Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Harmonic analysis of additive Lévy processes
Download PDF
Download PDF
  • Published: 22 August 2008

Harmonic analysis of additive Lévy processes

  • Davar Khoshnevisan1 &
  • Yimin Xiao2 

Probability Theory and Related Fields volume 145, pages 459–515 (2009)Cite this article

  • 188 Accesses

  • 30 Citations

  • Metrics details

Abstract

Let X 1, . . . ,X N denote N independent d-dimensional Lévy processes, and consider the N-parameter random field

$$\mathfrak{X}(t) := X_1(t_1)+\cdots+ X_N(t_N).$$

First we demonstrate that for all nonrandom Borel sets \({F\subseteq{{\bf R}^d}}\) , the Minkowski sum \({\mathfrak{X}({{\bf R}^{N}_{+}})\oplus F}\) , of the range \({\mathfrak{X}({{\bf R}^{N}_{+}})}\) of \({\mathfrak{X}}\) with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong by removing a certain condition of symmetry in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003). Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical (non-probabilistic) harmonic analysis that might be of independent interest. As was shown in Khoshnevisan et al. (Ann Probab 31(2):1097–1141, 2003), the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. Presently, we highlight a few new consequences.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Aizenman M.: The intersection of Brownian paths as a case study of a renormalization group method for quantum field theory. Comm. Math. Phys. 97(1-2), 91–110 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Albeverio S., Zhou X.Y.: Intersections of random walks and Wiener sausages in four dimensions. Acta Appl. Math. 45(2), 195–237 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berg C., Forst G.: Potential Theory on Locally Compact Abelian Groups. Springer, New York (1975)

    MATH  Google Scholar 

  4. Bertoin, J.: Subordinators: Examples and applications, In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717. Springer, Berlin (1999a)

  5. Bertoin J.: Intersection of independent regenerative sets. Probab. Theory Relat. Fields 114(1), 97–121 (1999b)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Csiszár I.: A note on limiting distributions on topological groups, English, with Russian summary. Magyar Tud. Akad. Mat. Kutató Int. Közl. 9, 595–599 (1965)

    Google Scholar 

  9. Dellacherie C., Meyer P.-A.: Probabilities and Potential, vol. 29. North-Holland, Amsterdam (1978)

    Google Scholar 

  10. Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart. Springer, Berlin, Reprint of the 1984 edn (2001)

  11. Dynkin E.B.: Self-intersection local times, occupation fields, and stochastic integrals. Adv. Math. 65(3), 254–271 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dynkin E.B.: Generalized random fields related to self-intersections of the Brownian motion. Proc. Nat. Acad. Sci. U.S.A. 83(11), 3575–3576 (1986)

    Article  MathSciNet  Google Scholar 

  13. Dynkin E.B.: Random fields associated with multiple points of the Brownian motion. J. Funct. Anal. 62(3), 397–434 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Dynkin, E.B.: Local times and quantum fields, In: Seminar on Stochastic Processes, 1983 (Gainesville, FL, 1983), Progr. Probab. Statist., vol. 7, pp. 69– 83. Birkhäuser Boston, Boston, MA, (1984)

  15. Dynkinm E.B.: Polynomials of the occupation field and related random fields. J. Funct. Anal. 58(1), 20–52 (1984)

    Article  MathSciNet  Google Scholar 

  16. Dynkin E.B.: Gaussian and non-Gaussian random fields associated with Markov processes. J. Funct. Anal. 55(3), 344–376 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  17. Dynkin, E.B.: Gaussian random fields and Gaussian evolutions, In: Theory and Application of Random Fields (Bangalore, 1982), Lecture Notes in Control and Inform. Sci., vol. 49, pp. 28–39. Springer, Berlin (1983)

  18. Dynkin E.B.: Markov processes as a tool in field theory. J. Funct. Anal. 50(2), 167–187 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Dynkin E.B.: Markov processes, random fields and Dirichlet spaces. Phys. Rep. 77(3), 239–247 (1981)

