Abstract
A 'chaos expansion' of the intersection local time functional of two independent Brownian motions in R d is given. The expansion is in terms of normal products of white noise (corresponding to multiple Wiener integrals). As a consequence of the local structure of the normal products, the kernel functions in the expansion are explicitly given and exhibit clearly the dimension dependent singularities of the local time functional. Their L p-properties are discussed. An important tool for deriving the chaos expansion is a computation of the 'S-transform' of the corresponding regularized intersection local times and a control about their singular limit.
Similar content being viewed by others
References
Albeverio, S., Bolthausen, E. and Zhou, X. Y.: A modified discrete Edwards model in three dimensions, In preparation.
Albeverio, S., Fenstad, J. E., Høegh-Krohn, R. and Lindstrøm, T.: Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.
Albeverio, S., Hu, Y. Z., Röckner, M. and Zhou, X. Y.: Stochastic quantization of the twodimensional polymer measure, Appl. Math. Optim. 40 (1999), 341–354.
Albeverio, S., Hu, Y. Z. and Zhou, X. Y.: A remark on non-smoothness of the self-intersection local time of planar Brownian motion, Statist. Probab. Lett. 32 (1997), 57–65.
Albeverio, S., Röckner, M. and Zhou, X. Y.: Stochastic quantization of the three-dimensional polymer measure, Inf. Dim. Anal. Quant. Prob. (1999), to appear.
Albeverio, S. and Zhou, X. Y.: Intersections of random walks and Wiener sausages in four dimensions, Acta Appl. Math. 45 (1996), 195–237.
Bass, R. F. and Khoshnevisan, D.: Intersection local times and Tanaka formulas, Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), 419–451.
Dvoretzky, A., Erdös, P., Kakutani, S. and Taylor, S. J.: Triple points of the Brownian motion in 3-space, Proc. Cambridge Philos. Soc. 53 (1957), 856–862.
Dvoretzky, A., Erdös, P. and Kakutani, S.: Double points of paths of Brownian motion in the plane, Bull. Res. Council Israel Sect. F 3 (1954), 364–371.
Dvoretzky, A., Erdös, P. and Kakutani, S.: Double points of paths of Brownian motion in n-space, Acta Sci. Math. Szeged 12 (1950), 75–81.
Dynkin, E. B.: Regularized self-intersection local times of planar Brownian motion, Ann. Probab. 16 (1988), 58–73.
Dynkin, E. B.: Self-intersection gauge for random walks and for Brownian motion, Ann. Probab. 16 (1988), 1–57.
Dynkin, E. B.: Polynomials of the occupation field and related random fields, J. Funct. Anal. 58 (1984), 20–52.
de Faria, M., Drumond, C. and Streit, L.: The renormalization of self intersection local times I: The chaos expansion, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000), 223–239.
de Faria, M., Hida, T., Streit, L. and Watanabe, H.: Intersection local times as generalized white noise functionals, Acta Appl. Math. 46 (1997), 351–362.
Geman, D., Horowitz. J. and Rosen, J.: A local time analysis of intersections of Brownian paths in the plane, Ann. Probab. 12 (1984), 86–107.
He, S. W., Yang, W. Q., Yao, R. Q. and Wang, J. G.: Local times of self-intersection for multidimensional Brownian motion, Nagoya Math. J. 138 (1995), 51–67.
Hida, T., Kuo, H. H., Potthoff, J. and Streit, L.: White Noise. An Infinite Dimensional Calculus, Kluwer Acad. Publ., Dordrecht, 1993.
Imkeller, P., Pérez-Abreu, V. and Vives, J.: Chaos expansions of double intersection local times of Brownian motion in ℝd and renormalization, Stochastic Process. Appl. 56 (1995), 1–34.
Imkeller, P. and Yan, J. A.: Multiple intersection local time of planar Brownian motion as a particular Hida distribution, J. Funct. Anal. 140 (1996), 256–273.
Kondratiev, Yu., Leukert, P., Potthoff, J., Streit, L. and Westerkamp, W.: Generalized functionals in Gaussian spaces: The characterization theorem revisited, J. Funct. Anal. 141 (1996), 301–318.
