×

Isoperimetric inequalities and Markov chains. (English) Zbl 0573.60059

Consider the stochastic matrix A of a reversible Markov chain as acting on a sequence space. An inequality between certain norms on this space is shown to be (nearly) equivalent to a certain rate of decay for powers of A. The results are related to probabilities on finite groups, to graph structures on compact manifolds, and to Sobolev and isoperimetric inequalities on groups. The proofs use Green’s functions, and an integral bound for Dirichlet forms. A criterion is obtained, using the norms, for transience of the Markov chain.
Reviewer: B.D.Craven

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
46A45 Sequence spaces (including Köthe sequence spaces)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Varopoulos, N. Th, Brownian motion and random walks on manifolds, Ann. Inst. Fourier (Grenoble), 34, II, 243-269 (1984) · Zbl 0523.60071
[2] Guivarch, Y., Marches aléatoires sur les groupes de Lie, (Lecture Notes No. 624 (1980), Springer-Verlag: Springer-Verlag New York/Berlin)
[3] Gromov, M., Groups of Polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math., 53 (1981)
[4] Varopoulos, N. Th, A potential theoretic property of soluble groups, Bull. Sci. Math., 108, No. 3, 263-273 (1984), (2) · Zbl 0546.60008
[5] Varopoulos, N. Th, C. R. Acad. Sci. Paris Ser. I Math., 297 (1984)
[6] Varopoulos, N. Th, C. R. Acad. Sci. Paris Ser. I Math., 297 (1984)
[8] Stein, E. M., Singular Integrals and Differentiability of Functions (1971), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J
[10] Osserman, R., Bull. Amer. Math. Soc., 84, 6, 1192-1198 (1978)
[11] Benedek, A.; Panzone, R., Duke Math. J., 28, 301 (1961)
[12] Deny, J., Potential Theory C.I.M.E. \(1^{er}\) cycle — Stresa — dal 2 al 10 Luglio (1969)
[13] Moser, J., Comm. Pure Appl. Math., 17 (1964)
[14] Moser, J., Comm. Pure Appl. Math., 24 (1971)
[15] Stillwell, J., Classical Topology and Combinatorial Group Theory (1980), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0453.57001
[16] Bollobás, B., Graph Theory (1979), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0418.05049
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.