    Article  MathSciNet  Google Scholar 

  20. Dynkin E.B.: Markov processes and random fields. Bull. Am. Math. Soc. (N.S.) 3(3), 975–999 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dvoretzky A., Erdős P., Kakutani S.: Multiple points of paths of Brownian motion in the plane. Bull. Res. Council Israel 3, 364–371 (1954)

    MathSciNet  Google Scholar 

  22. Dvoretzky A., Erdős P., Kakutani S.: Double points of paths of Brownian motion in n-space. Acta Sci. Math. Szeged 12, 75–81 (1950)

    MathSciNet  Google Scholar 

  23. Dvoretzky A., Erdős P., Kakutani S., Taylor S.J.: Triple points of Brownian paths in 3-space. Proc. Cambridge Philos. Soc. 53, 856–862 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  24. Evans S.N. Potential theory for a family of several Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 23(3), 499–530 (English, with French summary (1987a))

  25. Evans SN.: Multiple points in the sample paths of a Lévy process. Probab. Theory Relat. Fields 76(3), 359–367 (1987b)

    Article  Google Scholar 

  26. Farkas W., Jacob N., Schilling R.L.: Function spaces related to continuous negative definite functions: ψ-Bessel potential spaces. Dissertationes Math. (Rozprawy Mat.) 393, 62 (2001)

    MathSciNet  Google Scholar 

  27. Farkas W., Leopold H.-G.: Characterisations of function spaces of generalised smoothness. Ann. Mat. Pura Appl. (4) 185(1), 1–62 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  28. Felder G., Fröhlich J.: Intersection properties of simple random walks: a renormalization group approach. Comm. Math. Phys. 97(1-2), 111–124 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. Fitzsimmons P.J., Salisbury T.S.: Capacity and energy for multiparameter Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 25(3), 325–350 (1989) (English, with French summary)

    MATH  MathSciNet  Google Scholar 

  30. Fristedt, B.: Sample Functions of Stochastic Processes with Stationary, Independent Increments, In: Advances in Probability and Related Topics, vol. 3, pp. 241–396. Dekker, New York (1974)

  31. Fukushima M., Ōshima Y., Takeda M.: Dirichlet Forms and Symmetric Markov Processes, vol. 19. Walter de Gruyter & Co., Berlin (1994)

    Google Scholar 

  32. Getoor R.K.: Excessive Measures. Birkhäuser Boston Inc., Boston, MA (1990)

    MATH  Google Scholar 

  33. Hawkes, J.: Some geometric aspects of potential theory, Stochastic analysis and applications (Swansea, 1983), Lecture Notes in Math., 1095, Springer, Berlin, pp 130–154 (1984)

  34. Hawkes J.: Potential theory of Lévy processes. Proc. London Math. Soc. (3) 38(2), 335–352 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  35. Hawkes J.: Image and intersection sets for subordinators. J. London Math. Soc. (2) 17(3), 567–576 (1978a)

    Article  MATH  MathSciNet  Google Scholar 

  36. Hawkes J.: Multiple points for symmetric Lévy processes. Math. Proc. Cambridge Philos. Soc. 83(1), 83–90 (1978b)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hawkes J.: Local properties of some Gaussian processes. Z. Wahrsch. Verw. Gebiete 40(4), 309–315 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  38. Hawkes J.: Intersections of Markov random sets. Z. Wahrsch. Verw. Gebiete 37(3), 243–251 (1976/77)

    Article  MathSciNet  Google Scholar 

  39. Hendricks W.J.: Multiple points for transient symmetric Lévy processes in R d. Z. Wahrsch. Verw. Gebiete 49(1), 13–21 (1979)

    Article  MathSciNet  Google Scholar 

  40. Hendricks W.J.: Multiple points for a process in R 2 with stable components. Z. Wahrsche. Verw. Gebiete 28, 113–128 (1973/74)

    Article  MathSciNet  Google Scholar 

  41. Hendricks, W.J., Taylor, S.J.: Concerning some problems about polar sets for processes with stationary independent increments, (1979), unpublished manuscript

  42. Hirsch F.: Potential theory related to some multiparameter processes. Potential Anal. 4(3), 245–267 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  43. Hirsch, F., Song S.: Multiparameter Markov processes and capacity, In: Seminar on Stochastic Analysis, Random Fields and Applications (Ascona, 1996), Progr. Probab., vol. 45, pp. 189–200, Birkhäuser, Basel (1999)

