Upper and lower limits of doubly perturbed brownian motion

L. Chaumont; R. A. Doney; Y. Hu

Annales de l'I.H.P. Probabilités et statistiques (2000)

  • Volume: 36, Issue: 2, page 219-249
  • ISSN: 0246-0203

How to cite


Chaumont, L., Doney, R. A., and Hu, Y.. "Upper and lower limits of doubly perturbed brownian motion." Annales de l'I.H.P. Probabilités et statistiques 36.2 (2000): 219-249. <http://eudml.org/doc/77657>.

author = {Chaumont, L., Doney, R. A., Hu, Y.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Brownian motion; law of iterated logarithm},
language = {eng},
number = {2},
pages = {219-249},
publisher = {Gauthier-Villars},
title = {Upper and lower limits of doubly perturbed brownian motion},
url = {http://eudml.org/doc/77657},
volume = {36},
year = {2000},

AU - Chaumont, L.
AU - Doney, R. A.
AU - Hu, Y.
TI - Upper and lower limits of doubly perturbed brownian motion
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2000
PB - Gauthier-Villars
VL - 36
IS - 2
SP - 219
EP - 249
LA - eng
KW - Brownian motion; law of iterated logarithm
UR - http://eudml.org/doc/77657
ER -


  1. [1] Bingham N.N., Goldie C.M., Teugels J.L., Regular Variation, Cambridge University Press, 1987. Zbl0617.26001MR898871
  2. [2] Carmona Ph., Petit P., Yor M., Some extensions of the arc sine law as partial consequences of the scaling property for Brownian motion, Probab. Theory Related Fields100 (1994) 1-29. Zbl0808.60066MR1292188
  3. [3] Carmona Ph., Petit P., Yor M., Beta variables as times spent in [0, oo[ by certain perturbed Brownian motions, J. London Math. Soc.58 (1998) 239-256. Zbl0924.60067MR1670130
  4. [4] Chaumont L., Doney R.A., Pathwise uniqueness for perturbed versions of Brownian motion and reflected Brownian motion, Probab. Theory Related Fields113 (1999) 519-534. Zbl0945.60082MR1717529
  5. [5] Chaumont L., Doney R.A., Some calculations for doubly perturbed Brownian motion, Stoch. Proc. Appl. (1999), to appear. Zbl0997.60095MR1730618
  6. [6] Csáki E., On the lower limits of maxima and minima of Wiener process and partial sums, Z. Wahrsch. Verw. Gebiete.43 (1978) 205-221. Zbl0372.60113MR494527
  7. [7] Csáki E., An integral test for the supremum of Wiener local time, Probab. Theory Related Fields83 (1989) 207-217. Zbl0677.60087MR1012499
  8. [8] Csörgö M., Révész P., Strong Approximations in Probability and Statistics, Akadémiai Kiadó, Budapest and Academic Press, New York, 1981. Zbl0539.60029MR666546
  9. . [9] Davis B., Weak limits of perturbed Brownian motion and the equation Yt = Bt + σ sup{Ys: s ≤ t} + β inf{Ys: s ≤ t}, Ann. Probab.24 (1996) 2007-2017. Zbl0870.60076MR1415238
  10. [10] Davis B., Brownian motion and random walk perturbed at extrema, Probab. Theory Related Fields113 (1999) 501-518. Zbl0930.60041MR1717528
  11. [11] Doney R.A., Some calculations for perturbed Brownian motion, in: Azéma J., Émery M., Ledoux M., Yor M. (Eds.), Sém. Probab. XXXII, Lecture Notes Math., Vol. 1686, Springer, Berlin, 1998, pp. 231-236. Zbl0911.60067MR1655296
  12. [12] Doney R.A., Warren J., Yor M., Perturbed Bessel processes, in: Azéma J., Émery M., Ledoux M., Yor M. (Eds.), Sém. Probab. XXXII, Lecture Notes Math., Vol. 1686, Springer, Berlin, 1998, pp. 237-249. Zbl0924.60039MR1655297
  13. [13] Getoor R.K., The Brownian escape process, Ann. Probab.7 (1979) 864-867. Zbl0416.60086MR542136
  14. [14] Jeulin Th., Ray-Knight's theorem on Brownian local times and Tanaka's formula, in: Çinlar E., Chung K.L., Getoor R.K. (Eds.), Sem. Stoch. Proc., Birkhauser, Boston, 1984, pp. 131-142. Zbl0561.60077MR902415
  15. [15] Kochen S.B., Stone C.J., A note on the Borel—Cantelli lemma, Illinois J. Math.8 (1964) 248-251. Zbl0139.35401MR161355
  16. [16] Le Gall J.F., L'équation stochastique Yt = Bt + αMYt + βIYt comme limite des équations de Norris-Rogers-Williams, 1986, unpublished notes. 
  17. [17] Le Gall J.F., Yor M., Enlacement du mouvement brownien autour des courbes de l'espace, Trans. Amer. Math. Soc.317 (1990) 687-722. Zbl0696.60072MR946219
  18. [18] McGill P., Markov properties of diffusion local times: a martingale approach, Adv. Appl. Probab.14 (1982) 789-810. Zbl0502.60062MR677557
  19. [19] Norris J.R., Rogers L.C.G., Williams D., Self-avoiding random walk: a Brownian motion model with local time drift, Probab. Theory Related Fields74 (1987) 271- 287. Zbl0611.60052MR871255
  20. [20] Perman M., Werner W., Perturbed Brownian motions, Probab. Theory Related Fields108 (1997) 357-383. Zbl0884.60082MR1465164
  21. [21] Petit F., Sur le temps passé par le mouvement brownien au-dessus d'un multiple de son supremum et quelques extensions de la loi de l'arc sinus, Part of a Thèse de Doctorat, Université Paris VII, 1992. 
  22. [22] Revuz D., Yor M., Continuous Martingales and Brownian Motion, 2nd edn., Springer, Berlin, 1994. Zbl0804.60001MR1303781
  23. [23] Révész P., Random Walk in Random and Non-Random Environment, World Scientific Press, Singapore, London, 1990. Zbl0733.60091MR1082348
  24. [24] Shi Z., Werner W., Asymptotics for occupations times of half-lines by stable processes and perturbed reflecting Brownian motion, Stochastics55 (1995) 71-85. Zbl0886.60030MR1382286
  25. [25] Tóth B., The "true" self-avoiding walk with bond repulsion in Z: limit theorems, Ann. Probab.23 (1995) 1523-1556. Zbl0852.60083MR1379158
  26. [26] Tóth B., "True" self-avoiding walk with generalized bond repulsion in Z, J. Stat. Phys.77 (1994) 17-33. Zbl0838.60065MR1300526
  27. [27] Tucker H.G., On a necessary and sufficient condition that an infinitely divisible distribution be absolutely continuous, Amer. Math. Soc. Trans.118 (1965) 316- 330. Zbl0168.39102MR182061
  28. [28] Werner W., Some remarks on perturbed reflecting Brownian motion, in: Azéma J., Émery M., Meyer P.A., Yor M. (Eds.), Sém. Probab. XXIX, Lecture Notes Math., Vol. 1613, Springer, Berlin, 1995, pp. 37-43. Zbl0835.60072MR1459447
  29. [29] Yor M., Some Aspects of Brownian Motion, Part I: Some Special Functionals, Lecture Notes, ETH Zürich, Birkhäuser, Basel, 1992. Zbl0779.60070MR1193919
  30. [30] Yor M., Local Times and Excursions for Brownian Motion: A Concise Introduction, Lecciones en Metemáticas, Número I, Universidao Central de Venezuela, Caracas, 1995. 

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