  MATH  Google Scholar 

  13. Doob, J.L.: Classical Potential Theory and its Probabilistic Counterpart. Springer, Berlin (1984)

    MATH  Google Scholar 

  14. Ethier, S.N., Kurtz, T.G.: Markov Processes, Characterization and Convergence. Wiley, New York (1986)

    MATH  Google Scholar 

  15. Fitzsimmons, P.J.: On the equivalence of three potential principles for right Markov processes. Probab. Th. Rel. Fields 84, 251–265 (1990)

    MathSciNet  MATH  Google Scholar 

  16. Fitzsimmons, P.J.: On the quasi-regularity of semi-Dirichlet forms. Potential Anal. 15, 151–185 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Fitzsimmons, P.J.: Gross’ Bwownian motion fails to satisfy the polarity principle. Rev. Roum. Math. Pures Appl. 59, 87–91 (2014)

    MATH  Google Scholar 

  18. Fitzsimmons, P.J., Kanda, M.: On Choquet’s dichotomy of capacity for Markov processes. Ann. Probab. 20, 342–349 (1992)

    MathSciNet  MATH  Google Scholar 

  19. Forst, G.: The definition of energy in non-symmetric translation invariant Dirichlet spaces. Math. Ann. 216, 165–172 (1975)

    MathSciNet  MATH  Google Scholar 

  20. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov Processes (second revised and extended edition). De Gruyter (2011)

  21. Glover, J.: Topics in energy and potential theory. Seminar on Stochastic Processes, 1982. Birkhäuser, 195–202 (1983)

  22. Glover, J., Rao, M.: Hunt’s hypothesis (H) and Getoor’s conjecture. Ann. Probab. 14, 1085–1087 (1986)

    MathSciNet  MATH  Google Scholar 

  23. Glover, J., Rao, M.: Nonsymmetric Markov processes and hypothesis (H). J. Theor. Probab. l, pp 371–380 (1988)

  24. 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)

    MathSciNet  Google Scholar 

  25. Hansen, W., Netuka, I.: Hunt’s hypothesis (H) and the triangle property of the green function. Expo. Math. 34, 95–100 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Hawkes, J.: Potential theory of Lévy processes. Proc. London Math. Soc. 3, 335–352 (1979)

    MATH  Google Scholar 

  27. Hu, Z.-C., Sun, W.: Hunt’s hypothesis (H) and Getoor’s conjecture for Lévy processes. Stoch. Proc. Appl. 122, 2319–2328 (2012)

    MATH  Google Scholar 

  28. Hu, Z.-C., Sun, W.: Further study on hunt’s hypothesis (H) for Lévy processes. Sci. China Math. 59, 2205–2226 (2016)

    MathSciNet  MATH  Google Scholar 

  29. Hu, Z.-C., Sun, W.: Hunt’s hypothesis (H) for the sum of two independent Lévy processes. Comm. Math. Stat. 6, 227–247 (2018)

    MATH  Google Scholar 

  30. Hu, Z.-C., Sun, W., Zhang, J.: New results on hunt’s hypothesis (H) for Lévy processes. Potent. Anal. 42, 585–605 (2015)

    MATH  Google Scholar 

  31. Hunt, G.A.: Markoff processes and potentials. I. Illinois J. Math. 1, 44–93 (1957)

    MathSciNet  MATH  Google Scholar 

  32. Hunt, G.A.: Markoff processes and potentials. II. Illinois J. Math. 1, 316–369 (1957)

    MathSciNet  MATH  Google Scholar 

  33. Hunt, G.A.: Markoff processes and potentials. III. Illinois J. Math. 2, 151–213 (1958)

    MathSciNet  Google Scholar 

  34. Itô, K., Watanabe, S.: Transformation of Markov processes by multiplicative functionals. Ann. Inst. Fourier (Grenoble) 15, 13–30 (1965)

    MathSciNet  MATH  Google Scholar 

  35. Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)

    MATH  Google Scholar 

  36. Kabanov, Y., Liptser, R.S., Shiryaev, A.N.: On Absolute Continuity of Probability Measures for Markov-Itô Processes. Lecture Notes in Control and Inform. Sci, vol. 25, pp 114–128. Springer, New York (1980)

    Google Scholar 

  37. Kanda, M.: Two theorems on capacity for Markov processes with stationary independent increments. Z. Wahrsch. verw. Gebiete 35, 159–165 (1976)