Le Gall, J. F.: Sur le temps local d'intersection du mouvement brownien plan et la méthode de renormalisation de Varadhan, In: Sém. Prob. XIX, Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 314–331.
Le Gall, J. F.: Sur la saucisse de Wiener et les points multiples du mouvement brownien, Ann. Probab. 14 (1986), 1219–1244.
Lévy, P.: Le mouvement brownien plan, Amer. J. Math. 62 (1940), 487–550.
Lyons, T. J.: The critical dimension at which quasi-every Brownian motion is self-avoiding, Adv. Appl. Probab. (Spec. Suppl. 1986), 87–99.
Mendonça, S. and Streit, L.: Multiple intersection local times in terms of white noise, CCM, Preprint, 1999.
Meyer, P. A. and Yan, J. A.: Les fonctions caractéristiques des distributions sur l'espace de Wiener, In: Sém. Prob. XXV, Lecture Notes in Math. 1485, Springer, Berlin, 1991, pp. 61–78.
Nualart, D. and Vives, J.: Smoothness of Brownian local times and related functionals, Potential Anal. 1 (1992), 257–263.
Nualart, D. and Vives, J.: Chaos expansion and local times, Publ. Math. 36(2), 827–836.
Penrose, M. D.: On the existence of self-intersections for quasi-every Brownian path in space, Ann. Probab. 17 (1989), 482–502.
Pérez-Abreu, V.: Chaos expansions: A review, Proc. V Latin Amer. Congr. Prob. and Math. Stat., São Paulo 1993, RESHAS 1(2–3) (1994), 335–359.
Rosen, J.: A local time approach to the self-intersections of Brownian paths in space, Comm. Math. Phys. 88 (1983), 327–338.
Rosen, J.: Tanaka's formula and renormalisation for intersections of planar Brownian motion, Ann. Probab. 14 (1986), 1425–1251.
Rosen, J.: A renormalized local time for multiple intersections of planar Brownian motion, In: Sém. de Prob. XX, Lecture Notes in Math. 1204, Springer, Berlin, 1986, pp. 515–531.
Shieh, N. R.: White noise analysis and Tanaka formula for intersections of planar Brownian motion, Nagoya Math. J. 122 (1991), 1–17.
Stoll, A.: Invariance principle for Brownian local time and polymer measures, Math. Scand. 64 (1989), 133–160.
Streit, L. and Westerkamp, W.: A generalization of the characterization theorem for generalized functionals of white noise, In: Ph. Blanchard et al. (eds), Dynamics of Complex and Irregular Systems, World Scientific, Singapore, 1993.
Symanzik, K.: Euclidean quantum field theory, In: R. Jost (ed.), Local Quantum Theory, Academic Press, New York, 1969.
Varadhan, S. R. S.: Appendix to “Euclidean quantum field theory” by K. Symanzik, In: R. Jost (ed.), Local Quantum Theory, Academic Press, New York, 1969.
Watanabe, H.: The local time of self-intersections of Brownian motions as generalized Brownian functionals, Lett. Math. Phys. 23 (1991), 1–9.
Werner, W.: Sur les singularités des temps locaux d'intersection du mouvement brownien plan, Ann. Inst. H. Poincaré Probab. Statist. 29 (1993), 391–418.
Westwater, J.: On Edward's model for long polymer chains, Comm. Math. Phys. 72 (1980), 131–174.
Wolpert, R.: Wiener path intersection and local time, J. Funct. Anal. 30 (1978), 329–340.
Yor, M.: Compléments aux formules de Tanaka-Rosen, In: Sém. de Prob. XIX, Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 332–348.
Yor, M.: Renormalisation et convergence en loi pour les temps locaux d'intersection du mouvement brownien dans ℝ3, In: Séminaire de Probabilité, Lecture Notes in Math. 1123, Springer, Berlin, 1985, pp. 350–365.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Albeverio, S., Oliveira, M.J. & Streit, L. Intersection Local Times of Independent Brownian Motions as Generalized White Noise Functionals. Acta Applicandae Mathematicae 69, 221–241 (2001). https://doi.org/10.1023/A:1014212906782
Issue Date:
DOI: https://doi.org/10.1023/A:1014212906782