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Volume 29, Issue 3


Probabilistic trace and Poisson summation formulae on locally compact abelian groups

David Applebaum
  • Corresponding author
  • School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield, England, S3 7RH, United Kingdom
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Published Online: 2016-06-19 | DOI: https://doi.org/10.1515/forum-2016-0067


We investigate convolution semigroups of probability measures with continuous densities on locally compact abelian groups, which have a discrete subgroup such that the factor group is compact. Two interesting examples of the quotient structure are the d-dimensional torus, and the adèlic circle. Our main result is to show that the Poisson summation formula for the density can be interpreted as a probabilistic trace formula, linking values of the density on the factor group to the trace of the associated semigroup on L2-space. The Gaussian is a very important example. For rotationally invariant α-stable densities, the trace formula is valid, but we cannot verify the Poisson summation formula. To prepare to study semistable laws on the adèles, we first investigate these on the p-adics, where we show they have continuous densities which may be represented as series expansions. We use these laws to construct a convolution semigroup on the adèles whose densities fail to satisfy the probabilistic trace formula.

Keywords: Locally compact abelian group; discrete subgroup; Fourier transform; Poisson summation formula; convolution semigroup; adèles; idèles; Riemann–Roch theorem, Bruhat–Schwartz space; semistable; Gel’fand–Graev gamma function

MSC 2010: 60B15; 60E07; 11F85; 43A25; 11R56


  • [1]

    Albeverio S. and Karwowski W., Diffusion on p-adic numbers, Gaussian Random Fields (Nagoya 1991), World Scientific, Singapore (1991), 86–99. Google Scholar

  • [2]

    Albeverio S. and Karwowski W., A random walk on p-adics – the generator and its spectrum, Stochastic Process. Appl. 53 (1994), 1–22. Google Scholar

  • [3]

    Albeverio S., Karwowski W. and Yasuda K., Trace formula for p-adics, Acta Appl. Math. 71 (2002), 31–48. Google Scholar

  • [4]

    Albeverio S., Khrennikov A. Y. and Shelkovitch V. M., Theory of p-adic Distributions: Linear and Nonlinear Models, Cambridge University Press, Cambridge, 2010. Google Scholar

  • [5]

    Albeverio S. and Zhao X., On the relation between different types of construction of random walks on p-adics, Markov Process. Related Fields 6 (2000), 239–255. Google Scholar

  • [6]

    Aldous D. and Evans S. N., Dirichlet forms on totally disconnected spaces and bipartite Markov chains, J. Theoret. Probab. 12 (1999), 839–857. Google Scholar

  • [7]

    Applebaum D., Probability measures on compact groups which have square-integrable densities, Bull. Lond. Math. Soc. 40 (2008), 1038–1044; corrigendum, Bull. Lond. Math. Soc. 42 (2010), 948. Google Scholar

  • [8]

    Applebaum D., Some L2 properties of semigroups of measures on Lie groups, Semigroup Forum 79 (2009), 217–228. Google Scholar

  • [9]

    Applebaum D., Infinitely divisible central probability measures on compact Lie groups – regularity, semigroups and transition kernels, Ann. Probab. 39 (2011), 2474–2496. Google Scholar

  • [10]

    Applebaum D., Aspects of recurrence and transience for Levy processes in transformation groups and non-compact Riemannian symmetric pairs, J. Aust. Math. Soc. 94 (2013), 304–320. Google Scholar

  • [11]

    Applebaum D., Probability on Compact Lie Groups, Springer, Cham, 2014. Google Scholar

  • [12]

    Arthur J., An introduction to the trace formula, Harmonic Analysis, the Trace Formula, and Shimura varieties, Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 1–263. Google Scholar

  • [13]

    Berg C. and Forst G., Potential Theory on Locally Compact Abelian Groups, Springer, Berlin, 1975. Google Scholar

  • [14]

    Brekke L. and Freunde P. G. O., p-adic numbers in physics, Phys. Rep. 233 (1993), 1–66. Google Scholar

  • [15]

    Connes A., Consani C. and Marcolli M., The Weil proof and the geometry of the adèles class space, Algebra, Arithmetic, and Geometry. Vol. I, Progr. Math. 269, Birkhäuser, Boston (2009), 339–405. Google Scholar

  • [16]

    Davies E. B., Linear Operators and Their Spectra, Cambridge University Press, Cambridge, 2007. Google Scholar

  • [17]

    Del Muto M. and Figà-Talamanca A., Diffusion on locally compact ultrametric spaces, Expo. Math. 22 (2004), 197–211. Google Scholar

  • [18]

    Evans S. N., Local properties of Lévy processes on a totally disconnected group, J. Theoret. Probab. 2 (1989), 209–259. Google Scholar

  • [19]

    Feller W., An Introduction to Probability Theory and Its Applications, vol. 2, 2nd ed., Wiley & Sons, New York, 1971. Google Scholar

  • [20]

    Folland G. B., A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, Boca Raton, 2016. Google Scholar

  • [21]

    Gangolli R., Asymptotic behaviout of spectra of compact quotients of certain symmetric spaces, Acta Math. 121 (1968), 151–192. Google Scholar

  • [22]

    Gelbart S., An elementary introduction to the Langlands programme, Bull. Amer. Math. Soc. 10 (1984), 177–219. Google Scholar

