Brownian motion and transient groups
Annales de l'Institut Fourier, Volume 33 (1983) no. 2, pp. 241-261.

In this paper I consider M ˜→M a covering of a Riemannian manifold M. I prove that Green’s function exists on M ˜ if any and only if the symmetric translation invariant random walks on the covering group G are transient (under the assumption that M is compact).

Dans cet article, je considère M ˜→M un revêtement riemannien. Je démontre que l’existence de la fonction de Green sur M ˜ est équivalente au fait que G, le groupe de revêtement, est “transient" (à condition que M soit compacte).

@article{AIF_1983__33_2_241_0,
     author = {Varopoulos, Nicolas Th.},
     title = {Brownian motion and transient groups},
     journal = {Annales de l'Institut Fourier},
     pages = {241--261},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {33},
     number = {2},
     year = {1983},
     doi = {10.5802/aif.926},
     mrnumber = {84i:58130},
     zbl = {0498.60012},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.926/}
}
TY  - JOUR
AU  - Varopoulos, Nicolas Th.
TI  - Brownian motion and transient groups
JO  - Annales de l'Institut Fourier
PY  - 1983
SP  - 241
EP  - 261
VL  - 33
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - http://archive.numdam.org/articles/10.5802/aif.926/
DO  - 10.5802/aif.926
LA  - en
ID  - AIF_1983__33_2_241_0
ER  - 
%0 Journal Article
%A Varopoulos, Nicolas Th.
%T Brownian motion and transient groups
%J Annales de l'Institut Fourier
%D 1983
%P 241-261
%V 33
%N 2
%I Institut Fourier
%C Grenoble
%U http://archive.numdam.org/articles/10.5802/aif.926/
%R 10.5802/aif.926
%G en
%F AIF_1983__33_2_241_0
Varopoulos, Nicolas Th. Brownian motion and transient groups. Annales de l'Institut Fourier, Volume 33 (1983) no. 2, pp. 241-261. doi : 10.5802/aif.926. http://archive.numdam.org/articles/10.5802/aif.926/

[1] H. P. Mckean Jr, Stochastic Integrals, Academic Press, 1969. | Zbl

[2] N. Th. Varopoulos, Potential Theory and Diffusion on Riemannian manifolds, Zygmund 80th Birthday Volume, Chicago, 1981.

[3] D. Dacunha-Castell et al., Springer Verlag Lecture Notes, n° 678.

[4] Y. Guivarch et al., Springer Verlag Lecture Notes, n° 624.

[5] N. Th. Varopoulos, C. R. A. S., (1982), to appear.

[6] M. Gromov, Groups of polynomial growth and expanding maps, Publications Math. I.H.E.S., n° 53 (1981). | Numdam | MR | Zbl

[7] H. Reiter, Classical Harmonic Analysis and locally compact groups, Oxford Math. Monograph, (1968), O.U.P. | MR | Zbl

[8] P. Baldi, N. Lohoué et J. Peyrière, C. R. A. S., Paris, t. 285 (A), (1977), 1103-1104. | Zbl

[9] S. Y. Cheng, P. Li and S.-T. Yau, On the upper estimate of the heat kernel of a complete Riemannian manifold, American J. Math., Vol. 103, n° 5 (1981), 1021-1063. | MR | Zbl

[10] J. Cheeger and S. T. Yau, A lower Bound of the heat kernel, Comm. on Pure and Appl. Math., Vol. XXXIV (1981), 465-480. | MR | Zbl

[11] J. Milnor, A note on curvature and fundamental group., J. Diff. Geom., 2 (1968), 1-7. | MR | Zbl

[12] T. J. Lyons and H. P. Mckean, Winding of the plane Brownian motion (preprint). | Zbl

[13] W. Feller, An introduction to Probability Theory and its Applications (3rd Edition), Wiley. | Zbl

[14] R. Brooks, Amenability and the Spectrum of the Laplacian, Bull. Amer. Math. Soc., Vol. 6, 87-89 (1982), n° 1. | MR | Zbl

[15] N. Th. Varopoulos, Random walks on soluble Groups, Bull. Sci. Math. (to appear). | Zbl

Cited by Sources: