Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Brownian motion at a slow point


Authors: Martin T. Barlow and Edwin A. Perkins
Journal: Trans. Amer. Math. Soc. 296 (1986), 741-775
MSC: Primary 60J65; Secondary 60H05, 60J60
DOI: https://doi.org/10.1090/S0002-9947-1986-0846605-2
MathSciNet review: 846605
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $ c > 1$ there are points $ T(\omega)$ such that the piece of a Brownian path $ B,X(t) = B(T + t) - B(T)$, lies within the square root boundaries $ \pm c\sqrt t $. We study probabilistic and sample path properties of $ X$. In particular, we show that $ X$ is an inhomogeneous Markov process satisfying a certain stochastic differential equation, and we analyze the local behaviour of its local time at zero.


References [Enhancements On Off] (What's this?)

  • [1] M. T. Barlow, Study of a filtration expanded to include an honest time, Z. Wahrsch. Verw. Gebiete 44 (1978), no. 4, 307–323. MR 509204, https://doi.org/10.1007/BF01013194
  • [2] Martin T. Barlow and Edwin A. Perkins, Sample path properties of stochastic integrals, and stochastic differentiation, Stochastics Stochastics Rep. 27 (1989), no. 4, 261–293. MR 1011661, https://doi.org/10.1080/17442508908833579
  • [3] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0233396
  • [4] Burgess Davis, On Brownian slow points, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 359–367. MR 716492, https://doi.org/10.1007/BF00532967
  • [5] Burgess Davis and Edwin Perkins, Brownian slow points: the critical case, Ann. Probab. 13 (1985), no. 3, 779–803. MR 799422
  • [6] Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel, Hermann, Paris, 1975 (French). Chapitres I à IV; Édition entièrement refondue; Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. XV; Actualités Scientifiques et Industrielles, No. 1372. MR 0488194
  • [7] J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258
  • [8] Temps locaux, Astérisque, vol. 52, Société Mathématique de France, Paris, 1978 (French). Exposés du Séminaire J. Azéma-M. Yor; Held at the Université Pierre et Marie Curie, Paris, 1976–1977; With an English summary. MR 509476
  • [9] Peter Ney and Sidney Port (eds.), Advances in probability and related topics. Vol. 3, Marcel Dekker, Inc., New York, 1974. MR 0358885
  • [10] Priscilla Greenwood and Edwin Perkins, A conditioned limit theorem for random walk and Brownian local time on square root boundaries, Ann. Probab. 11 (1983), no. 2, 227–261. MR 690126
  • [11] P. Greenwood and E. Perkins, Limit theorems for excursions from a moving boundary, Teor. Veroyatnost. i Primenen. 29 (1984), no. 4, 703–714. MR 773439
  • [12] Donald L. Iglehart, Functional central limit theorems for random walks conditioned to stay positive, Ann. Probability 2 (1974), 608–619. MR 0362499
  • [13] Kiyosi Itô, Poisson point processes attached to Markov processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 225–239. MR 0402949
  • [14] Thierry Jeulin, Semi-martingales et grossissement d’une filtration, Lecture Notes in Mathematics, vol. 833, Springer, Berlin, 1980 (French). MR 604176
  • [15] Jean-Pierre Kahane, Sur l’irrégularité locale du mouvement brownien, C. R. Acad. Sci. Paris Sér. A 278 (1974), 331–333 (French). MR 0345187
  • [16] Harry Kesten, An iterated logarithm law for local time, Duke Math. J. 32 (1965), 447–456. MR 0178494
  • [17] Frank B. Knight, Essentials of Brownian motion and diffusion, Mathematical Surveys, vol. 18, American Mathematical Society, Providence, R.I., 1981. MR 613983
  • [18] P. A. Meyer, Un cours sur les intégrales stochastiques, Séminaire de Probabilités, X (Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg, année universitaire 1974/1975), Springer, Berlin, 1976, pp. 245–400. Lecture Notes in Math., Vol. 511 (French). MR 0501332
  • [19] Edwin Perkins, On the Hausdorff dimension of the Brownian slow points, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 369–399. MR 716493, https://doi.org/10.1007/BF00532968
  • [20] A. V. Skorokhod (1961), Stochastic equations for diffusion processes in a bounded region, Theory Probab. Appl. 6, 264-274.
  • [21] Kōhei Uchiyama, Brownian first exit from and sojourn over one-sided moving boundary and application, Z. Wahrsch. Verw. Gebiete 54 (1980), no. 1, 75–116. MR 595482, https://doi.org/10.1007/BF00535355
  • [22] David Williams, Diffusions, Markov processes, and martingales. Vol. 1, John Wiley & Sons, Ltd., Chichester, 1979. Foundations; Probability and Mathematical Statistics. MR 531031
  • [23] Toshio Yamada and Shinzo Watanabe, On the uniqueness of solutions of stochastic differential equations., J. Math. Kyoto Univ. 11 (1971), 155–167. MR 0278420, https://doi.org/10.1215/kjm/1250523691
  • [24] M. Yor (1978), Sur la continuité des temps locaux associés a certaines semi-martingales, Temps Locaux (J. Azema and M. Yor, eds.), Astérisque 52-53, pp. 23-35.
  • [25] Toshio Yamada and Yukio Ogura, On the strong comparison theorems for solutions of stochastic differential equations, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 1, 3–19. MR 612157, https://doi.org/10.1007/BF00531971
  • [26] H. Kunita (1982), Stochastic differential equations and stochastic flows of diffeomorphisms, Cours a l'école d'été de probabilités de Saint-Flour XII.
  • [27] Paul Lévy, Sur certains processus stochastiques homogènes, Compositio Math. 7 (1939), 283–339 (French). MR 0000919
  • [28] P. A. Meyer, R. T. Smythe, and J. B. Walsh, Birth and death of Markov processes, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 295–305. MR 0405600

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60J65, 60H05, 60J60

Retrieve articles in all journals with MSC: 60J65, 60H05, 60J60


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1986-0846605-2
Keywords: Brownian motion, slow point, stochastic differential equation, grossissement d'une filtration, local time, stochastic integral
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society