  44. Hirsch F., Song S.: Inequalities for Bochner’s subordinates of two-parameter symmetric Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 32(5), 589–600 (1996) (English, with English and French summaries)

    MathSciNet  Google Scholar 

  45. Hirsch F., Song S.: Markov properties of multiparameter processes and capacities. Probab. Theory Relat. Fields 103(1), 45–71 (1995a)

    Article  MATH  MathSciNet  Google Scholar 

  46. Hirsch F., Song S.: Symmetric Skorohod topology on n-variable functions and hierarchical Markov properties of n-parameter processes. Probab. Theory Relat. Fields 103(1), 25–43 (1995b)

    Article  MATH  MathSciNet  Google Scholar 

  47. Hirsch F., Song S.: Une inégalité maximale pour certains processus de Markov à plusieurs paramètres. II. C. R. Acad. Sci. Paris Sér. I Math. 320(7), 867–870 (1995) (French)

    MATH  MathSciNet  Google Scholar 

  48. Hirsch F., Song S.: Une inégalité maximale pour certains processus de Markov à plusieurs paramètres. I. C. R. Acad. Sci. Paris Sér. I Math. 320(6), 719–722 (1995) (French)

    MATH  MathSciNet  Google Scholar 

  49. Hirsch F., Song S.: Propriétés de Markov des processus à plusieurs paramètres et capacités. C. R. Acad. Sci. Paris Sér. I Math. 319(5), 483–488 (1994) (French)

    MATH  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Google Scholar 

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

    MathSciNet  Google Scholar 

  53. Hunt G.A.: Markoff processes and potentials. Proc. Nat. Acad. Sci. U.S.A. 42, 414–418 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  54. Jacob N.: Pseudo Differential Operators and Markov Processes, vol. III. Imperial College Press, London (2005)

    MATH  Google Scholar 

  55. Jacob N.: Pseudo Differential Operators & Markov Processes, vol. II. Imperial College Press, London (2002)

    MATH  Google Scholar 

  56. Jacob N.: Pseudo-Differential Operators and Markov Processes, vol. I. Imperial College Press, London (2001)

    Google Scholar 

  57. Jacob, N., Schilling, R.L.: Function spaces as Dirichlet spaces (about a paper by W. Maz′ya and J. Nagel). Comment on: “On equivalent standardization of anisotropic functional spaces H μ(R n)” (German) [Beiträge Anal. No. 12 (1978), 7–17], Z. Anal. Anwendungen 24(1), 3–28 (2005)

  58. Kahane J.-P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  59. Kakutani S: On Brownian motions in n-space. Proc. Imp. Acad. Tokyo 20, 648–652 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  60. Kakutani S.: Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo 20, 706–714 (1944)

    Article  MATH  MathSciNet  Google Scholar 

  61. Kesten, H.: Hitting Probabilities of Single Points for Processes with Stationary Independent Increments, Memoirs of the American Mathematical Society, No. 93. American Mathematical Society, Providence, RI (1969)

  62. Khoshnevisan D.: Intersections of Brownian motions. Expo. Math. 21(2), 97–114 (2003)

    MATH  MathSciNet  Google Scholar 

  63. Khoshnevisan D.: Brownian sheet images and Bessel–Riesz capacity. Trans. Am. Math. Soc. 351(7), 2607–2622 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  64. Khoshnevisan D.: Multiparameter Processes. Springer, New York (2002)

    MATH  Google Scholar 

  65. Khoshnevisan D., Shieh N.-R., Xiao Y.: Hausdorff dimension of the contours of symmetric additive Lévy processes. Probab. Theory Relat. Fields 140(1), 129–167 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  66. Khoshnevisan D., Xiao Y.: Level sets of additive Lévy processes. Ann. Probab. 30(2), 62–100 (2002)

    MATH  MathSciNet  Google Scholar 

  67. Khoshnevisan, D., Xiao, Y.: Additive Lévy processes: capacity and Hausdorff dimension, In: Proc. of Inter. Conf. on Fractal Geometry and Stochastics III., Progress in Probability, vol. 57, pp. 62–100. Birkhäuser, Basel (2004)

