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On the collision local time of fractional Brownian motions. (English) Zbl 1124.60036

Summary: The existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through \(L^{2}\) convergence and chaos expansion. Furthermore, the regularity of the collision local time process is studied.

MSC:

60G15 Gaussian processes
60G18 Self-similar stochastic processes
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[1] Rosen, J., The intersection local time of fractional Brownian motion in the plane, J. Multivariate Anal., 23(1), 1987, 37–46 · Zbl 0633.60057 · doi:10.1016/0047-259X(87)90176-X
[2] Nualart, D. and Vives, J., Chaos expansion and local time, Publ. Mat., 36(2), 1992, 827–836 · Zbl 0787.60060
[3] Imkeller, P., Abreu, V. and Vives, J., Chaos expansions of double intersection local time of Brownian motion in Rd and renormalization, Stoch Process. Appl., 56, 1995, 1–34 · Zbl 0822.60048 · doi:10.1016/0304-4149(94)00041-Q
[4] Hu, Y. Z., Self-intersection of fractional Brownian motions – via chaos expansion, J. Math Kyoto Univ., 41(2), 2001, 233–250 · Zbl 1008.60091
[5] Hu, Y. Z. and Nualart, D., Renormalized Self-intersection local time for fractional Brownian motion, Ann. Prob., 33(3), 2005, 948–983 · Zbl 1093.60017 · doi:10.1214/009117905000000017
[6] Xiao, Y. M. and Zhang, T. S., Local time of fractional Brownian sheets, Probab. Theory Relat. Fields, 124, 2002, 121–139 · Zbl 1009.60024 · doi:10.1007/s004400200210
[7] Simon, B., The P(_)2 Euclidean Field Theory, Princeton University Press, Princeton, New Jersey, 1974 · Zbl 1175.81146
[8] Meyer, P. A., Quantum for Probabilists, Lecture Notes in Mathematics, 1538, Springer, Heidelberg, 1993 · Zbl 0773.60098
[9] Watanabe, S., Stochastic Di_eretial Equation and Malliavin Calculus, Tata Institute of Fundamental Research, Springer, New York, 1984
[10] Berman, S. M., Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J., 23, 1973, 69–94 · Zbl 0264.60024 · doi:10.1512/iumj.1973.23.23006
[11] Xiao, Y. M., Properties of local nondeterminism of Gaussian and stable random fields and their applications, Ann. Fac. Sci. Toulouse Math., XV, 2006, 157–193 · Zbl 1128.60041
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