## Random walks driven by low moment measures.(English)Zbl 1262.60005

Random walks on a locally compact unimodular group $$G$$ are considered. The authors investigate the rate of decay of the probability to return to a neighborhood of the origin (identity element $$e$$) at the $$2n$$-th step. This probability corresponds to the value at $$e$$ of the density $$\varphi^{(2n)}$$ of $$2n$$-fold convolution of the one-step density $$\varphi$$ (w.r.t. the Haar measure). It is assumed that $$\varphi$$ belongs to the class $$S_{G,\rho}$$ of symmetric continuous densities with finite $$\int\rho\varphi$$ for some fixed moment function $$\rho: G\to [0,\infty)$$.
For different types of moment functions $$\rho$$ and groups $$G$$, the fastest possible rates of decay $\Phi_{G,\rho}(n)=\inf\{\varphi^{(2n)}(e)\;:\;\varphi\in S_{G,\rho}\}$ are derived, e.g., for polycyclic groups with exponential volume growth and $$\rho(g)=\exp(c[\log(1+|g|]^\alpha)$$ ($$|g|$$ being a natural word length measuring the distance from $$g$$ to $$e$$), $\log \Phi_{G,\rho}(n)\approx n\exp(-c_1[\log n]^\alpha).$

### MSC:

 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60G50 Sums of independent random variables; random walks
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### References:

 [1] Alexopoulos, G. (1992). A lower estimate for central probabilities on polycyclic groups. Canad. J. Math. 44 897-910. · Zbl 0762.31003 [2] Bendikov, A., Pittet, C. and Sauer, R. (2012). Spectral distribution and $$\mathrm{L}^{2}$$-isoperimetric profile of Laplace operators on groups. Math. Annal. 354 43-72. · Zbl 1268.43001 [3] Bendikov, A. and Saloff-Coste, L. (2010). On the stability of group invariants associated with random walks driven by low moment measures. Unpublished manuscript, Cornell Univ. [4] Bendikov, A. and Saloff-Coste, L. (2012). Random walks on groups and discrete subordination. Math. Nachr. 285 580-605. · Zbl 1251.60004 [5] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Encyclopedia of Mathematics and Its Applications 27 . Cambridge Univ. Press, Cambridge. · Zbl 0667.26003 [6] Brown, L. G. and Kosaki, H. (1990). Jensen’s inequality in semi-finite von Neumann algebras. J. Operator Theory 23 3-19. · Zbl 0718.46026 [7] Dixmier, J. (1981). Von Neumann Algebras. North-Holland Mathematical Library 27 . North-Holland, Amsterdam. With a preface by E. C. Lance, Translated from the second French edition by F. Jellett. · Zbl 0473.46040 [8] Donsker, M. D. and Varadhan, S. R. S. (1979). On the number of distinct sites visited by a random walk. Comm. Pure Appl. Math. 32 721-747. · Zbl 0418.60074 [9] Erschler, A. (2003). On isoperimetric profiles of finitely generated groups. Geom. Dedicata 100 157-171. · Zbl 1049.20024 [10] Erschler, A. (2006). Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks. Probab. Theory Related Fields 136 560-586. · Zbl 1105.60009 [11] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II . Wiley, New York. · Zbl 0138.10207 [12] Feller, W. (1967). On regular variation and local limit theorems. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability ( Berkeley , Calif. , 1965 / 66), Vol. II : Contributions to Probability Theory , Part 1 373-388. Univ. California Press, Berkeley, CA. · Zbl 0214.17303 [13] Gretete, D. (2008). Stabilité du comportement des marches aléatoires sur un groupe localement compact. Ann. Inst. Henri Poincaré Probab. Stat. 44 129-142. · Zbl 1181.60012 [14] Griffin, P. S. (1983). Probability estimates for the small deviations of $$d$$-dimensional random walk. Ann. Probab. 11 939-952. · Zbl 0519.60044 [15] Griffin, P. S., Jain, N. C. and Pruitt, W. E. (1984). Approximate local limit theorems for laws outside domains of attraction. Ann. Probab. 12 45-63. · Zbl 0539.60022 [16] Hebisch, W. and Saloff-Coste, L. (1993). Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 673-709. · Zbl 0776.60086 [17] Jacob, N. (2001). Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups . Imperial College Press, London. · Zbl 0987.60003 [18] Milnor, J. (1968). Growth of finitely generated solvable groups. J. Differential Geom. 2 447-449. · Zbl 0176.29803 [19] Montgomery, R. (2002). A Tour of Subriemannian Geometries , Their Geodesics and Applications. Mathematical Surveys and Monographs 91 . Amer. Math. Soc., Providence, RI. · Zbl 1044.53022 [20] Pittet, C. and Saloff-Coste, L. (2000). On the stability of the behavior of random walks on groups. J. Geom. Anal. 10 713-737. · Zbl 0985.60043 [21] Pittet, C. and Saloff-Coste, L. (2002). On random walks on wreath products. Ann. Probab. 30 948-977. · Zbl 1021.60004 [22] Pittet, C. and Saloff-Coste, L. (2003). Random walks on finite rank solvable groups. J. Eur. Math. Soc. ( JEMS ) 5 313-342. · Zbl 1057.20026 [23] Revelle, D. (2003). Rate of escape of random walks on wreath products and related groups. Ann. Probab. 31 1917-1934. · Zbl 1051.60047 [24] Saloff-Coste, L. (1989). Sur la décroissance des puissances de convolution sur les groupes. Bull. Sci. Math. (2) 113 3-21. · Zbl 0682.43004 [25] Saloff-Coste, L. (2001). Probability on groups: Random walks and invariant diffusions. Notices Amer. Math. Soc. 48 968-977. · Zbl 0987.60018 [26] Saloff-Coste, L. (2004). Analysis on Riemannian co-compact covers. In Surveys in Differential Geometry. Vol. IX. Surv. Differ. Geom. IX 351-384. International Press, Somerville, MA. · Zbl 1082.31006 [27] Schilling, R. L., Song, R. and Vondraček, Z. (2010). Bernstein Functions : Theory and Applications. de Gruyter Studies in Mathematics 37 . de Gruyter, Berlin. [28] Spitzer, F. (1964). Principles of Random Walk . Van Nostrand, Princeton, NJ. · Zbl 0119.34304 [29] Varopoulos, N. T. (1987). Convolution powers on locally compact groups. Bull. Sci. Math. (2) 111 333-342. · Zbl 0626.22004 [30] Varopoulos, N. T. (1991). Groups of superpolynomial growth. In Harmonic Analysis ( Sendai , 1990) 194-200. Springer, Tokyo. · Zbl 0802.43002 [31] Varopoulos, N. T., Saloff-Coste, L. and Coulhon, T. (1992). Analysis and Geometry on Groups. Cambridge Tracts in Mathematics 100 . Cambridge Univ. Press, Cambridge. · Zbl 0813.22003 [32] Wolf, J. A. (1968). Growth of finitely generated solvable groups and curvature of Riemanniann manifolds. J. Differential Geom. 2 421-446. · Zbl 0207.51803
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