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Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes

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Abstract

We prove uniform Hausdorff and packing dimension results for the inverse images of a large class of real-valued symmetric Lévy processes. Our main result for the Hausdorff dimension extends that of Kaufman (C R Acad Sci Paris Sér I Math 300:281–282, 1985) for Brownian motion and that of Song et al. (Electron Commun Probab 23:10, 2018) for \(\alpha \)-stable Lévy processes with \(1<\alpha <2\). Along the way, we also prove an upper bound for the uniform modulus of continuity of the local times of these processes.

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References

  1. Barlow, M.T.: Necessary and sufficient conditions for the continuity of local time of Lévy processes. Ann. Probab. 16, 1389–1427 (1988)

    Article  MathSciNet  Google Scholar 

  2. Barlow, M.T., Hawkes, J.: Application de l’entropie métrique à la continuité des temps locaux des processus de Lévy. C. R. Acad. Sci. Paris Sér. I Math. 301, 237–239 (1985)

    MathSciNet  MATH  Google Scholar 

  3. Barlow, M.T., Perkins, E.A., Taylor, S.J.: Two uniform intrinsic constructions for the local time of a class of Lévy processes. Ill. J. Math. 30(1), 19–65 (1986)

    Article  Google Scholar 

  4. Blumenthal, R.M., Getoor, R.K.: Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10, 493–516 (1961)

    MathSciNet  MATH  Google Scholar 

  5. Böttcher, B., Schilling, R., Wang, J.: Lévy Matters III. Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Lecture Notes in Mathematics, 2099. Lévy Matters, p. xviii+1999. Springer, Cham (2013)

    Book  Google Scholar 

  6. Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266, 3543–3571 (2014)

    Article  MathSciNet  Google Scholar 

  7. Falconer, K.J.: Fractal Geometry-Mathematical Foundations and Applications, 2nd edn. Wiley, New York (2003)

    Book  Google Scholar 

  8. Grzywny, T.: On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 41, 1–29 (2014)

    Article  MathSciNet  Google Scholar 

  9. Grzywny, T., Ryznar, M.: Hitting times of points and intervals for symmetric Lévy processes. Potential Anal. 46, 739–777 (2017)

    Article  MathSciNet  Google Scholar 

  10. Hawkes, J.: On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set. Z. Wahrsch. Verw. Gebiete 1(9), 90–102 (1971)

    Article  MathSciNet  Google Scholar 

  11. Hawkes, J.: Local times as stationary processes. In: David Elworthy, K. (ed.) From Local Times to Global Geometry, Control and Physics (Coventry, 1984/85). Pitman Research Notes in Mathematics Series, vol. 150, pp. 111–120. Longman Sci. Tech, Harlow (1986)

    Google Scholar 

  12. Hawkes, J.: Exact capacity results for stable processes. Probab. Theory Relat. Fields 112, 1–11 (1998)

    Article  MathSciNet  Google Scholar 

  13. Hu, X., Taylor, S.J.: Multifractal structure of a general subordinator. Stoch. Process. Appl. 88(2), 245–258 (2000)

    Article  MathSciNet  Google Scholar 

  14. Kaufman, R.: Temps locaux et dimensions. (French) local times and dimensions. C. R. Acad. Sci. Paris Sér. I Math. 300(10), 281–282 (1985)

    MathSciNet  MATH  Google Scholar 

  15. Khoshnevisan, D., Xiao, Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33, 841–878 (2005)

    Article  MathSciNet  Google Scholar 

  16. Khoshnevisan, D., Xiao, Y., Zhong, Y.: Local times of additive Lévy processes. Stoch. Process. Appl. 104, 193–216 (2003)

    Article  Google Scholar 

  17. Knopova, V., Schilling, R.L.: On level and collision sets of some Feller processes. ALEA Lat. Am. J. Probab. Math. Stat. 12, 1001–1029 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Marcus, M.B., Rosen, J.: Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes. Ann. Probab. 20, 1603–1684 (1992)

    Article  MathSciNet  Google Scholar 

  19. Marsalle, L.: Slow points and fast points of local times. Ann. Probab. 27, 150–165 (1999)

    Article  MathSciNet  Google Scholar 

  20. Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  21. Seuret, S., Yang, X.: Multifractal analysis for the occupation measure of stable-like processes. Electron. J. Probab. 22(47), 36 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Song, R., Xiao, Y., Yang, X.: Uniform Hausdorff dimension result for the inverse images of stable Lévy processes. Electron. Commun. Probab. 23(75), 10 (2018)

    MATH  Google Scholar 

  23. Sun, X., Xiao, Y., Xu, L., Zhai, J.: Uniform dimension results for a family of Markov processes. Bernoulli 24, 3924–3951 (2018)

    Article  MathSciNet  Google Scholar 

  24. Taylor, S.J.: The measure theory of random fractals. Math. Proc. Camb. Philos. Soc. 100, 383–406 (1986)

    Article  MathSciNet  Google Scholar 

  25. Watanabe, T.: The isoperimetric inequality for isotropic unimodal Lévy processes. Z. Wahrsch. Verw. Gebiete 63(4), 487–499 (1983)

    Article  MathSciNet  Google Scholar 

  26. Xiao, Y.: Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields. Probab. Theory Relat. Fields 109, 129–157 (1997)

    Article  Google Scholar 

  27. Xiao, Y.: Random fractals and Markov processes. In: Lapidus, M.L., van Frankenhuijsen, M. (eds.) Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Proceedings of Symposia in Pure Mathematics, vol. 72, 2nd edn, pp. 261–338. American Mathematical Society, Providence (2004)

    Chapter  Google Scholar 

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Acknowledgements

Research of Yimin Xiao was partially supported in part by the NSF Grants DMS-1607089 and DMS-1855185. Xiaochuan Yang was supported in part by Luxembourg’s National Foundation for Research (FNR), through the AFR project MiSSILe. The authors thank the anonymous referee for providing some useful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, State University of New York at New Paltz, New Paltz, NY, 12561, USA

    Hyunchul Park

  2. Department of Statistics and Probability, Michigan State University, East Lansing, MI, 48824, USA

    Yimin Xiao & Xiaochuan Yang

  3. Mathematics Research Unit, University of Luxembourg, Esch-sur-Alzette, Grand Duchy of Luxembourg

    Xiaochuan Yang

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  1. Hyunchul Park
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  2. Yimin Xiao
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  3. Xiaochuan Yang
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Correspondence to Xiaochuan Yang.

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Park, H., Xiao, Y. & Yang, X. Uniform Dimension Results for the Inverse Images of Symmetric Lévy Processes. J Theor Probab 33, 2213–2232 (2020). https://doi.org/10.1007/s10959-019-00956-3

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  • DOI: https://doi.org/10.1007/s10959-019-00956-3

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