Electronic Journal of Probability

The argmin process of random walks, Brownian motion and Lévy processes

Jim Pitman and Wenpin Tang

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In this paper we investigate the argmin process of Brownian motion $B$ defined by $\alpha _t:=\sup \left \{s \in [0,1]: B_{t+s}=\inf _{u \in [0,1]}B_{t+u} \right \}$ for $t \geq 0$. The argmin process $\alpha $ is stationary, with invariant measure which is arcsine distributed. We prove that $(\alpha _t; t \geq 0)$ is a Markov process with the Feller property, and provide its transition kernel $Q_t(x,\cdot )$ for $t>0$ and $x \in [0,1]$. Similar results for the argmin process of random walks and Lévy processes are derived. We also consider Brownian extrema of a given length. We prove that these extrema form a delayed renewal process with an explicit path construction. We also give a path decomposition for Brownian motion at these extrema.

Article information

Electron. J. Probab., Volume 23 (2018), paper no. 60, 35 pp.

Received: 22 August 2017
Accepted: 6 June 2018
First available in Project Euclid: 20 June 2018

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Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60G51: Processes with independent increments; Lévy processes 60J65: Brownian motion [See also 58J65]

arcsine law argmin process Brownian extrema Feller semigroup Brownian excursion theory jump process Lévy process Lévy system Markov property space-time shift process path decomposition random walks renewal property sample path property stable process stationary process

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Pitman, Jim; Tang, Wenpin. The argmin process of random walks, Brownian motion and Lévy processes. Electron. J. Probab. 23 (2018), paper no. 60, 35 pp. doi:10.1214/18-EJP185. https://projecteuclid.org/euclid.ejp/1529460158