  • [23]

    Gel’fand I. M., Graev M. I. and Pyatetskii-Shapiro I. I., Representation Theory and Automorphic Functions, W. B. Saunders, Philadelphia, 1969. Google Scholar

  • [24]

    Hazod W. and Siebert E., Stable Probability Measures on Euclidean Spaces and on Locally Compact Groups. Structural Properties and Limit Theorems, Math. Appl. 531, Kluwer, Dordrecht, 2001. Google Scholar

  • [25]

    Heyer H., Probability Measures on Locally Compact Groups, Springer, Berlin, 1977. Google Scholar

  • [26]

    Heyer H., Recent contributions to the embedding problem for probability measures on a locally compact group, J. Multivariate Anal. 19 (1986), 119–131. Google Scholar

  • [27]

    Karwowski W. and Vilela Mendes R., Hierarchical structures and asymmetric stochastic processes on p-adics and adeles, J. Math. Phys. 35 (1994), 4637–4650. Google Scholar

  • [28]

    Katznelson Y., An Introduction to Harmonic Analysis, 3rd ed., Cambridge University Press, Cambridge, 2004. Google Scholar

  • [29]

    Khrennikov A., p-adic analogues of the law of large numbers and the central limit theorem, Indag. Math. (N.S.) 8 (1997), 61–77. Google Scholar

  • [30]

    Koblitz N., p-adic Numbers, padic Analysis and Zeta Functions, Springer, Berlin, 1984. Google Scholar

  • [31]

    Kochubei A. N., Limit theorems for sums of p-adic random variables, Expo. Math. 16 (1998), 425–439. Google Scholar

  • [32]

    Lang S., Algebraic Number Theory, Addison–Wesley, Reading, 1970. Google Scholar

  • [33]

    Mackey G. W., Induced representations and the applications of harmonic analysis, The Scope and History of Commutative and Noncommutative Harmonic Analysis, Hist. Math. 5, American Mathematical Society, Providence (1992), 275–310. Google Scholar

  • [34]

    Parthasarathy K. R., Probability Measures on Metric Spaces, Academic Press, New York, 1967. Google Scholar

  • [35]

    Pruitt W. E. and Taylor S. J., The potential kernel and hitting probabilities for the general stable process in n, Trans. Amer. Math. Soc. 146 (1969), 299–321. Google Scholar

  • [36]

    Ramakrishnan D. and Valenza R. J., Fourier Analysis on Number Fields, Grad. Texts in Math. 186, Springer, New York, 1999. Google Scholar

  • [37]

    Reiter H., Classical Harmonic Analysis and Locally Compact Groups, Oxford University Press, Oxford, 1968. Google Scholar

  • [38]

    Rosenberg S., The Laplacian on a Riemannian Manifold, London Math. Soc. Stud. Texts 31, Cambridge University Press, Cambridge, 1997. Google Scholar

  • [39]

    Sato K.-I., Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, 1999. Google Scholar

  • [40]

    Shah R., Infinitely divisible measures on p-adic groups, J. Theoret. Probab. 4 (1991), 261–284. Google Scholar

  • [41]

    Shah R., Semistable measures and limit theorems on real and p-adic groups, Monatsh. Math. 115 (1993), 191–223. Google Scholar

  • [42]

    Stein E. M. and Shakarchi R., Fourier Analysis: An Introduction, Princeton University Press, Princeton, 2003. Google Scholar

  • [43]

    Sztonyk P., Transition density estimates for jump Lévy processes, Stochastic Process. Appl. 121 (2011), 1245–1265. Google Scholar

  • [44]

    Tate J. T., Fourier analysis in number fields and Hecke’s zeta–functions, Algebraic Number Theory, Academic Press, New York (1967), 305–347. Google Scholar

  • [45]

    Tenenbaum G. and Mendès-France M., The Prime Numbers and Their Distributions, American Mathematical Society, Providence, 2000. Google Scholar

  • [46]

    Terras A., Harmonic Analysis on Symmetric Spaces – Euclidean Space, The Sphere and the Poincaré Upper Half-Plane, 2nd ed., Springer, New York, 2013. Google Scholar

  • [47]

    Urban R., Markov processes on the adeles and Dedekind’s zeta function, Statist. Probab. Lett. 82 (2012), 1583–1589. Google Scholar

  • [48]

    Yasuda K., Additive processes on local fields, J. Math. Sci. Univ. Tokyo 3 (1996), 629–654. Google Scholar

  • [49]

    Yasuda K., On infinitely divisible distributions on locally compact groups, J. Theoret. Probab. 13 (2000), 635–657. Google Scholar

  • [50]

    Yasuda K., Semi-stable processes on local fields, Tohoku Math. J. (2) 58 (2006), 419–431. Google Scholar

  • [51]

    Yasuda K., Markov processes on the Adeles and representations of Euler products, J. Theoret. Probab. 23 (2010), 748–769. Google Scholar

  • [52]

    Yasuda K., Markov processes on the adeles and Chebychev function, Statist. Probab. Lett. 83 (2013), 238–244. Google Scholar

About the article

Received: 2016-03-09

Revised: 2016-05-23

Published Online: 2016-06-19

Published in Print: 2017-05-01

Citation Information: Forum Mathematicum, Volume 29, Issue 3, Pages 501–517, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2016-0067.

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