Last zero time or maximum time of the winding number of Brownian motions

Izumi Okada (Tokyo Institute of Technology)

Abstract


In this paper we consider the winding number, $\theta(s)$, of planar Brownian motion and study asymptotic behavior of the process of the maximum time, the time when $\theta(s)$ attains the maximum in the interval $0\le s \le t$. We find the limit law of its logarithm with a suitable normalization factor and the upper growth rate of the maximum time process itself. We also show that the process of the last zero time of $\theta(s)$ in $[0,t]$ has the same law as the maximum time process.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-8

Publication Date: September 18, 2014

DOI: 10.1214/ECP.v19-3485

References

  • Bertoin, Jean; Werner, Wendelin. Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. Seminaire de Probabilites, XXVIII, 138--152, Lecture Notes in Math., 1583, Springer, Berlin, 1994. MR1329109
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8 MR2722836
  • Lamperti, John. Wiener's test and Markov chains. J. Math. Anal. Appl. 6 1963 58--66. MR0143258
  • Lawler, Gregory F. Hausdorff dimension of cut points for Brownian motion. Electron. J. Probab. 1 (1996), no. 2, approx. 20 pp. (electronic). MR1386294
  • Morters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8 MR2604525
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1991. x+533 pp. ISBN: 3-540-52167-4 MR1083357
  • Shi, Z. Liminf behaviours of the windings and L�vy's stochastic areas of planar Brownian motion. Seminaire de Probabilites, XXVIII, 122--137, Lecture Notes in Math., 1583, Springer, Berlin, 1994. MR1329108
  • Shi, Zhan. Windings of Brownian motion and random walks in the plane. Ann. Probab. 26 (1998), no. 1, 112--131. MR1617043
  • Spitzer, Frank. Some theorems concerning $2$-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 1958 187--197. MR0104296
  • Spitzer, Frank. Principles of random walk. The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp. MR0171290
  • Vakeroudis, S. On Hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity. Teor. Veroyatn. Primen. 56 (2011), no. 3, 566--591; translation in Theory Probab. Appl. 56 (2012), no. 3, 485--507 MR3136465
  • Williams,D.: A Simple Geometric Proof of Spitzer's Winding Number Formula for 2-dimensional Brownian Motion. preprint, (1974), phUniversity College, Swansea.
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.