Bernoulli

  • Bernoulli
  • Volume 24, Number 4B (2018), 3924-3951.

Uniform dimension results for a family of Markov processes

Xiaobin Sun, Yimin Xiao, Lihu Xu, and Jianliang Zhai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we prove uniform Hausdorff and packing dimension results for the images of a large family of Markov processes. The main tools are the two covering principles in Xiao (In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2 (2004) 261–338 Amer. Math. Soc.). As applications, uniform Hausdorff and packing dimension results for certain classes of Lévy processes, stable jump diffusions and non-symmetric stable-type processes are obtained.

Article information

Source
Bernoulli, Volume 24, Number 4B (2018), 3924-3951.

Dates
Received: July 2017
Revised: September 2017
First available in Project Euclid: 18 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1524038773

Digital Object Identifier
doi:10.3150/17-BEJ994

Mathematical Reviews number (MathSciNet)
MR3788192

Zentralblatt MATH identifier
06869895

Keywords
cover principles Markov processes uniform Hausdorff dimension

Citation

Sun, Xiaobin; Xiao, Yimin; Xu, Lihu; Zhai, Jianliang. Uniform dimension results for a family of Markov processes. Bernoulli 24 (2018), no. 4B, 3924--3951. doi:10.3150/17-BEJ994. https://projecteuclid.org/euclid.bj/1524038773


