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On the collision local time of sub-fractional Brownian motions. (English) Zbl 1185.60040

Summary: Let S H i ={S t H i ,t0},i=1,2, be two independent sub-fractional Brownian motions with respective indices H i (0,1). We consider the so-called collision local time

T = 0 T δ(S t H 1 -S t H 2 )dt,T>0,

where δ denotes the Dirac delta function. By an elementary method we show that T is smooth in the sense of Meyer and Watanabe if and only if min{H 1 ,H 2 }<1/3·

MSC:
60G22Fractional processes, including fractional Brownian motion
60G15Gaussian processes
60G18Self-similar processes
60F25L p -limit theorems (probability)
References:
[1]Berman, S. M.: Local nondeterminism and local times of Gaussian processes, Indiana univ. Math. J. 23, 69-94 (1973) · Zbl 0264.60024 · doi:10.1512/iumj.1973.23.23006
[2]Berman, S. M.: Self-intersections and local nondeterminism of Gaussian processes, Ann. probab. 19, 160-191 (1991) · Zbl 0728.60037 · doi:10.1214/aop/1176990539
[3]Bingham, N. H.; Goldie, C. M.; Teugels, J. L.: Regular variation, (1987)
[4]Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A.: Sub-fractional Brownian motion and its relation to occupation times, Statist. probab. Lett. 69, 405-419 (2004) · Zbl 1076.60027 · doi:10.1016/j.spl.2004.06.035
[5]Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A.: Limit theorems for occupation time fluctuations of branching systems (I): long-range dependence, Stochastic process. Appl. 116, 1-18 (2006) · Zbl 1082.60024 · doi:10.1016/j.spa.2005.07.002
[6]Bojdecki, T.; Gorostiza, L. G.; Talarczyk, A.: Some extension of fractional Brownian motion and sub-fractional Brownian motion related to particle systems, Electron. comm. Probab. 12, 161-172 (2007) · Zbl 1128.60025 · doi:http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=1880&layout=abstract
[7]Cuzick, J.; Dupreez, J.: Joint continuity of Gaussian local time, Ann. probab. 10, 810-817 (1982) · Zbl 0492.60032 · doi:10.1214/aop/1176993789
[8]Geman, D.; Horowitz, J.; Rosen, J.: A local time analysis of intersections of Brownian paths in the plane, Ann. probab. 12, 86-107 (1984) · Zbl 0536.60046 · doi:10.1214/aop/1176993375
[9]Hu, Y.: Self-intersection local time of fractional Brownian motions–via chaos expansion, J. math. Kyoto univ. 41, 233-250 (2001) · Zbl 1008.60091
[10]Hu, Y.; Nualart, D.: Renormalized self-intersection local time for fractional Brownian motions, Ann. probab. 33, 948-983 (2005) · Zbl 1093.60017 · doi:10.1214/009117905000000017
[11]Imkeller, P.; Perez-Abreu, V.; Vives, J.: Chaos expansion of double intersection local time of Brownian motion in rd and renormalization, Stochastic process. Appl. 56, 1-34 (1995) · Zbl 0822.60048 · doi:10.1016/0304-4149(94)00041-Q
[12]Jiang, Y.; Wang, Y.: On the collision local time of fractional Brownian motion, Chin. ann. Math. 28, 311-320 (2007) · Zbl 1124.60036 · doi:10.1007/s11401-006-0029-3
[13]Nualart, D.: Malliavin calculus and related topics, (2006)
[14]Pitman, E. J. G.: On the behavior of the characteristic function of a probability distribution in the neighbourhood of the origin, J. aust. Math. soc. A 8, 422-443 (1968) · Zbl 0164.48502 · doi:10.1017/S1446788700006121
[15]Rosen, J.: The intersection local time of fractional Brownian motion in the plane, J. multivariate anal. 23, 37-46 (1987) · Zbl 0633.60057 · doi:10.1016/0047-259X(87)90176-X
[16]Shen, G., Chen, C., Yan, L., 2009. Remarks on sub-fractional Bessel processes, preprint
[17]Tudor, C. A.; Xiao, Y.: Some path properties of bi-fractional Brownian motion, Bernoulli 13, 1023-1052 (2007) · Zbl 1132.60034 · doi:10.3150/07-BEJ6110 · doi:euclid:bj/1194625601
[18]Tudor, C.: Some properties of the sub-fractional Brownian motion, Stochastics 79, 431-448 (2007) · Zbl 1124.60038 · doi:10.1080/17442500601100331
[19]Tudor, C.: Inner product spaces of integrands associated to sub-fractional Brownian motion, Statist. probab. Lett. 78, 2201-2209 (2008)
[20]Tudor, C.: On the Wiener integral with respect to a sub-fractional Brownian motion on an interval, J. math. Anal. appl. 351, 456-468 (2009) · Zbl 1154.60041 · doi:10.1016/j.jmaa.2008.10.041
[21]Varadhan, S. R. S.: Appendix to Euclidean quantum field theory, by K. Symanzik, Local quantum theory (1968)
[22]Watanabe, S.: Stochastic differential equation and Malliavin calculus, (1984)
[23]Wolpert, R.: Wiener path intersections and local time, J. funct. Anal. 30, 329-340 (1978) · Zbl 0403.60069 · doi:10.1016/0022-1236(78)90061-7
[24]Xiao, Y.: Strong local nondeterminism and the sample path properties of Gaussian random fields, Asymptotic theory in probability and statistics with applications, 136-176 (2007)
[25]Yan, L., Shen, G., 2009. Itô’s formula for the sub-fractional Brownian motion (submitted for publication)