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  • [1] Joshua Abramson and Steven N. Evans. Lipschitz minorants of Brownian motion and Lévy processes. Probab. Theory Related Fields, 158(3–4):809–857, 2014.
  • [2] Glen Baxter. Combinatorial methods in fluctuation theory. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 1:263–270, 1962/1963.
  • [3] Vladimir Belitsky and Pablo A. Ferrari. Ballistic annihilation and deterministic surface growth. J. Statist. Phys., 80(3–4):517–543, 1995.
  • [4] Albert Benveniste and Jean Jacod. Systèmes de Lévy des processus de Markov. Invent. Math., 21:183–198, 1973.
  • [5] Jean Bertoin. Lévy processes. Cambridge: Cambridge Univ. Press, 1996.
  • [6] Jean Bertoin and Jim Pitman. Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci. Math., 118(2):147–166, 1994.
  • [7] Philippe Biane and Marc Yor. Quelques précisions sur le méandre brownien. Bull. Sci. Math. (2), 112(1):101–109, 1988.
  • [8] Erwin Bolthausen. On a functional central limit theorem for random walks conditioned to stay positive. Ann. Probability, 4(3):480–485, 1976.
  • [9] P. J. Brockwell, S. I. Resnick, and R. L. Tweedie. Storage processes with general release rule and additive inputs. Adv. in Appl. Probab., 14(2):392–433, 1982.
  • [10] L. Chaumont and R. A. Doney. Invariance principles for local times at the maximum of random walks and Lévy processes. Ann. Probab., 38(4):1368–1389, 2010.
  • [11] E. Çinlar and M. Pinsky. A stochastic integral in storage theory. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 17:227–240, 1971.
  • [12] E. Çinlar and M. Pinsky. On dams with additive inputs and a general release rule. J. Appl. Probability, 9:422–429, 1972.
  • [13] Erhan Çinlar. A local time for a storage process. Ann. Probability, 3(6):930–950, 1975.
  • [14] D. J. Daley and D. Vere-Jones. An introduction to the theory of point processes. Vol. I. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2003. Elementary theory and methods.
  • [15] M. H. A. Davis. Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. Roy. Statist. Soc. Ser. B, 46(3):353–388, 1984. With discussion.
  • [16] I. V. Denisov. A random walk and a wiener process near a maximum. Theory of Probability & Its Applications, 28(4):821–824, 1984.
  • [17] Richard T. Durrett, Donald L. Iglehart, and Douglas R. Miller. Weak convergence to Brownian meander and Brownian excursion. Ann. Probability, 5(1):117–129, 1977.
  • [18] E. B. Dynkin. Markov processes. Vols. I, II, volume 122 of Translated with the authorization and assistance of the author by J. Fabius, V. Greenberg, A. Maitra, G. Majone. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. Academic Press Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1965.
  • [19] Steven N. Evans and Jim Pitman. Stationary Markov processes related to stable Ornstein-Uhlenbeck processes and the additive coalescent. Stochastic Process. Appl., 77(2):175–185, 1998.
  • [20] Alessandra Faggionato. The alternating marked point process of $h$-slopes of drifted Brownian motion. Stochastic Process. Appl., 119(6):1765–1791, 2009.
  • [21] William Feller. An introduction to probability theory and its applications. Vol. I. Third edition. John Wiley & Sons Inc., New York, 1968.
  • [22] William Feller. An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney, 1971.
  • [23] Priscilla Greenwood and Jim Pitman. Fluctuation identities for Lévy processes and splitting at the maximum. Adv. in Appl. Probab., 12(4):893–902, 1980.
  • [24] Piet Groeneboom. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields, 81(1):79–109, 1989.
  • [25] J. Hoffmann-Jørgensen. Markov sets. Math. Scand., 24:145–166 (1970), 1969.
  • [26] Kyosi Itô. Poisson point processes attached to markov processes. In Proc. 6th Berk. Symp. Math. Stat. Prob, volume 3, pages 225–240, 1971.
  • [27] J. Jacod and A. V. Skorokhod. Jumping Markov processes. Ann. Inst. H. Poincaré Probab. Statist., 32(1):11–67, 1996.
  • [28] Olav Kallenberg. Foundations of modern probability. Probability and its Applications (New York). Springer-Verlag, New York, second edition, 2002.
  • [29] Ioannis Karatzas and Steven E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
  • [30] J. F. C. Kingman. Homecomings of Markov processes. Advances in Appl. Probability, 5:66–102, 1973.
  • [31] N. V. Krylov and A. A. Juškevič. Markov random sets. Trudy Moskov. Mat. Obšč., 13:114–135, 1965.
  • [32] Christophe Leuridan. Un processus ponctuel associé aux maxima locaux du mouvement brownien. Probab. Theory Related Fields, 148(3–4):457–477, 2010.
  • [33] Paul Lévy. Processus Stochastiques et Mouvement Brownien. Suivi d’une note de M. Loève. Gauthier-Villars, Paris, 1948.
  • [34] Bernard Maisonneuve. Exit systems. Ann. Probability, 3(3):399–411, 1975.
  • [35] P. A. Meyer. Une mise au point sur les systèmes de Lévy. Remarques sur l’exposé de A. Benveniste. In Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972), pages 25–32. Lecture Notes in Math., Vol. 321. Springer, Berlin, 1973.
  • [36] P. W. Millar. A path decomposition for Markov processes. Ann. Probability, 6(2):345–348, 1978.
  • [37] Ditlev Monrad and Martin L. Silverstein. Stable processes: sample function growth at a local minimum. Z. Wahrsch. Verw. Gebiete, 49(2):177–210, 1979.
  • [38] J. Neveu and J. Pitman. Renewal property of the extrema and tree property of the excursion of a one-dimensional Brownian motion. In Séminaire de Probabilités, XXIII, volume 1372 of Lecture Notes in Math., pages 239–247. Springer, Berlin, 1989.
  • [39] J. W. Pitman. Lévy systems and path decompositions. In Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981), volume 1 of Progr. Prob. Statist., pages 79–110. Birkhäuser, Boston, Mass., 1981.
  • [40] Jim Pitman. The distribution of local times of a Brownian bridge. In Séminaire de Probabilités, XXXIII, volume 1709 of Lecture Notes in Math., pages 388–394. Springer, Berlin, 1999.
  • [41] Jim Pitman and Wenpin Tang. Patterns in random walks and Brownian motion. In Catherine Donati-Martin, Antoine Lejay, and Alain Rouault, editors, In Memoriam Marc Yor - Séminaire de Probabilités XLVII, volume 2137 of Lecture Notes in Mathematics, pages 49–88. Springer International Publishing, 2015.
  • [42] Jim Pitman and Marc Yor. The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. Ann. Probab., 25(2):855–900, 1997.
  • [43] Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, third edition, 1999.
  • [44] L. C. G. Rogers and J. W. Pitman. Markov functions. Ann. Probab., 9(4):573–582, 1981.
  • [45] Ken-iti Sato. Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author.
  • [46] Michael Sharpe. General theory of Markov processes, volume 133 of Pure and Applied Mathematics. Academic Press, Inc., Boston, MA, 1988.
  • [47] Erik Sparre Andersen. On sums of symmetrically dependent random variables. Skand. Aktuarietidskr., 36:123–138, 1953.
  • [48] Frank Spitzer. A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc., 82:323–339, 1956.
  • [49] Hermann Thorisson. On time- and cycle-stationarity. Stochastic Process. Appl., 55(2):183–209, 1995.
  • [50] Boris Tsirelson. Brownian local minima, random dense countable sets and random equivalence classes. Electron. J. Probab., 11:no. 7, 162–198 (electronic), 2006.
  • [51] Gerónimo Uribe Bravo. Bridges of Lévy processes conditioned to stay positive. Bernoulli, 20(1):190–206, 2014.
  • [52] Shinzo Watanabe. On discontinuous additive functionals and Lévy measures of a Markov process. Japan. J. Math., 34:53–70, 1964.
  • [53] V. M. Zolotarev. One-dimensional stable distributions, volume 65 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1986. Translated from the Russian by H. H. McFaden, Translation edited by Ben Silver.