Large deviations for local time fractional Brownian motion and applications

Journal of Mathematical Analysis and Applications

Volumn 346, Issue 2, 2008, Pages 432-445

  Meerschaert, Mark M ^a   Nane, Erkan ^a   Xiao, Yimin ^a  

^a MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Fractional Brownian motion; Large deviation; Law of the iterated logarithm; L vy process; Local time; Modulus of continuity; Self similarity; Strictly stable process

Indexed keywords

EID: 45049085578     PISSN: 0022247X     EISSN: 10960813     Source Type: Journal    
DOI: 10.1016/j.jmaa.2008.05.087     Document Type: Article

References (41)

1. Allouba H. Brownian-time processes: The PDE connection and the corresponding Feynman-Kac formula. Trans. Amer. Math. Soc. 354 (2002) 4627-4637
2. Allouba H., and Zheng W. Brownian-time processes: The PDE connection and the half-derivative generator. Ann. Probab. 29 (2001) 1780-1795
3. Baeumer B., and Meerschaert M.M. Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4 (2001) 481-500
4. B. Baeumer, M.M. Meerschaert, E. Nane, Brownian subordinators and fractional Cauchy problems, Trans. Amer. Math. Soc., in press
5. Bañuelos R., and DeBlassie R.D. The exit distribution for iterated Brownian motion in cones. Stochastic Process. Appl. 116 (2006) 36-69
6. Becker-Kern P., Meerschaert M.M., and Scheffler H.P. Limit theorems for coupled continuous time random walks. Ann. Probab. 32 (2004) 730-756
7. Bertoin J. Lévy Processes (1996), Cambridge University Press
8. Borodin A.N. On the character of convergence to Brownian local time I. Probab. Theory Related Fields 72 (1986) 231-250
9. Borodin A.N. On the character of convergence to Brownian local time II. Probab. Theory Related Fields 72 (1986) 251-277
10. Burdzy K. Some path properties of iterated Brownian motion. In: Çinlar E., Chung K.L., and Sharpe M.J. (Eds). Seminar on Stochastic Processes (1993), Birkhäuser, Boston 67-87
11. Burdzy K. Variation of iterated Brownian motion. In: Dawson D.A. (Ed). Workshops and Conference on Measure-valued Processes, Stochastic Partial Differential Equations and Interacting Particle Systems (1994), Amer. Math. Soc., Providence, RI 35-53
12. Burdzy K., and Khoshnevisan D. The level set of iterated Brownian motion. In: Azéma J., Emery M., Meyer P.-A., and Yor M. (Eds). Séminarie de probabilités XXIX. Lecture Notes in Math. vol. 1613 (1995), Springer, Berlin 231-236
13. Csáki E., and Csörgo{combining double acute accent} M. Inequalities for increments of stochastic processes and moduli of continuity. Ann. Probab. 20 (1992) 1031-1052
14. Csáki E., Csörgo{combining double acute accent} M., Földes A., and Révész P. On the occupation time of an iterated process having no local time. Stochastic Process. Appl. 70 (1997) 199-217
15. Csáki E., Csörgo{combining double acute accent} M., Földes A., and Révész P. Global Strassen-type theorems for iterated Brownian motions. Stochastic Process. Appl. 59 (1995) 321-341
16. Csáki E., Csörgo{combining double acute accent} M., Földes A., and Révész P. The local time of iterated Brownian motion. J. Theoret. Probab. 9 (1996) 717-743
17. Csáki E., Földes A., and Révész P. Strassen theorems for a class of iterated processes. Trans. Amer. Math. Soc. 349 (1997) 1153-1167
18. Davies L. Tail probabilities for positive random variables with entire characteristic functions of every regular growth. ZAMM Z. Angew. Math. Mech. 56 (1976) T334-T336
19. DeBlassie R.D. Iterated Brownian motion in an open set. Ann. Appl. Probab. 14 (2004) 1529-1558
20. Dembo A., and Zeitouni O. Large Deviation Techniques and Applications. second ed. (1998), Springer
21. Geman D., and Horowitz J. Occupation densities. Ann. Probab. 8 (1980) 1-67
22. Gradshteyn I.S., and Ryzhik I.M. Table of Integrals, Series and Products (2000), Academic Press, New York
23. Hawkes J. A lower Lipschitz condition for stable subordinator. Z. Wahrsch. Verw. Gebiete 17 (1971) 23-32
24. Hu Y. Hausdorff and packing measures of the level sets of iterated Brownian motion. J. Theoret. Probab. 12 (1999) 313-346
25. Hu Y., Pierre-Loti-Viaud D., and Shi Z. Laws of iterated logarithm for iterated Wiener processes. J. Theoret. Probab. 8 (1995) 303-319
26. Ikeda N., and Watanabe S. Stochastic Differential Equations and Diffusion Processes (1981), North-Holland, Amsterdam
27. Kasahara Y. Limit theorems for Lévy processes and Poisson point processes and their applications to Brownian excursions. J. Math. Kyoto Univ. 24 (1984) 521-538
28. Khoshnevisan D., and Lewis T.M. Chung's law of the iterated logarithm for iterated Brownian motion. Ann. Inst. H. Poincare Probab. Statist. 32 (1996) 349-359
29. König W., and Mörters P. Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002) 1605-1656
30. Lacey M. Large deviations for the maximum local time of stable Lévy processes. Ann. Probab. 18 (1990) 1669-1675
31. Meerschaert M.M., and Scheffler H.P. Limit theorems for continuous time random walks with infinite mean waiting times. J. Appl. Probab. 41 (2004) 623-638
32. Móricz F.A., Serfling R.J., and Stout W.F. Moment and probability bounds with quasisuperadditive structure for the maximum partial sum. Ann. Probab. 10 (1982) 1032-1040
33. Nane E. Iterated Brownian motion in parabola-shaped domains. Potential Anal. 24 (2006) 105-123
34. n. Stochastic Process. Appl. 116 (2006) 905-916
35. Nane E. Higher order PDE's and iterated processes. Trans. Amer. Math. Soc. 360 (2008) 2681-2692
36. n. Statist. Probab. Lett. 78 (2008) 90-95
37. n. ESAIM Probab. Stat. 11 (2007) 147-160
38. Papanicolaou G.C., Stroock D.W., and Varadhan S.R.S. Martingale approach to some limit theorems. Papers from the Duke Turbulence Conference. Duke Univ., Durham, NC, 1976. Duke Univ. Math. Series III vol. 6 (1977)
39. Samorodnitsky G., and Taqqu M.S. Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance (1994), Chapman and Hall, New York
40. Valiron G. Lectures on the General Theory of Integral Functions (1949), Chelsea Publishing Company, New York
41. Xiao Y. Local times and related properties of multi-dimensional iterated Brownian motion. J. Theoret. Probab. 11 (1998) 383-408