Brownian motion and transient groups

Nicolas Th. Varopoulos

Annales de l'institut Fourier (1983)

  • Volume: 33, Issue: 2, page 241-261
  • ISSN: 0373-0956

Abstract

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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).

How to cite

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Varopoulos, Nicolas Th.. "Brownian motion and transient groups." Annales de l'institut Fourier 33.2 (1983): 241-261. <http://eudml.org/doc/74588>.

@article{Varopoulos1983,
abstract = {In this paper I consider $\widetilde\{M\}\rightarrow M$ a covering of a Riemannian manifold $M$. I prove that Green’s function exists on $\widetilde\{M\}$ 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).},
author = {Varopoulos, Nicolas Th.},
journal = {Annales de l'institut Fourier},
keywords = {translation invariant random walks on covering groups},
language = {eng},
number = {2},
pages = {241-261},
publisher = {Association des Annales de l'Institut Fourier},
title = {Brownian motion and transient groups},
url = {http://eudml.org/doc/74588},
volume = {33},
year = {1983},
}

TY - JOUR
AU - Varopoulos, Nicolas Th.
TI - Brownian motion and transient groups
JO - Annales de l'institut Fourier
PY - 1983
PB - Association des Annales de l'Institut Fourier
VL - 33
IS - 2
SP - 241
EP - 261
AB - In this paper I consider $\widetilde{M}\rightarrow M$ a covering of a Riemannian manifold $M$. I prove that Green’s function exists on $\widetilde{M}$ 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).
LA - eng
KW - translation invariant random walks on covering groups
UR - http://eudml.org/doc/74588
ER -

References

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  1. [1] H. P. MCKEAN Jr, Stochastic Integrals, Academic Press, 1969. Zbl0191.46603
  2. [2] N. Th. VAROPOULOS, Potential Theory and Diffusion on Riemannian manifolds, Zygmund 80th Birthday Volume, Chicago, 1981. 
  3. [3] D. DACUNHA-CASTELL et al., Springer Verlag Lecture Notes, n° 678. 
  4. [4] Y. GUIVARCH et al., Springer Verlag Lecture Notes, n° 624. 
  5. [5] N. Th. VAROPOULOS, C. R. A. S., (1982), to appear. 
  6. [6] M. GROMOV, Groups of polynomial growth and expanding maps, Publications Math. I.H.E.S., n° 53 (1981). Zbl0474.20018MR83b:53041
  7. [7] H. REITER, Classical Harmonic Analysis and locally compact groups, Oxford Math. Monograph, (1968), O.U.P. Zbl0165.15601MR46 #5933
  8. [8] P. BALDI, N. LOHOUÉ et J. PEYRIÈRE, C. R. A. S., Paris, t. 285 (A), (1977), 1103-1104. Zbl0376.60072
  9. [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. Zbl0484.53035MR83c:58083
  10. [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. Zbl0481.35003MR82i:58065
  11. [11] J. MILNOR, A note on curvature and fundamental group., J. Diff. Geom., 2 (1968), 1-7. Zbl0162.25401MR38 #636
  12. [12] T. J. LYONS and H. P. MCKEAN, Winding of the plane Brownian motion (preprint). Zbl0541.60075
  13. [13] W. FELLER, An introduction to Probability Theory and its Applications (3rd Edition), Wiley. Zbl0155.23101
  14. [14] R. BROOKS, Amenability and the Spectrum of the Laplacian, Bull. Amer. Math. Soc., Vol. 6, 87-89 (1982), n° 1. Zbl0489.58033MR83f:58076
  15. [15] N. Th. VAROPOULOS, Random walks on soluble Groups, Bull. Sci. Math. (to appear). Zbl0532.60009

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