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Exact joint distributions of three global characteristic times for Brownian motion

Alexander K. Hartmann1 and Satya N. Majumdar2

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Phys. Rev. E 111, 044134 – Published 25 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.044134

Abstract

We consider three global chracteristic times for a one-dimensional Brownian motion x(τ) in the interval τ∈[0,t]: the occupation time to denoting the cumulative time where x(τ)>0, the time tm at which the process achieves its global maximum in [0,t], and the last-passage time tl through the origin before t. All three random variables have the same marginal distribution given by Lévy's arcsine law. We compute exactly the pairwise joint distributions of these three times and show that they are quite different from each other. The joint distributions display rather rich and nontrivial correlations between these times. Our analytical results are verified by numerical simulations.

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References (104)

  1. P. Fornasini, The Uncertainty in Physical Measurements: An Introduction to Data Analysis in the Physics Laboratory (Springer, New York, 2008).
  2. V. L. Smith, Am. Econ. Rev. 66, 274 (1976).
  3. D. Houle, C. Pélabon, G. Wagner, and T. Hansen, Q. Rev. Biol. 86, 3 (2011).
  4. P. D. Delmas, Osteoporosis Int. 9, S33 (1999).
  5. L. Bennani, F. Allali, S. Rostom, I. Hmamouchi, H. Khazzani, L. El Mansouri, L. Ichchou, F. Z. Abourazzak, R. Abouqal, and N. Hajjaj-Hassouni, Clin. Rheumatol. 28, 1283 (2009).
  6. T. A. Hillier, L. Lui, D. M. Kado, E. LeBlanc, K. K. Vesco, D. C. Bauer, J. A. Cauley, K. E. Ensrud, D. M. Black, M. C. Hochberg, and S. R. Cummings, J. Bone Miner. Res. 27, 153 (2012).
  7. W. Pluskiewicz, P. Adamczyk, A. Werner, M. Bach, and B. Drozdzowska, Biomedicines 11, 2231 (2023).
  8. D. M. Ragan, Structural Geology (Cambridge University Press, Cambridge, 2012).
  9. S. D. Sloan, G. P. Tsoflias, D. W. Steeples, and P. D. Vincent, J. Appl. Geophys. 62, 281 (2007).
  10. H. Yuan, A. Z. Abdu, and L. Nielsen, Geophys. 88, MR141 (2023).
  11. J. A. Hildebrand, S. M. Wiggins, J. L. Driver, and M. R. Waters, Archaeol. Prospect. 14, 245 (2007).
  12. R. Serfozo, Basics of Applied Stochastic Processes (Springer, Berlin, 2009).
  13. W. Feller, Introduction to Probability Theory and Its Applications (John Wiley & Sons, New York, 1957).
  14. P. Mörters, Y. Peres, Brownian Motion (Cambridge University Press, Cambridge, 2010), Vol. 30.
  15. R. M. Mazo, Brownian Motion: Fluctuations, Dynamics, and Applications (Oxford University Press, Oxford, 2008).
  16. P. Doyle and J. Snell, Random Walks and Electric Networks (Mathematical Association of America, Washington, DC, 1984).
  17. J. Baz and G. Chacko, Financial Derivatives: Pricing, Applications, and Mathematics (Cambridge University Press, Cambridge, 2004).
  18. A. Hirsa and S. N. Neftci, An Introduction to the Mathematics of Financial Derivatives (Academic Press, Amsterdam, 2014).
  19. F. Schweitzer, Brownian Agents and Active Particles: Collective Dynamics in the Natural and Social Sciences (Springer, Berlin, 2007).
  20. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, 2001).
  21. S. N. Majumdar, Curr. Sci. 89, 2075 (2005).
  22. S. N. Majumdar, Physica A 389, 4299 (2010).