Export citation

References

  • [1] Alili, L., Chaumont, L., Graczyk, P. and Żak, T. (2017). Inversion, duality and Doob $h$-transforms for self-similar Markov processes. Electron. J. Probab. 22 Paper No. 20. DOI:10.1214/17-EJP33.
  • [2] Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics 93. Cambridge: Cambridge Univ. Press.
  • [3] Bass, R. (1988). Uniqueness in law for pure jump Markov processes. Probab. Theory Related Fields 79 271–287.
  • [4] Benjamini, I., Chen, Z.-Q. and Rohde, S. (2004). Boundary trace of reflecting Brownian motions. Probab. Theory Related Fields 129 1–17. DOI:10.1007/s00440-003-0318-7.
  • [5] Berg, C. and Forst, G. (1975). Potential Theory on Locally Compact Abelian Groups. Ergebnisse der Mathematik und ihrer Grenzgebiete 87. New York: Springer.
  • [6] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191–212. DOI:10.1214/154957805100000122.
  • [7] Blumenthal, R.M. and Getoor, R. (1960). A dimension theorem for sample functions of stable processes. Illinois J. Math. 4 370–375.
  • [8] Böttcher, B., Schilling, R. and Wang, J. (2013). Lévy Matters. III. Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Lecture Notes in Math. 2099. Cham: Springer.
  • [9] Chaumont, L., Pantí, H. and Rivero, V. (2013). The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19 2494–2523.
  • [10] Chen, Z.Q. (2009). Symmetric jump processes and their heat kernel estimates. Sci. China Math. 52 1423–1445.
  • [11] Chen, Z.Q. and Kumagai, T. (2003). Heat kernel estimates for stable-like processes on $d$-sets. Stoch. Model. Appl. 108 27–62.
  • [12] Chen, Z.Q. and Zhang, X. (2016). Heat kernels and analyticity of non-symmetric jump diffusion semigroups. Probab. Theory Related Fields 165 267–312.
  • [13] Falconer, K.J. (2003). Fractal Geometry–Mathematical Foundations and Applications, 2nd ed. New York: Wiley & Sons.
  • [14] Gikhman, I.I. and Skorohod, A.V. (1974). The Theory of Stochastic Processes, Vol. 1. Berlin: Springer.
  • [15] Graversen, S.E. and Vuolle-Apiala, J. (1986). $\alpha$-Self-similar Markov processes. Probab. Theory Related Fields 71 149–158.
  • [16] Hawkes, J. (1970/71). Some dimension theorems for the sample functions of stable processes. Indiana Univ. Math. J. 20 733–738. DOI:10.1512/iumj.1971.20.20058.
  • [17] Hawkes, J. (1971). On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set. Z. Wahrsch. Verw. Gebiete 19 90–102.
  • [18] Hawkes, J. and Pruitt, W.E. (1973/74). Uniform dimension results for processes with independent increments. Z. Wahrsch. Verw. Gebiete 28 277–288.
  • [19] Jacob, N. (2001). Pseudo Differential Operators and Markov Processes, Vol. I: Fourier Analysis and Semigroups. London: Imperial College Press.
  • [20] Jacob, N. and Schilling, R.L. (2001). Lévy-type processes and pseudodifferential operators. In Lévy Processes 139–168. Boston, MA: Birkhäuser.
  • [21] Kaleta, K. and Sztonyk, P. (2015). Estimates of transition densities and their derivatives for jump Lévy processes. J. Math. Anal. Appl. 431 260–282.
  • [22] Kaufman, R. (1968). Une propriété métrique du mouvement brownien. C. R. Acad. Sci. Paris 268 727–728.
  • [23] Khoshnevisan, D. (1997). Escape rates for Lévy processes. Studia Sci. Math. Hungar. 33 177–183.
  • [24] Khoshnevisan, D., Schilling, R.L. and Xiao, Y. (2012). Packing dimension profiles and Lévy processes. Bull. Lond. Math. Soc. 44 931–943.
  • [25] Khoshnevisan, D. and Xiao, Y. (2003). Weak unimodality of finite measures, and an application to potential theory of additive Lévy processes. Proc. Amer. Math. Soc. 131 2611–2616. DOI:10.1090/S0002-9939-02-06778-3.
  • [26] Khoshnevisan, D. and Xiao, Y. (2005). Lévy processes: Capacity and Hausdorff dimension. Ann. Probab. 33 841–878.
  • [27] Khoshnevisan, D. and Xiao, Y. (2008). Packing dimension of the range of a Lévy process. Proc. Amer. Math. Soc. 136 2597–2607. DOI:10.1090/S0002-9939-08-09163-6.
  • [28] Khoshnevisan, D. and Xiao, Y. (2009). Harmonic analysis of additive Lévy processes. Probab. Theory Related Fields 145 459–515.
  • [29] Khoshnevisan, D. and Xiao, Y. (2015). Brownian motion and thermal capacity. Ann. Probab. 43 405–434.
  • [30] Kinney, J.R. (1953). Continuity properties of sample functions of Markov processes. Trans. Amer. Math. Soc. 74 280–302. DOI:10.2307/1990883.
  • [31] Kiu, S.W. (1980). Semi-stable Markov processes in $\mathbf{R}^{n}$. Stoch. Model. Appl. 10 183–191.
  • [32] Knopova, V. and Kulik, A. (2017). Intrinsic compound kernel estimates for the transition probability density of a Lévy-type process and their applications. Probab. Math. Statist. 37 53–100.
  • [33] Knopova, V. and Schilling, R. (2015). On level and collision sets of some Feller processes. ALEA Lat. Am. J. Probab. Math. Stat. 12 1001–1029.
  • [34] Knopova, V., Schilling, R.L. and Wang, J. (2015). Lower bounds of the Hausdorff dimension for the images of Feller processes. Statist. Probab. Lett. 97 222–228. DOI:10.1016/j.spl.2014.11.027.
  • [35] Kolokoltsov, V. (2000). Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80 725–768.
  • [36] Kolokoltsov, V. (2010). Nonlinear Markov Processes and Kinetic Equations. Cambridge Tracts in Mathematics 182. Cambridge: Cambridge Univ. Press.
  • [37] Kühn, F. (2017). Existence and estimates of moments for Lévy-type processes. Stochastic Process. Appl. 127 1018–1041.
  • [38] Kühn, F. (2017). Lévy Matters IV. Lévy-Type Processes: Moments, Construction and Heat Kernel Estimates. Lecture Notes in Math. 2187. Berlin: Springer.
  • [39] Lamperti, J. (1972). Semi-stable Markov processes. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [40] Liu, L. and Xiao, Y. (1998). Hausdorff dimension theorems for self-similar Markov processes. Probab. Math. Statist. 18 369–383.
  • [41] Manstavičius, M. (2004). $p$-Variation of strong Markov processes. Ann. Probab. 32 2053–2066.
  • [42] Mörters, P. and Peres, P. (2010). Brownian Motion. Cambridge: Cambridge Univ. Press.
  • [43] Negoro, A. (1994). Stable-like processes: Construction of the transition density and the behavior of sample paths near ${t=0}$. Osaka J. Math. 31 189–214.
  • [44] Perkins, E.A. and Taylor, S.J. (1987). Uniform measure results for the image of subsets under Brownian motion. Probab. Theory Related Fields 76 257–289. DOI:10.1007/BF01297485.
  • [45] Pruitt, W.E. (1975). Some dimension results for processes with independent increments. In Stochastic Processes and Related Topics (Proc. Summer Res. Inst. on Statist. Inference for Stochastic Processes) 133–165. New York: Academic Press.
  • [46] Pruitt, W.E. (1981). The growth of random walks and Lévy processes. Ann. Probab. 9 948–956.
  • [47] Revuz, D. and Yor, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Berlin: Springer.
  • [48] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. New York: Chapman & Hall.
  • [49] Schilling, R.L. (1998). Growth and Hölder conditions for the sample paths of Feller processes. Probab. Theory Related Fields 112 565–611.
  • [50] Schilling, R.L. and Partzsch, L. (2014). Brownian Motion. An Introduction to Stochastic Processes, 2nd ed. Berlin: De Gruyter. xvi+408 pp.
  • [51] Schilling, R.L. and Schnurr, A. (2010). The symbol associated with the solution of a stochastic differential equation. Electron. J. Probab. 15 1369–1393.
  • [52] Taylor, S.J. (1986). The measure theory of random fractals. Math. Proc. Cambridge Philos. Soc. 100 383–406. DOI:10.1017/S0305004100066160.
  • [53] Vuolle-Apiala, J. (1994). Itô excursion theory for self-similar Markov processes. Ann. Probab. 22 546–565.
  • [54] Vuolle-Apiala, J. and Graversen, S.E. (1988). Duality theory for self-similar Markov processes. Ann. Inst. Henri Poincaré Probab. Stat. 22 376–392.
  • [55] Xiao, Y. (1998). Asymptotic results for self-similar Markov processes. In Asymptotic Methods in Probability and Statistics 323–340. Amsterdam: North-Holland.
  • [56] Xiao, Y. (2004). Random fractals and Markov processes. In Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot, Part 2. Proc. Sympos. Pure Math., Part 2 72 261–338. Providence, RI: Amer. Math. Soc.
  • [57] Xiao, Y. (2008). Strong local nondeterminism and sample path properties of Gaussian random fields. In Asymptotic Theory in Probability and Statistics with Applications (T.L. Lai, Q. Shao, L. Qian, eds.). Adv. Lect. Math. (ALM) 2 136–176. Somerville, MA: Int. Press.
  • [58] Yang, X. (2005). Sharp value for the Hausdorff dimension of the range and the graph of stable-like processes. Bernoulli. To appear.