Large deviations for Riesz potentials of additive processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, p. 626-666
Nous étudions les fonctionelles de la forme ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|-σ ds1 ⋯ dsp, où X1(t), …, Xp(t) sont des processus stables symétriques indépendants et identiquement distribués d'ordre 0<β≤2. Nous obtenons des résultats sur les grandes déviations et les lois du logarithme itéré.
We study functionals of the form ζt=∫0t⋯∫0t|X1(s1)+⋯+Xp(sp)|-σ ds1 ⋯ dsp, where X1(t), …, Xp(t) are i.i.d. d-dimensional symmetric stable processes of index 0<β≤2. We obtain results about the large deviations and laws of the iterated logarithm for ζt.
@article{AIHPB_2009__45_3_626_0,
     author = {Bass, Richard and Chen, Xia and Rosen, Jay},
     title = {Large deviations for Riesz potentials of additive processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {3},
     year = {2009},
     pages = {626-666},
     doi = {10.1214/08-AIHP181},
     zbl = {1181.60035},
     mrnumber = {2548497},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_3_626_0}
}
Bass, Richard; Chen, Xia; Rosen, Jay. Large deviations for Riesz potentials of additive processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 3, pp. 626-666. doi : 10.1214/08-AIHP181. http://www.numdam.org/item/AIHPB_2009__45_3_626_0/

[1] X. Chen. Large deviations and laws of the iterated logarithm for the local time of additive stable processes. Ann. Probab. 35 (2007) 602-648. | MR 2308590 | Zbl 1121.60025

[2] X. Chen, W. Li and J. Rosen. Large deviations for local times of stable processes and random walks in 1 dimension. Electron. J. Probab. 10 (2005) 577-608. | MR 2147318 | Zbl 1109.60016

[3] R. C. Dalang and J. B. Walsh. Geography of the level set of the Brownian sheet. Probab. Theory Related Fields 96 (1993) 153-176. | MR 1227030 | Zbl 0792.60038

[4] R. C. Dalang and J. B. Walsh. The structure of a Brownian bubble. Probab. Theory Related Fields 96 (1993) 475-501. | MR 1234620 | Zbl 0794.60047

[5] W. Donoghue. Distributions and Fourier Transforms. Academic Press, New York, 1969. | Zbl 0188.18102

[6] M. Donsker and S. R. S. Varadhan. Asymtotics for the polaron. Comm. Pure Appl. Math. 36 (1983) 505-528. | MR 709647 | Zbl 0538.60081

[7] P.-J. Fitzsimmons and T. S. Salisbury. Capacity and energy for multiparameter stable processes. Ann. Inst. H. Poincaré Probab. Statist. 25 (1989) 325-350. | Numdam | MR 1023955 | Zbl 0689.60071

[8] F. Hirsch and S. Song. Symmetric Skorohod topology on n-variable functions and hierarchical Markov properties of n-parameter processes. Probab. Theory Related Fields 103 (1995) 25-43. | MR 1347169 | Zbl 0833.60074

[9] J.-P. Kahane. Some Random Series of Functions. Health and Raytheon Education Co., Lexington, MA, 1968. | MR 254888 | Zbl 0192.53801

[10] W. S. Kendall. Contours of Brownian processes with several-dimensional times. Z. Wahrsch. Verw. Gebiete 52 (1980) 267-276. | MR 576887 | Zbl 0431.60056

[11] D. Khoshnevisan. Brownian sheet images and Bessel-Riesz capacity. Trans. Amer. Math. Soc. 351 (1999) 2607-2622. | MR 1638246 | Zbl 0930.60055

[12] D. Khoshnevisan and Z. Shi. Brownian sheet and capacity. Ann. Probab. 27 (1999) 1135-1159. | MR 1733143 | Zbl 0962.60066

[13] D. Khoshnevisan and Y. Xiao. Level sets of additive Lévy processes. Ann. Probab. 30 (2002) 62-100. | MR 1894101 | Zbl 1019.60049

[14] V. Kolokoltsov. Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. 80 (2000) 725-768. | MR 1744782 | Zbl 1021.60011

[15] W. König and P. Mörters. Brownian intersection local times: Upper tail asymptotics and thick points. Ann. Probab. 30 (2002) 1605-1656. | MR 1944002 | Zbl 1032.60073

[16] J.-F. Le Gall, J. Rosen and N.-R. Shieh. Multiple points of Lévy processes. Ann. Probab. 17 (1989) 503-515. | MR 985375 | Zbl 0684.60057

[17] M. Marcus and J. Rosen. Markov Processes, Gaussian Processes, and Local Times. Cambridge Studies in Advanced Mathematics 100. Cambridge Univ. Press, NY, 2006. | MR 2250510 | Zbl 1129.60002