×

A uniform law of iterated logarithm for Brownian motion on compact Riemannian manifolds. (English) Zbl 0649.60037

Using a general result of W. Philipp [Mem. Amer. math. Soc. 114 (1971; Zbl 0224.10052)] we prove that \[ \limsup _{T\to \infty}T D_ T(x(t))/\sqrt{2T \log \log T}=const>0 \] for almost all trajectories x(t) starting at a given point \(w=x(0)\); \(D_ T(x(t))\) denotes the discrepancy of x(t) on the compact connected Riemannian manifold X. Thus a theorem of O. Stackelberg [Indiana Univ. Math. J. 21, 515-528 (1971; Zbl 0218.60034)] concerning the one-dimensional torus is generalized.
Reviewer: M.Blümlinger

MSC:

60F15 Strong limit theorems
60J65 Brownian motion
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Berard, P.H.: Spectral Geometry: Direct and invers problems. Lect. Notes Math., vol. 1207. Berlin Heidelberg New York: Springer 1986
[2] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une Variété Riemannienne. Lect. Notes Math., vol. 194. Berlin Heidelberg New York: Springer 1971 · Zbl 0223.53034
[3] Bishop, R.L., Crittenden, R.J.: Geometry of manifolds. New York: Academic Press 1964 · Zbl 0132.16003
[4] Blümlinger, M., Drmota, M., Tichy, R.: Metrische Sätze derC-Gleichverteilung. Hlawka, E. (ed.) Zahlentheoretische Analysis II. (Lect. Notes Math., vol 1262). Berlin Heidelberg New York: Springer 1987
[5] Blümlinger, M., Drmota, M., Tichy, R.: Asymptotic distribution of functions on compact homogeneous spaces. Ann. Mat. Pura Appl. (to appear) · Zbl 0668.10054
[6] Chavel, I.: Eigenvalues in Riemannian Geometry. Orlando: Academic Press 1984 · Zbl 0551.53001
[7] Hlawka, E.: ÜberC-Gleichverteilung. Ann. Mat. Pura Appl. (IV)49, 331-366 (1960)
[8] Hlawka, E.: Theorie der Gleichverteilung. Wien Mannheim Zürich: Bibliographisches Institut 1979 · Zbl 0406.10001
[9] Itô, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin Heidelberg New York: Springer 1975 · Zbl 0837.60001
[10] Kuipers, L., Niederreiter, H.: Uniform distribution of sequences. New York: Wiley 1974 · Zbl 0281.10001
[11] Phillipp, W.: Mixing sequences of random variables and probabilistic number theory. A.M.S. Memoirs 114, Providence 1971
[12] Stackelberg, O.: A uniform law of the iterated logarithm for functions C-uniformly distributed mod 1. Indiana Univ. Math. J.21, 515-528 (1971) · Zbl 0218.60034 · doi:10.1512/iumj.1971.21.21041
[13] Warner, F.: Foundations of differential manifolds and Lie groups. Berlin Heidelberg New York: Springer 1983 · Zbl 0516.58001
[14] Yosida, K.: Functional analysis. Berlin Heidelberg New York: Springer 1974 · Zbl 0286.46002
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.