Varopoulos, Nicolas Th.
Brownian motion and transient groups
Annales de l'institut Fourier, Tome 33 (1983) no. 2 , p. 241-261
Zbl 0498.60012 | MR 84i:58130 | 2 citations dans Numdam
doi : 10.5802/aif.926
URL stable :

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).
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).


[1] H. P. Mckean Jr, Stochastic Integrals, Academic Press, 1969. Zbl 0191.46603

[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 83b:53041 | Zbl 0474.20018

[7] H. Reiter, Classical Harmonic Analysis and locally compact groups, Oxford Math. Monograph, (1968), O.U.P. MR 46 #5933 | Zbl 0165.15601

[8] P. Baldi, N. Lohoué et J. Peyrière, C. R. A. S., Paris, t. 285 (A), (1977), 1103-1104. Zbl 0376.60072

[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 83c:58083 | Zbl 0484.53035

[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 82i:58065 | Zbl 0481.35003

[11] J. Milnor, A note on curvature and fundamental group., J. Diff. Geom., 2 (1968), 1-7. MR 38 #636 | Zbl 0162.25401

[12] T. J. Lyons and H. P. Mckean, Winding of the plane Brownian motion (preprint). Zbl 0541.60075

[13] W. Feller, An introduction to Probability Theory and its Applications (3rd Edition), Wiley. Zbl 0155.23101

[14] R. Brooks, Amenability and the Spectrum of the Laplacian, Bull. Amer. Math. Soc., Vol. 6, 87-89 (1982), n° 1. MR 83f:58076 | Zbl 0489.58033

[15] N. Th. Varopoulos, Random walks on soluble Groups, Bull. Sci. Math. (to appear). Zbl 0532.60009