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Hausdorff dimension theorems for self-similar Markov processes

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Języki publikacji
EN
Abstrakty
EN
Let X(t) (t ∈R+) be an α-self-similar Markov process on Rd or Rd+. The Hausdorff dimension of the image, graph and zero set of X (t) are obtained under certain mild conditions. Similar results are also proved for a class of elliptic diffusions.
Rocznik
Strony
369--383
Opis fizyczny
Bibliogr. 29 poz.
Twórcy
autor
  • Department of Mathematics, Wuhan University, Wuhan 430072, China
autor
  • Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, U.S.A.
Bibliografia
  • [1] R. J. Adler, The Geometry of Random Fields, Wiley, New York 1981.
  • [2] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc. 73 (1967), pp. 890-896.
  • [3] M. Chaleyat-Maurel and J.-F. LeGall, Green functions, capacity and sample path properties for a class of hypoelliptic diffusion processes, Probab. Theory Related Fields 83 (1989), pp. 219-264.
  • [4] A. Dvoretsky and P. Erdös, Some problems on random walk in space, Proceedings of the Second Berkeley Symposium, Univ. of California Press, Berkeley 1950, pp. 353-367.
  • [5] K. J. Falconer, Fractal Geometry - Mathematical Foundations and Applications, Wiley, 1990.
  • [6] S. E. Graversen and J. Vuolle-Apiala, α-self-similar Markov processes, Probab. Theory Related Fields 71 (1986), pp. 149-158.
  • [7] J.-P. Kahane, Some Random Series of Functions, 2nd edition, Cambridge University Press, 1985.
  • [8] S. W. Kiu, Semi-stable Markov processes in Rn, Stochastic Process. Appl. 10 (1980), pp. 183-191.
  • [9] X Lamperti, Semi-stable Markov processes, Z. Wahrsch. Verw. Gebiete 22 (1972), pp. 205-225.
  • [10] Bingzhang Li, Lihu Huang and Luqin Liu, The exact measure functions of the images for a class of self-similar processes, Preprint, 1997.
  • [11] Luqin Liu, The Hausdorff dimensions of the image, graph and level sets of a self-similar process, in: Stochastic and Quantum Mechanics, World Scientific Press, 1992, pp. 178-188.
  • [12] M. B. Marcus, Capacity of level sets of certain stochastic processes, Z, Wahrsch. Verw. Gebiete 34 (1976), pp. 279-284.
  • [13] X Neveu, Mathematical Foundations of the Calculus of Probability, Holden-Day, San Francisco 1965.
  • [14] W. E. Pruitt and S. J. Taylor, Sample path properties of processes with stable components, Z. Wahrsch. Verw. Gebiete 12 (1969), pp. 267-289.
  • [15] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 2nd edition, Springer, Berlin 1994.
  • [16] N.-R. Shieh, Multiple points of sample paths of Markov processes, Ann. Probab. 20 (1992), pp. 553-562.
  • [17] F. Spitzer, Some theorems concerning 2-dimensional Brownian motion, Trans. Amer. Math. Soc. 87 (1958), pp. 187-197.
  • [18] C. Stone, The set of zeros of a semistable process, Illinois X Math. 7 (1963), pp. 631-660.
  • [19] D. W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilité XXII, Lecture Notes in Math. 1321, Berlin 1988, pp. 316-347.
  • [20] A. S. Sznitman, Some bounds and limiting results for the measure of Wiener sausage of small radius associated with elliptic diffusions, Stochastic Process. Appl. 25 (1987), pp. 1-25.
  • [21] S. J. Taylor, The measure theory of random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), pp. 383-406.
  • [22] F. Testard, Polarité, points multiples et géométrie de certains processus gaussiens, These doctorat, Orsay 1987. -
  • [23] J. Vuolle-Apiala, Time-change of self-similar Markov processes, Ann. Inst. H. Poincare Probab. Statist 25 (1989), pp. 581-587.
  • [24] — Itô excursion theory for self-similar Markov processes, Ann. Probab. 22 (1994), pp. 546-565.
  • [25] — and S. E. Graversen, Duality theory for self-similar Markov processes, Ann. Inst. H. Poincare Probab. Statist 22 (1988), pp. 376-392.
  • [26] Yimin Xiao, Dimension results for Gaussian vector fields and index α-stable fields, Ann. Probab. 23 (1995), pp. 273-291.
  • [27] — Fractal measures of the sets associated to Gaussian random fields, to appear in: Trends in Probability and Related Analysis: Proceedings of the Symposium on Analysis and Probability 1996, N. Kono and N.-R. Shieh (eds.), World Scientific.
  • [28] — Asymptotic results for self-similar Markov processes, to appear in : Asymptotic Methods in Probability and Statistics (ICAMPS’97), B. Szyszkowicz (ed.), Elsevier Science.
  • [29] — and Luqin Liu, Packing dimension results for certain Levy processes and self-similar Markov processes, Chinese Ann. Math. 17A (1996), pp. 392-400.
Typ dokumentu
Bibliografia
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