  23. A. J. Bray, S. N. Majumdar, and G. Schehr, Adv. Phys. 62, 225 (2013).
  24. L. A. Shepp, J. Appl. Probab. 16, 423 (1979).
  25. E. Buffet, J. Appl. Math. Stoch. Anal. 16, 201 (2003).
  26. J. Randon-Furling and S. N. Majumdar, J. Stat. Mech. (2007) P10008.
  27. S. N. Majumdar, J. Randon-Furling, M. J. Kearney, and M. Yor, J. Phys. A: Math. Theor. 41, 365005 (2008).
  28. S. N. Majumdar and J.-P. Bouchaud, Quant. Finance 8, 753 (2008).
  29. S. N. Majumdar, A. Comtet, and J. Randon-Furling, J. Stat. Phys. 138, 955 (2010).
  30. G. Schehr and P. Le Doussal, J. Stat. Mech. (2010) P01009.
  31. J. Rambeau and G. Schehr, Phys. Rev. E 83, 061146 (2011).
  32. F. Mori, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett. 123, 200201 (2019).
  33. F. Mori, S. N. Majumdar, and G. Schehr, Phys. Rev. E 101, 052111 (2020).
  34. P. Lévy, Compos. Math. 7, 283 (1940).
  35. E. S. Andersen, Math. Scand. 1, 263 (1954).
  36. J. Lamperti, Trans. Am. Math. Soc. 88, 380 (1958).
  37. C. Godrèche and J. M. Luck, J. Stat. Phys. 104, 489 (2001).
  38. S. Burov and E. Barkai, Phys. Rev. Lett. 107, 170601 (2011).
  39. F. den Hollander, S. N. Majumdar, J. M. Meylahn, and H. Touchette, J. Phys. A: Math. Theor. 52, 175001 (2019).
  40. A. Dhar and S. N. Majumdar, Phys. Rev. E 59, 6413 (1999).
  41. G. de Smedt, C. Godrèche, and J. M. Luck, J. Phys. A: Math. Gen. 34, 1247 (2001).
  42. T. J. Newman and Z. Toroczkai, Phys. Rev. E 58, R2685 (1998).
  43. J.-M. Drouffe and C. Godrèche, J. Phys. A: Math. Gen. 31, 9801 (1998).
  44. Z. Toroczkai, T. J. Newman, and S. Das Sarma, Phys. Rev. E 60, R1115 (1999).
  45. A. Baldassarri, J.-P. Bouchaud, I. Dornic, and C. Godrèche, Phys. Rev. E 59, R20 (1999).
  46. J. T. Cox and D. Griffeath, Ann. Probab. 11, 876 (1983).
  47. G. Margolin and E. Barkai, Phys. Rev. Lett. 94, 080601 (2005).
  48. S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 89, 060601 (2002).
  49. S. Sabhapandit, S. N. Majumdar, and A. Comtet, Phys. Rev. E 73, 051102 (2006).
  50. M. Radice, M. Onofri, R. Artuso, and G. Pozzoli, Phys. Rev. E 101, 042103 (2020).
  51. T. Kay and L. Giuggioli, J. Phys. A: Math. Theor. 56, 345002 (2023).
  52. S. N. Majumdar and D. S. Dean, Phys. Rev. E 66, 041102 (2002).
  53. G. Bel and E. Barkai, Phys. Rev. Lett. 94, 240602 (2005).
  54. E. Barkai, J. Stat. Phys. 123, 883 (2006).
  55. G. Del Vecchio Del Vecchio and S. N. Majumdar, J. Stat. Mech. (2025) 023207.
  56. T. Sadhu, M. Delorme, and K. J. Wiese, Phys. Rev. Lett. 120, 040603 (2018).
  57. H. J. O. Boutcheng, T. B. Bouetou, T. W. Burkhardt, A. Rosso, A. Zoia, and K. T. Crepin, J. Stat. Mech. (2016) 053213.
  58. P. Singh and A. Kundu, J. Stat. Mech. (2019) 083205.
  59. P. C. Bressloff, Phys. Rev. E 102, 042135 (2020).
  60. S. Mukherjee and N. R. Smith, Phys. Rev. E 107, 064133 (2023).
  61. X. Fang, H. L. Gan, S. Holmes, H. Huang, E. Peköz, A. Röllin, and W. Tang, J. Appl. Probab. 58, 851 (2021).
  62. I. N. Burenev, S. N. Majumdar, and A. Rosso, Phys. Rev. E 109, 044150 (2024).
  63. X. Brokmann, J.-P. Hermier, G. Messin, P. Desbiolles, J.-P. Bouchaud, and M. Dahan, Phys. Rev. Lett. 90, 120601 (2003).
  64. F. D. Stefani, J. P. Hoogenboom, and E. Barkai, Phys. Today 62(2), 34 (2009).
  65. A. C. Barato, E. Roldán, I. A. Martínez, and S. Pigolotti, Phys. Rev. Lett. 121, 090601 (2018).
  66. V. G. Ramesh, K. J. H. Peters, and S. R. K. Rodriguez, Phys. Rev. Lett. 132, 133801 (2024).
  67. S. N. Majumdar, A. Pal, and G. Schehr, Phys. Rep. 840, 1 (2020).
  68. S. N. Majumdar and G. Schehr, Statistics of Extremes and Records in Random Sequences (Oxford University Press, Oxford, 2024).
  69. S. N. Majumdar, A. Rosso, and A. Zoia, Phys. Rev. Lett. 104, 020602 (2010).
  70. F. Mori, S. N. Majumdar, and G. Schehr, Europhys. Lett. 135, 30003 (2021).
  71. S. N. Majumdar, A. Rosso, and A. Zoia, J. Phys. A: Math. Theor. 43, 115001 (2010).
  72. P. Singh, Phys. Rev. E 105, 024113 (2022).
  73. M. Delorme and K. J. Wiese, Phys. Rev. E 94, 052105 (2016).
  74. P. Singh and A. Pal, Phys. Rev. E 103, 052119 (2021).
  75. F. Mori, S. N. Majumdar, and G. Schehr, Phys. Rev. E 106, 054110 (2022).
  76. J. Randon-Furling, S. N. Majumdar, and A. Comtet, Phys. Rev. Lett. 103, 140602 (2009).
  77. A. Reymbaut, S. N. Majumdar, and A. Rosso, J. Phys. A: Math. Theor. 44, 415001 (2011).
  78. E. Dumonteil, S. N. Majumdar, A. Rosso, and A. Zoia, Proc. Natl. Acad. Sci. USA 110, 4239 (2013).
  79. A. K. Hartmann, S. N. Majumdar, H. Schawe, and G. Schehr, J. Stat. Mech. (2020) 053401.
  80. S. N. Majumdar, F. Mori, H. Schawe, and G. Schehr, Phys. Rev. E 103, 022135 (2021).
  81. P. Singh, A. Kundu, S. N. Majumdar, and H. Schawe, J. Phys. A: Math. Theor. 55, 225001 (2022).
  82. C. Dale and R. Workman, Financ. Anal. J. 36, 71 (1980).
  83. R. Chicheportiche and J.-P. Bouchaud, First-Passage Phenomena and Their Applications, edited by R. Metzler, G. Oshanin, and S. Redner (World Scientific, Singapore, 2014), pp. 447–476.
  84. A. Clauset, M. Kogan, and S. Redner, Phys. Rev. E 91, 062815 (2015).
  85. J. D. Bao and Y. Jia, J. Stat. Phys. 123, 861 (2006).
  86. A. Comtet, F. Cornu, and G. Schehr, J. Stat. Phys. 181, 1565 (2020).
  87. B. H. Leung, IEEE Trans. Circuits Syst. I 51, 471 (2004).
  88. S. Robson, B. Leung, and G. Gong, IEEE Trans. Circuits Syst. II: Express Briefs 61, 937 (2014).
  89. J. D. Bao and Y. Jia, Phys. Rev. C 69, 027602 (2004).
  90. C.-O. Hwang and J. A. Given, Phys. Rev. E 74, 027701 (2006).
  91. U. Yu, Y.-M. Lee, and C.-O. Hwang, J. Sci. Comput. 88, 82 (2021).
  92. C. T. Barkar and M. J. Newby, Reliab. Eng. Syst. Saf. 94, 33 (2009).
  93. A. Nikeghbali and E. Platen, Finance Stoch. 17, 615 (2013).
  94. M. H. Bin Azami, N. C. Orger, V. H. Schulz, T. Oshiro, and M. Cho, Remote Sensing 14, 1874 (2022).
  95. V. Mazák, Mamm. Species  152 , 1 (1981).
  96. S. N. Majumdar and A. Comtet, Phys. Rev. Lett. 92, 225501 (2004).
  97. S. N. Majumdar and A. Comtet, J. Stat. Phys. 119, 777 (2005).
  98. A. Perret, A. Comtet, S. N. Majumdar, and G. Schehr, Phys. Rev. Lett. 111, 240601 (2013).
  99. A. K. Hartmann, Big Practical Guide to Computer Simulations (World Scientific, Singapore, 2015).
  100. B. Walter and K. J. Wiese, Phys. Rev. E 101, 043312 (2020).
  101. MPFR multi precision floating point library, https://www.mpfr.org/.
  102. S. N. Majumdar and A. J. Bray, Phys. Rev. E 65, 051112 (2002).
  103. W. Feller, An Introduction to Probability Theory and its Applications (Wiley, New York, 1957).
  104. A. Perret, A. Comtet, S. N. Majumdar, and G. Schehr, J. Stat. Phys. 161, 1112 (2015).

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Phys. Rev. E 111, 044134– Published 25 April, 2025
  • Received 3 December 2024
  • Revised 7 February 2025
  • Accepted 10 April 2025
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