  68. Khoshnevisan D., Xiao Y.: Lévy processes: capacity and Hausdorff dimension. Ann. Probab. 33(3), 841–878 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  69. Khoshnevisan D., Xiao Y.: Images of the Brownian sheet. Trans. Am. Math. Soc. 359(7), 3125–3151 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  70. Khoshnevisan D., Xiao Y., Zhong Y.: Measuring the range of an additive Lévy process. Ann. Probab. 31(2), 1097–1141 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  71. Lawler G.F.: Intersections of random walks with random sets. Israel J. Math. 65(2), 113–132 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  72. Lawler G.F.: Intersections of random walks in four dimensions. II . Comm. Math. Phys. 97(4), 583–594 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  73. Lawler G.F.: The probability of intersection of independent random walks in four dimensions. Comm. Math. Phys. 86(4), 539–554 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  74. Le Gall, J.-F.: Some properties of planar Brownian motion, École d’Été de Probabilités de Saint-Flour XX—1990, Lecture Notes in Math., vol. 1527, pp. 111–235. Springer, Berlin (1992)

  75. Le Gall J.-F.: Le comportement du mouvement brownien entre les deux instants où il passe par un point double. J. Funct. Anal. 71(2), 246–262 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  76. Le Gall J.-F., Rosen J.S., Shieh N.-R.: Multiple points of Lévy processes. Ann. Probab. 17(2), 503–515 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  77. Lévy P.: Le mouvement brownien plan (French). Am. J. Math. 62, 487–550 (1940)

    Article  Google Scholar 

  78. Marcus M.B., Rosen J.: Multiple Wick product chaos processes. J. Theor. Probab. 12(2), 489–522 (1999b)

    Article  MATH  MathSciNet  Google Scholar 

  79. Marcus, M.B., Rosen, J.: Renormalized self-intersection local times and Wick power chaos processes. Mem. Am. Math. Soc. 142(675), (1999a)

  80. Masja W., Nagel J.: Über äquivalente Normierung der anisotropen Funktionalräume H μ(R n). Beiträge Anal. 12, 7–17 (1978) (German)

    MathSciNet  Google Scholar 

  81. Mattila P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  82. Orey, S.: Polar sets for processes with stationary independent increments, In: Markov Processes and Potential Theory (Proc. Sympos. Math. Res. Center, Madison, Wis., 1967). Wiley, New York, (1967)

  83. Pemantle R., Peres Y., Shapiro J.W.: The trace of spatial Brownian motion is capacity-equivalent to the unit square. Probab. Theory Relat. Fields 106(3), 379–399 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  84. Peres, Y.: Probability on trees: an introductory climb, In: Lectures on Probability Theory and Statistics (Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, pp. 193– 280. Springer, Berlin, (1999)

  85. Peres Y.: Remarks on intersection-equivalence and capacity-equivalence. Ann. Inst. H. Poincaré Phys. Théor. 64(3), 339–347 (1996) (English, with English and French summaries)

    MATH  MathSciNet  Google Scholar 

  86. Peres Y.: Intersection-equivalence of Brownian paths and certain branching processes. Comm. Math. Phys. 177(2), 417–434 (1996b)

    Article  MATH  MathSciNet  Google Scholar 

  87. Ren J.G.: Topologie p-fine sur l’espace de Wiener et théorème des fonctions implicite. Bull. Sci. Math. 114(2), 99–114 (1990) (French, with English summary)

    MATH  MathSciNet  Google Scholar 

  88. Röckner, M.: General Theory of Dirichlet Forms and Applications, In: Dirichlet forms, Varenna, 1992, Lecture Notes in Math., vol. 1563, pp. 129–193. Springer, Berlin (1993)

  89. Rogers, L.C.G.: Multiple points of Markov processes in a complete metric space, In: Séminaire de Probabilités XXIII, Lecture Notes in Math., vol. 1372, pp. 186– 197. Springer, Berlin (1989)

