The Annals of Probability

On the sample paths of Brownian motions on compact infinite dimensional groups

Alexander Bendikov and Laurent Saloff-Coste

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Abstract

We study the regularity of the sample paths of certain Brownian motions on the infinite dimensional torus ${\mathbb T}^\infty$ and other compact connected groups in terms of the associated intrinsic distance. For each $\lambda\in (0,1)$, we give examples where the intrinsic distance $d$ is continuous and defines the topology of ${\mathbb T}^\infty$ and where the sample paths satisfy \[ 0<\liminf_{t\ra 0} \frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}\le \limsup_{t\ra 0} \frac{d(X_0,X_t)}{t^{(1-\lambda)/2}}<\infty \] and \[ 0<\lim_{\varepsilon\to 0} \sup_{0<t<s<1 \atop t-s\le \varepsilon}\frac{d(X_s,X_t)}{(t-s )^{(1-\lambda)/2}} <\infty. \]

Article information

Source
Ann. Probab., Volume 31, Number 3 (2003), 1464-1493.

Dates
First available in Project Euclid: 12 June 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1055425787

Digital Object Identifier
doi:10.1214/aop/1055425787

Mathematical Reviews number (MathSciNet)
MR1989440

Zentralblatt MATH identifier
1043.60064

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60B99: None of the above, but in this section 31C25: Dirichlet spaces 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

Keywords
Invariant diffusions path regularity Gaussian convolution semigroups.

Citation

Bendikov, Alexander; Saloff-Coste, Laurent. On the sample paths of Brownian motions on compact infinite dimensional groups. Ann. Probab. 31 (2003), no. 3, 1464--1493. doi:10.1214/aop/1055425787. https://projecteuclid.org/euclid.aop/1055425787


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References

  • [1] BENDIKOV, A. (1976). On harmonic functions for a category of Markov processes and projective limits in it. Russian Math. Surv. 31 209-210.
  • [2] BENDIKOV, A. (1995). Potential Theory on Infinite-Dimensional Abelian Groups. de Gruy ter, Berlin.
  • [3] BENDIKOV, A. (1995). Sy mmetric stable semigroups on the infinite-dimensional torus. Exposiones Math. 13 39-79.
  • [4] BENDIKOV, A. and SALOFF-COSTE, L. (1997). Elliptic diffusions on infinite products. J. Reine Angew. Math. 493 171-220.
  • [5] BENDIKOV, A. and SALOFF-COSTE, L. (1999). Potential theory on infinite products and groups. Potential Anal. 11 325-358.
  • [6] BENDIKOV, A. and SALOFF-COSTE, L. (2000). Onand off-diagonal heat kernel behaviors on certain infinite-dimensional local Dirichlet spaces. Amer. J. Math. 122 1205-1263.
  • [7] BENDIKOV, A. and SALOFF-COSTE, L. (2001). Central Gaussian semigroups of measures with continuous density. J. Funct. Anal. 186 206-268.
  • [8] BENDIKOV, A. and SALOFF-COSTE, L. (2001). On the absolute continuity of Gaussian measures on locally compact groups. J. Theoret. Probab. 14 887-898.
  • [9] BENDIKOV, A. and SALOFF-COSTE, L. (2000). Gaussian semigroups of measures on locally compact locally connected groups. Preprint.
  • [10] BENDIKOV, A. and SALOFF-COSTE, L. (2002). Some properties of the paths of subelliptic diffusions. Unpublished manuscript.
  • [11] BENDIKOV, A. and SALOFF-COSTE, L. (2002). On the sample paths of Brownian motions on T. Preprint.
  • [12] BINGHAM, N. H., GOLDIE, C. M. and TEUGELS, J. L. (1989). Regular Variation. Ency clopedia of Mathematics and Its Applications. Cambridge Univ. Press.
  • [13] BLUMENTHAL, R. and GETOOR, R. (1968). Markov Processes and Potential Theory. Academic Press, New York.
  • [14] BRUHAT, F. (1961). Distributions sur un groupe localement compact et application à l'étude des représentations des groupes p-adiques. Bull. Soc. Math. France 89 43-75.
  • [15] DAVIES, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • [16] DVORETZKY, A. and ERDÖS, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Sy mp. Math. Statist. Probab. 353-367. Univ. California Press, Berkeley.
  • [17] FUKUSHIMA M., OSHIMA Y. and TAKEDA, M. (1994). Dirichlet Forms and Sy mmetric Markov Processes. de Gruy ter, Berlin.
  • [18] GRIGOR'YAN, A. (1999). Escape rate of Brownian motion on Riemannian manifolds. Appl. Anal. 71 63-89.
  • [19] GRIGOR'YAN, A. and KELBERT, M. (1998). Range of fluctuation of Brownian on a complete Riemannian manifold. Ann. Probab. 26 78-111.
  • [20] HEy ER, H. (1977). Probability Measures on Locally Compact Groups. Springer, Berlin.
  • [21] HOFMANN, K. and MORRIS, S. (1998). The Structure of Compact Groups. de Gruy ter, Berlin.
  • [22] HÖRMANDER, L. (1967). Hy poelliptic second order differential equations. Acta Math. 119 147-171.
  • [23] HUNT, G. A. (1956). Semi-groups of measures on Lie groups. Trans. Amer. Math. Soc. 81 264-293.
  • [24] ITÔ, K. and MCKEAN, P. (1974). Diffusion Processes and Their Sample Paths. Springer, Berlin.
  • [25] KUELBS, J. and LEPAGE, R. (1973). The law of iterated logarithm for Brownian motion on a Banach space. Trans. Amer. Math. Soc. 185 253-264.
  • [26] LEDOUX, M. and TALAGRAND, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • [27] REVUZ, D. and YOR, M. (1994). Continuous Martingales and Brownian Motion, 2nd ed. Springer, Berlin.
  • [28] SIEBERT, E. (1982). Absolute continuity, singularity and supports of Gaussian semigroups on a Lie group. Monatshefte für Math. 93 239-253.
  • [29] STURM, K.-T. (1995). On the geometry defined by Dirichlet forms. In Seminar on Stochastic Processes, Random Fields and Applications (E. Bolthausen, M. Dozzi and F. Russo, eds.) 231-242. Birkhäuser, Boston.
  • [30] VAROPOULOS, N., SALOFF-COSTE, L. and COULHON, T. (1993). Analy sis and Geometry on Groups. Cambridge Univ. Press.
  • ITHACA, NEW YORK 14853-4201 E-MAIL: lsc@math.cornell.edu bendikov@math.cornell.edu