Uniform oscillations of the local time of iterated Brownian motion

Bernoulli

Volumn 5, Issue 1, 1999, Pages 49-65

  Eisenbaum, Nathalie ^a   Shi, Z H A N ^a  

^a University of Paris VII LPSM partially supported by ERC Starting Grant 680275 MALIG   (France)

Author keywords

Iterated brownian motion; Local time; Modulus of continuity; Modulus of nondifferentiability

Indexed keywords

EID: 33645672071     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.2307/3318612     Document Type: Article

References (25)

1. Bertoin, J. and Shi, Z. (1996) Hirsch's integral test for the iterated Brownian motion. In J. Azéma, M. Emery and M. Yor (eds), Séminaire de Probabilités XXX, pp. 361-368. Lecture Notes Math., 1626. Berlin: Springer-Verlag.
2. Biane, P. and Yor, M. (1987) Valeurs principales associées aux temps locaux browniens. Bull. Sei. Math., III, 23-101.
3. Burdzy, K. (1993) Some path properties of iterated Brownian motion. In E. Çinlar, K.L. Chung and M. Sharpe (eds), Seminar on Stochastic Processes '92, pp. 67-87. Boston, MA: Birkhäuser.
4. Burdzy, K. and Khoshnevisan, D. (1995) The level sets of iterated Brownian motion. In J. Azéma, M. Emery, P.-A. Meyer and M. Yor (eds), Séminaire de Probabilités XXIX, pp. 231-236. Lecture Notes Math., 1613. Berlin: Springer-Verlag.
5. Ciesielski, Z. and Taylor, S.J. (1962) First passage times and sojourn times for Brownian motion in space and the exact Hausdorff measure of the sample path. Trans. Amer. Math. Soc., 103, 434-450.
6. Csâki, E., Csörgö, M., Feldes, A. and Révész, P. (1995) Global Strassen type theorems for iterated Brownian motion. Stochastic Pmcesses Applic., 59, 321-341.
7. Csâki, E., Csörgö, M., Földes, A. and Révész, P. (1996) The local time of iterated Brownian motion. J Theor. Probab., 9, 717-743.
8. Csâki, E., Csörgö, M., Földes, A. and Révész, P. (1997a) On thé occupation time of an iterated process having no local time. Stochastic Processes Applic., 70, 199-217.
9. Csâki, E. and Földes, A. (1986) How small are the increments of the local time of a Wiener process? Ann. Probab., 14, 533-546.
10. Csâki, E. and Földes, A. (1997) On the logarithmic average of iterated processes. Statist. Probab. Lett., 33, 347-358.
11. Csâki, E., Földes, A. and Révész, P. (1997b) Strassen theorems for a class of iterated processes. Trans. Amer. Math. Soc., 349, 1153-1167.
12. Csörgö, M. and Révész, P. (1979) How small are the increments of a Wiener process? Stochastic Processes Applic., 8, 119-129.
13. Eisenbaum, N. (1990) Un théorème de Ray-Knight lié au supremum des temps locaux browniens. Probab. Theory Related Fields, 87, 27-34.
14. Getoor, R.K. (1979) The Brownian escape process. Ann. Probab., 5, 864-867.
15. Hu, Y. (1996) Sur le mouvement brownien: calcul de lois; études asymptotiques; filtrations; relations avec certaines équations paraboliques. Thèse de Doctorat, Université Paris VI.
16. Kesten, H. (1965) An iterated logarithm law for local time. Duke Math. J., 32, 447-456.
17. Khoshnevisan, D. and Lewis, T.M. (1997) Stochastic calculus for Brownian motion on a Brownian fracture. Ann. Appl. Probab. To appear.
18. Khoshnevisan, D., Lewis, T.M. and Li, W.V. (1994) On the future infima of some transient processes. Probab. Theory Related Fields, 99, 337-360.
19. Levy, P. (1939) Theorie de l'Addition des Variables Aléatoires. Paris: Gauthier-Villars.
20. Révész, P. (1990) Random Walk in Random -and Non-Random Environments. Singapore: World Scientific.
21. Revuz, D. and Yor, M. (1994) Continuous Martingales and Brownian Motion, 2nd edn. Berlin: Springer-Verlag.
22. Rogers, L.C.G. and Williams, D. (1987) Diffusions, Markov Processes, and Martingales, Vol. II, Itô Calculus. Chichester, West Sussex: Wiley.
23. Shi, Z. and Yor, M. (1997) Integrability and lower limits of the local time of iterated Brownian motion. Stadia Sei. Math. Hung., 33, 279-298.
24. Xiao, Y. (1997) Local times and related properties of multi-dimensional iterated Brownian motion. J. Theor. Probab., 11, 383-408.
25. Yor, M. (1992) Some Aspects of Brownian Motion, Part I, Some Special Fwictionals. Lectures in Math. Eidgenössiche Technische Hochschule Zürich. Basel: Birkhäuser.