  90. Rosen J.: Self-intersections of random fields. Ann. Probab. 12(1), 108–119 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  91. Rosen J.: A local time approach to the self-intersections of Brownian paths in space. Comm. Math. Phys. 88(3), 327–338 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  92. Salisbury, T.S.: Energy, and intersections of Markov chains, Random discrete structures, Minneapolis, MN, 1993,IMA Vol. Math. Appl., vol. 76, pp. 213–225. Springer, New York (1996)

  93. Salisbury T.S.: A low intensity maximum principle for bi-Brownian motion. Illinois J. Math. 36(1), 1–14 (1992)

    MathSciNet  Google Scholar 

  94. Salisbury, T.S.: Brownian bitransforms, Seminar on Stochastic Processes, 1987 (Princeton, NJ, 1987), Progr. Probab. Statist., vol. 15, pp. 249–263. Birkhäuser Boston, Boston, MA (1988)

  95. Sato, Ken-iti.: Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge (1999) Translated from the 1990 Japanese original, Revised by the author

  96. Schoenberg I.J.: Metric spaces and positive definite functions. Trans. Am. Math. Soc. 44, 522–536 (1938)

    Article  MATH  MathSciNet  Google Scholar 

  97. Slobodeckiĭ, L.N., S.L.: Sobolev’s spaces of fractional order and their application to boundary problems for partial differential equations, Russian, Dokl. Akad. Nauk SSSR (N.S.), 118, 243–246 (1958)

  98. Tongring N.: Which sets contain multiple points of Brownian motion?. Math. Proc. Cambridge Philos. Soc. 103(1), 181–187 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  99. Ville J.: Sur un problème de géométrie suggéré par l’étude du mouvement brownien. C. R. Acad. Sci Paris 215, 51–52 (1942) (French)

    MathSciNet  Google Scholar 

  100. Walsh, J.B.: Martingales with a Multidimensional Parameter and Stochastic Integrals in the Plane, Lectures in Probability and Statistics, Santiago de Chile, 1986, Lecture Notes in Math., vol. 1215, pp. 329–491. Springer, Berlin (1986)

  101. Westwater J.: On Edwards’ model for long polymer chains. Comm. Math. Phys. 72(2), 131–174 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  102. Westwater J.: On Edwards’ model for polymer chains. II. The self-consistent potential. Comm. Math. Phys. 79(1), 53–73 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  103. Westwater J.: On Edwards’ model for polymer chains. III. Borel summability. Comm. Math. Phys. 84(4), 459–470 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  104. Wolpert R.L.: Local time and a particle picture for Euclidean field theory. J. Funct. Anal. 30(3), 341–357 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  105. Yang M.: Hausdorff dimension of the image of additive processes. Stoch. Process Appl. 118(4), 681–702 (2008)

    Article  MATH  Google Scholar 

  106. Yang M.: On a theorem in multi-parameter potential theory. Electron. Comm. Probab. 12, 267–275 (2007) (electronic)

    MathSciNet  Google Scholar 

  107. Yang M.: A short proof of the dimension formula for Lévy processes. Electron. Comm. Probab. 11, 217–219 (2006) (electronic)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Utah, 155 S. 1400 E., Salt Lake City, UT, 84112-0090, USA

    Davar Khoshnevisan

  2. Department of Statistics and Probability, Michigan State University, A-413 Wells Hall, East Lansing, MI, 48824, USA

    Yimin Xiao

Authors
  1. Davar Khoshnevisan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yimin Xiao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Davar Khoshnevisan.

Additional information

Research supported in part by a grant from the National Science Foundation (DMS-0706728).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Khoshnevisan, D., Xiao, Y. Harmonic analysis of additive Lévy processes. Probab. Theory Relat. Fields 145, 459–515 (2009). https://doi.org/10.1007/s00440-008-0175-5

Download citation

  • Received: 27 June 2007

  • Revised: 27 July 2008

  • Published: 22 August 2008

  • Issue Date: November 2009

  • DOI: https://doi.org/10.1007/s00440-008-0175-5

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Keywords

  • Additive Lévy processes
  • Multiplicative Lévy processes
  • Capacity
  • Intersections of regenerative sets

Mathematics Subject Classification (2000)

  • 60G60
  • 60J55
  • 60J45
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature