Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T23:58:53.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Brownian motion parametrized with metric space of constant curvature

Published online by Cambridge University Press:  22 January 2016

Shigeo Takenaka ,
Izumi Kubo  and
Hajime Urakawa
Shigeo Takenaka
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University
Izumi Kubo
Affiliation:
Department of Mathematics, Faculty of Science, Nagoya University
Hajime Urakawa
Affiliation:
Department of Mathematics, College of General Education, Tôhoku University
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.

P. Lévy introduced a generalized notion of Brownian motion in his monograph “Processus stochastiques et mouvement brownien” by taking the time parameter space to be a general metric space. Let (M, d) be a metric space and let O be a fixed point of M called the origin. Following his definition, a Brownian motion parametrized with the metric space (M, d) is a Gaussian system ℬ = {B(m); m ∈ M} such that the difference B(m) − B(m′) is a random variable with mean zero and variance d(m, m′), and that B(O) = 0.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 82 , June 1981 , pp. 131 - 140
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Blaschke, W., Vorlesungen über Differentialgeometrie III, Springer (1929).Google Scholar
[2] Doob, J. L., Stochastic Processes, John Wiley & Sons (1953).Google Scholar
[3] Gangolli, R., Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters, Ann. Inst. Henri Poincaré 3 sec. B (1967), 121–225.Google Scholar
[4a] Hida, T., Canonical representations of Gaussian processes and their applications, Mem. of the College of Sci. Univ. of Kyoto, ser. A 33 (1960), 109–155.Google Scholar
[4b] Hida, T., Brownian Motion (in Japanese), Iwanami, Tokyo (1975), English edition, Springer (1980).Google Scholar
[5] Hida, T. and Hitsuda, M., Gaussian Processes (in Japanese), Kinokuniya, Tokyo (1976).Google Scholar
[6] Kubo, I., Topics on Random Fields (in Japanese), Seminar on Probability, 26 (1967).Google Scholar
[7] Kurita, M., Integral Geometry (in Japanese), Kyôritsu, Tokyo (1956).Google Scholar
[8] Lévy, P., Processus stochastiques et mouvement brownien, Gauthier-Villars (1965).Google Scholar
[9] McKean, Jr., , H. P., Brownian motion with a several-dimensional time. 8 (1963), 355–378.Google Scholar
[10] 221 (1975), 1276–1279.Google Scholar
[11] Takenaka, S., On projective invariance of multi-parameter Brownian motion, Nagoya Math. J., 67 (1977), 89–120.Google Scholar
[12] 2 (1956), 281–282.Google Scholar
[13] Inoue, K. and Noda, A., Independence of the increments of Gaussian random fields, to appear in Nagoya Math. J., 85 (1982).Google Scholar
[14] Noda, A., On Lévy’s Brownian motion (in Japanese), Epsilon—Journal of Math. Aichi Univ. of Education, 22 (1980), 35–40.Google Scholar