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Equidistribution of mass for random processes on finite-volume spaces

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Abstract

Let G be a real Lie group, Λ ⊆ G a lattice, and X = G/Λ. We fix a probability measure μ on G and consider the left random walk induced on X. It is assumed that μ is aperiodic, has a finite first moment, spans a semisimple algebraic group without compact factors, and has two non mutually singular convolution powers. We show that for every starting point x ∈ X, the n-th step distribution μn*δx of the walk weak-* converges toward some homogeneous probability measure on X.

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References

  1. T. Bénard and N. de Saxcé, Random walks with bounded first moment on finite-volume spaces, Geometric and Functional Analysis 32 (2022), 687–724.

    MathSciNet  MATH  Google Scholar 

  2. Y. Benoist and J.-F. Quint, Introduction to random walks on homogeneous spaces, Japanese Journal of Mathematics 7 (2012), 135–166.

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Benoist and J.-F. Quint. Stationnary measures and invariant subsets of homogeneous spaces. III, Annals of Mathematics 178 (2013), 1017–1059.

    Article  MathSciNet  MATH  Google Scholar 

  4. S. Foguel, Iterates of a convolution on a non abelian group, Annales de l’Institut Henri Poincaré. Section B. Calcul des Probabilités et Statistique 11 (1975), 199–202.

    MathSciNet  MATH  Google Scholar 

  5. D. Ornstein and L. Sucheston, An operator theorem on L1convergence to zero with applications to Markov kernels, Annals of Mathematical Statistics 41 (1970), 1631–1639.

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Prohaska, Aspects of convergence of random walks on finite volume homogeneous spaces. https://arxiv.org/abs/1910.11639.

  7. M. Ratner, Distribution rigidity for unipotent actions on homogeneous spaces, Bulletin of the American Mathematical Society 24 (1991), 321–325.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre for Mathematical Sciences, University of Cambridge, CB3 0WB, Cambridge, UK

    Timothée Bénard

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  1. Timothée Bénard
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Correspondence to Timothée Bénard.

Additional information

The author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 803711)

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Bénard, T. Equidistribution of mass for random processes on finite-volume spaces. Isr. J. Math. 255, 417–422 (2023). https://doi.org/10.1007/s11856-022-2422-3

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  • DOI: https://doi.org/10.1007/s11856-022-2422-3

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