    MathSciNet  MATH  Google Scholar 

  38. Kanda, M.: Characterisation of semipolar sets for processes with stationary independent increments. Z. Wahrsch. verw. Gebiete 42, 141–154 (1978)

    MATH  Google Scholar 

  39. 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)

    Google Scholar 

  40. Kunita, H.: Absolute continuity of Markov processes and generators. Nagaya Math. J. 36, 1–26 (1969)

    MathSciNet  MATH  Google Scholar 

  41. Kunita, H.: Absolute Continuity of Markov Processes. Séminaire De Probabilités X. Leture Notex in Math, vol. 511, pp 44–77. Springer, Berlin (1976)

    Google Scholar 

  42. Kunita, H., Watanabe, S.: On square integrable martingales. Nagoya Math. J. 30, 209–245 (1967)

    MathSciNet  MATH  Google Scholar 

  43. Liptser, R.S., Shiryaev, A.N.: Statistics for Random Processes I, 2nd edn. Springer, New York (1977). (2nd edn. 2001)

    Google Scholar 

  44. Ma, Z.-M., Röckner, M.: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin (1992)

    MATH  Google Scholar 

  45. Newman, C.M.: The inner product of path space measures corresponding to random processes with independent increments. Bull. Amer. Math. Soc. 78, 268–271 (1972)

    MathSciNet  MATH  Google Scholar 

  46. Newman, C.M.: On the orthogonality of independent increment processes, Topics in Probability Theory. In: Stroock, D.W., Varadhan, S.R.S. (eds.) Courant Inst. Math. Sci., New York Univ., New York), pp. 93–111 (1973)

  47. Palmowski, Z., Rolski, T.: A technique for exponential change of measure for Markov processes. Bernoulli 8, 767–785 (2002)

    MathSciNet  MATH  Google Scholar 

  48. Port, S.C., Stone, C.J.: The asymmetric Cauchy process on the line. Ann. Math. Statist. 40, 137–143 (1969)

    MathSciNet  MATH  Google Scholar 

  49. Port, S.C., Stone, C.J.: Brownian Motion and Classical Potential Theory. Academic Press, New York (1978)

    MATH  Google Scholar 

  50. Rao, M.: On a result of M. Kanda. Z. Wahrsch. verw. Gebiete 41, 35–37 (1977)

    MathSciNet  MATH  Google Scholar 

  51. Rao, M.: Hunt’s hypothesis for lévy processes. Proc. Amer. Math. Soc. 104, 621–624 (1988)

    MathSciNet  MATH  Google Scholar 

  52. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Univ. Press, Cambridge (1999)

    MATH  Google Scholar 

  53. Silverstein, M.L.: The sector condition implies that semipolar sets are quasi-polar. Z. Wahrsch. verw. Gebiete 41, 13–33 (1977)

    MathSciNet  MATH  Google Scholar 

  54. Skorohod, A.V.: On the differentiability of measures which correspond to stochastic processes, I. Processes with independent increments. Theory Probab. Appl. 2, 407–432 (1957)

    MathSciNet  Google Scholar 

  55. Skorohod, A.V.: Studies in the Theory of Random Processes, Addison-Wesley, Reading, Mass. (Russian Original 1961) (1965)

  56. Stroock, D.W.: Diffusion processes associated with lévy generators. Z. Wahrsch. verw. Gebiete 32, 209–244 (1975)

    MATH  Google Scholar 

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Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771309 and 11871184), the Natural Science and Engineering Research Council of Canada and the Scientific Research Foundation of Hebei Education Department (Grant No. QN2017094). We thank the referee for constructive comments which helped improve the quality of the paper.

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Authors and Affiliations

  1. College of Mathematics, Sichuan University, Chengdu, 610065, China

    Ze-Chun Hu

  2. Department of Mathematics and Statistics, Concordia University, Montreal, QC, H3G 1M8, Canada

    Wei Sun

  3. Postdoctoral Research Station of Mathematics & School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China

    Li-Fei Wang

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  1. Ze-Chun Hu
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Correspondence to Li-Fei Wang.

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Hu, ZC., Sun, W. & Wang, LF. Two Theorems on Hunt’s Hypothesis (H) for Markov Processes. Potential Anal 55, 29–52 (2021). https://doi.org/10.1007/s11118-020-09848-2

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