Hostname: page-component-7f64f4797f-g5lq8 Total loading time: 0.001 Render date: 2025-11-04T00:12:48.384Z Has data issue: false hasContentIssue false
  • English
  • Français

White noise analysis and Tanaka formula for intersections of planar Brownian motion

Published online by Cambridge University Press:  22 January 2016

Narn-Rueih Shieh
Narn-Rueih Shieh*
Affiliation:
Department of Mathematics, National Taiwan University, Taipei, Taiwan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we shall use Hida’s [5, 7, 9] theory of generalized Brownian functionals (or named white noise analysis) to establish a stochastic integral formula concerning the multiple intersection local times of planar Brownian motion B(t).

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 122 , June 1991 , pp. 1 - 17
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[1] Dvoretzky, A., Erdös, P. and Kakutani, S., Multiple points of paths of Brownian motion in the plane, Bull. Res. Council Israel, F3 (1954), 364–371.Google Scholar
[2] Dynkin, E. B., Self-intersection guage for random walks and for Brownian motion, Ann. Probab., 16 (1988), 1–57.Google Scholar
[3] Dynkin, E. B., Regularized self-intersection local times of planar Brownian motion, Ann. Probab., 16 (1988), 58–74.Google Scholar
[4] 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.Google Scholar
[5] Hida, T., Analysis of Brownian Functionals, Carleton Math. Lecture Notes, no. 13 (1975), Carleton Univ. Ottawa.Google Scholar
[6] Hida, T., Generalized multiple Wiener integrals, Proc. Japan Acad., 54A (1978), 55–58.Google Scholar
[7] Hida, T., Brownian Motion, Application of Mathematics, vol. 11 (1980), Springer-Verlag.Google Scholar
[8] Hida, T., Generalized Brownian functions and stochastic integrals, Appl. Math. Optim., 12 (1984), 115–123.Google Scholar
[9] Hida, T., Lévy’s Functional Analysis and Stochastic Analysis, Nagoya Univ. Lecture Note, (1986).Google Scholar
[10] Kallianpur, G. and Kuo, H. -H., Regularity property of Donsker’s delta function, Appl. Math. Optim., 12 (1984), 89–95.Google Scholar
[11] Kubo, I., Ito formula for generalized Brownian functionals, Lecture Notes in Control and Information Sciences, vol. 49 (1983), 156–166, Springer-Verlag.Google Scholar
[12] Kuo, H. -H., Donsker’s delta function as a generalized Brownian functionals and its application, Lecture Notes in Control and Information Sciences, vol. 49 (1983), 167–178 Springer-Verlag.Google Scholar
[13] Kuo, H. -H. and Shieh, N. R., A generalized Ito’s formula for multi-dimensional Brownian motions and its applications, Chinese J. Math., 15 (1987), 163–174.Google Scholar
[14] Major, P., Multiple Wiener-Ito Integrals, Lecture Notes in Math., vol. 849 (1981), Springer-Verlag.Google Scholar
[15] Rosen, J., Self intersections of random fields, Ann. Probab., 12 (1984), 108–119.Google Scholar
[16] Rosen, J., A representation for the intersection local time of Brownian motion in space, Ann. Probab., 13 (1985), 145–153.Google Scholar
[17] Rosen, J., Tanaka’s formula for multiple intersections of planar Brownian motion, Stoch. Processes and Appl., 23 (1986), 131–141.Google Scholar
[18] Rosen, J., Tanaka’s formula and renormalization for intersections of planar Brownian motion, Ann. Probab., 14 (1986), 1245–1251.Google Scholar
[19] Rosen, J., A renormalized local time for multiple intersections of planar Brownian motion, Lecture Notes in Math., vol. 1204 (1986), 515–531.Google Scholar
[20] Rosen, J. and Yor, M., Tanaka formulae and renormalization for triple intersections of Brownian motion in the plane, preprint 1988.Google Scholar
[21] Szegö, G., Orthogonal Polynomians, Amer. Math. Soc. Colloq. Publ., vol. XXIII (1939).Google Scholar
[22] Varadhan, S. R. S., Appendix to Euclidean Quantum field theory by Symanzik, K.. In: Local Quantum Theory (Jost, R., ed.) A.P. 1969.Google Scholar
[23] Yor, M., Comploments aux formules des Tanaka-Rosen, Lecture Notes in Math., vol. 1123 (1986), 332–349, Springer-Verlag.Google Scholar