Intersection local time for two independent fractional Brownian motions. (English) Zbl 1154.60028

For two independent \(d\)-dimensional fractional Brownian motions \(B^1, B^2\) with Hurst parameter \(H\in(0,1)\) it is shown that, for \(d\geq 2\) the mutual intersection local time exists as \(L^2\)-limit \[ \lim_{\epsilon\downarrow 0} \int_0^T\int_0^T p_\epsilon(B_t^1-B_s^2)\, ds dt, \] where \(p_\epsilon\) is the Gaussian density with variance \(\epsilon\), if and only if \(dH<2\).


60G15 Gaussian processes
60F25 \(L^p\)-limit theorems
60G18 Self-similar stochastic processes
60J55 Local time and additive functionals
Full Text: DOI arXiv


[1] Doukhan, P., Oppenheim, G., Taqqu, M.S.: Theory and Applications of Long Range Dependence. Birkhäuser, Boston (2003) · Zbl 1005.00017
[2] 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
[3] Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33, 948–983 (2005) · Zbl 1093.60017 · doi:10.1214/009117905000000017
[4] LeGall, J.F.: Sur le temps local d’intersection du mouvement Browniren plan et la méthode de renormalisation de varadhan. In: Séminaire de Probabilités XIX. Lecture Notes in Mathematics, vol. 1123, pp. 314–331. Springer, Berlin (1985)
[5] Marcus, M.B., Rosen, J.: Additive functionals of several levy processes and intersection local times. Ann. Probab. 27, 1643–1678 (1999) · Zbl 0963.60072 · doi:10.1214/aop/1022677544
[6] Muirhead, R.J.: Aspects of multivariate statistical theory. Wiley Series in Probability and Mathematical Statistics. Wiley, New York (1982) · Zbl 0556.62028
[7] Nualart, D.: Stochastic integration with respect to fractional Brownian motion and applications. Contemp. Math. 336, 3–39 (2003) · Zbl 1063.60080
[8] Nualart, D., Rovira, C., Tindel, S.: Probabilistic models for vortex filaments based on fractional Brownian motion. Ann. Probab. 31, 1862–1899 (2003) · Zbl 1047.76013 · doi:10.1214/aop/1068646369
[9] Rosen, J.: The intersection local time of fractional Brownian motion in the plane. J. Multivar. Anal. 23, 37–46 (1987) · Zbl 0633.60057 · doi:10.1016/0047-259X(87)90176-X
[10] Varadhan, S.R.S.: Appendix to Euclidean quantum field theory, by K. Symanzik. In: Jost, R. (ed.) Local Quantum Theory. Academic Press, New York (1969)
[11] Wolpert, R.: Local time and a particle picture for Euclidean field theory. J. Funct. Anal. 30, 341–357 (1978) · Zbl 0464.70016 · doi:10.1016/0022-1236(78)90062-9
[12] 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.