Revista Matemática Iberoamericana

Properties of centered random walks on locally compact groups and Lie groups

Nick Dungey

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The basic aim of this paper is to study asymptotic properties of the convolution powers $K^{(n)} = K*K* \cdots *K$ of a possibly non-symmetric probability density $K$ on a locally compact, compactly generated group $G$. If $K$ is centered, we show that the Markov operator $T$ associated with $K$ is analytic in $L^p(G)$ for $1 < p < \infty$, and establish Davies-Gaffney estimates in $L^2$ for the iterated operators $T^n$. These results enable us to obtain various Gaussian bounds on $K^{(n)}$. In particular, when $G$ is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of centered densities and for the heat kernels of centered sublaplacians. Finally, in case $G$ is amenable, we discover that the properties of analyticity or Davies-Gaffney estimates hold only if $K$ is centered.

Article information

Rev. Mat. Iberoamericana, Volume 23, Number 2 (2007), 587-634.

First available in Project Euclid: 26 September 2007

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Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G50: Sums of independent random variables; random walks 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX] 22D05: General properties and structure of locally compact groups 35B40: Asymptotic behavior of solutions

locally compact group Lie group amenable group random walk probability density heat kernel gaussian estimates convolution powers


Dungey , Nick. Properties of centered random walks on locally compact groups and Lie groups. Rev. Mat. Iberoamericana 23 (2007), no. 2, 587--634.

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  • Alexopoulos, G.: Sous-laplaciens et densités centrés sur les groupes de Lie à croissance polynomiale du volume. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 539-542.
  • Alexopoulos, G.: Sub-Laplacians with drift on Lie groups of polynomial volume growth. Mem. Amer. Math. Soc. 155 (2002), no. 739.
  • Alexopoulos, G.: Random walks on discrete groups of polynomial volume growth. Ann. Probab. 30 (2002), 723-801.
  • Alexopoulos, G.: Centered densities on Lie groups of polynomial volume growth. Probab. Theory Relat. Fields 124 (2002), 112-150.
  • Blunck, S.: Perturbation of analytic operators and temporal regularity of discrete heat kernels. Colloq. Math. 86 (2000), 189-201.
  • Blunck, S.: Analyticity and discrete maximal regularity on $L_p$-spaces. J. Funct. Anal. 183 (2001), 211-230.
  • Bounechada, N.: Distorsion des distances dans les groupes de Lie nilpotents. Bull. Sci. Math. 127 (2003), 797-813.
  • Carne, T. K.: A transmutation formula for Markov chains. Bull. Sci. Math. (2) 109 (1985), 399-405.
  • Coulhon, T., Grigor'yan, A. and Zucca, F.: The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 57 (2005), 559-587.
  • Coulhon, T. and Saloff-Coste, L.: Puissances d'un opérateur régularisant. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990), 419-436.
  • Coifman, R. R. and Weiss, G.: Transference methods in analysis. CBMS Regional Conference Series in Mathematics 31. American Mathematical Society, Providence, 1976.
  • Davies, E. B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92. Cambridge University Press, Cambridge, 1989.
  • Davies, E. B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132 (1995), 141-169.
  • Dungey, N.: On Gaussian kernel estimates on groups. Colloq. Math. 100 (2004), 77-90.
  • Dungey, N.: Heat kernel and semigroup estimates for sublaplacians with drift on Lie groups. Publ. Mat. 49 (2005), 375-391.
  • Dungey, N.: Properties of random walks on discrete groups: time regularity and off-diagonal estimates. To appear in Bull. Sci. Math.
  • Dungey, N.: A note on time regularity for discrete time heat kernels. Semigroup Forum 72 (2006), no. 3, 404-410.
  • Grigor'yan, A.: Estimates of heat kernels on Riemannian manifolds. In Spectral theory and geometry (Edinburgh, 1998), 140-225. London Math. Soc. Lecture Note Ser. 273. Cambridge Univ. Press, Cambridge, 1999.
  • Grigorchuk, R.: Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk. SSSR Ser. Mat. 48 (1984), no. 5, 939-985.
  • Guivarc'h, Y.: Croissance polynomiale et périodes des fonctions harmoniques. Bull. Soc. Math. France 101 (1973), 333-379.
  • Hebisch, W.: On heat kernels on Lie groups. Math. Z. 210 (1992), no. 4, 593-605.
  • Hebisch, W. and Saloff-Coste, L.: Gaussian estimates for Markov chains and random walks on groups. Ann. Probab. 21 (1993), 673-709.
  • Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis. I. Structure of topological groups, integration theory, group representations. Fundamental Principles of Mathematical Sciences 115. Springer-Verlag, Berlin-New York, 1979.
  • Kato, T.: Perturbation theory for linear operators. Grundlehren der Mathematischen Wissenschaften 132. Springer-Verlag, Berlin, 1976.
  • Mackey, G.: Induced representations of locally compact groups I. Ann. of Math. (2) 55 (1952), 101-139.
  • Melzi, C.: Large time estimates for heat kernels in nilpotent Lie groups. Bull. Sci. Math. 126 (2002), 71-86.
  • Montgomery, D. and Zippin, L.: Topological transformation groups. Interscience Publishers, New York-London, 1955.
  • Mustapha, S.: Gaussian estimates for heat kernels on Lie groups. Math. Proc. Cambridge Philos. Soc. 128 (2000), 45-64.
  • Nevanlinna, O.: Convergence of iterations for linear equations. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1993.
  • Nevanlinna, O.: On the growth of the resolvent operators for power bounded operators. In Linear Operators (Warsaw, 1994), 247-264. Banach Center Publ. 38. Polish Acad. Sci., Warsaw, 1997,
  • Paterson, A.: Amenability. Mathematical Surveys and Monographs 29. American Mathematical Society, Providence, 1988.
  • Robinson, D. W.: Elliptic operators and Lie groups. Oxford Mathematical Monographs. Oxford University Press, New York, 1991
  • Varadarajan, V. S.: Lie groups, Lie algebras, and their representations. Graduate Texts in Mathematics 102. Springer-Verlag, New York, 1984.
  • Varopoulos, N. T.: Long range estimates for Markov chains. Bull. Sci. Math. (2) 109 (1985), 225-252.
  • Varopoulos, N. T.: Analysis on Lie groups. Rev. Mat. Iberoamericana 12 (1996), 791-917.
  • Varopoulos, N. T.: Distance distortion on Lie groups. In: Random walks and discrete potential theory (Cortona, 1997), 320-357. Sympos. Math. XXXIX. Cambridge Univ. Press, Cambridge, 1999.
  • Varopoulos, N. T.: Geometric and potential theoretic results on Lie groups. Canad. J. Math. 52 (2000), 412-437.
  • Varopoulos, N. T., Saloff-Coste, L. and Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts in Mathematics 100. Cambridge University Press, Cambridge, 1992.