Spread out random walks on homogeneous spaces

Prohaska, Roland

doi:10.1017/etds.2020.98

Cited by 4 publications

(4 citation statements)

References 26 publications

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“…Their approach was recently refined and generalized to some nilmanifolds in the works [45, 46, 47] of He–de Saxcé and He–Lakrec–Lindenstrauss. Outside the realm of nilmanifolds, quantitative results on the convergence of include the work of Buenger [21, §3] and Khalil–Luethi [53], who consider some classes of measures supported on compact-by-solvable groups, and work of the first-named author [78] on spread-out measures.…”

Section: Recurrence Equidistribution Topology Of Homogeneous Measuresmentioning

confidence: 99%

Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska

Sert

Shi

2023

Forum of Mathematics, Sigma

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Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu $ -stationary probability measures on $G/\Lambda $ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on $G/\Lambda $ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda $ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.

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Section: Recurrence Equidistribution Topology Of Homogeneous Measuresmentioning

confidence: 99%

Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska

Sert

Shi

2023

Forum of Mathematics, Sigma

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“…Their approach was recently refined and generalized to some nilmanifolds in the works [42,43,44] of He-de Saxcé and He-Lakrec-Lindenstrauss. Outside the realm of nilmanifolds, quantitative results on the convergence of µ * n * δ x include the work of Buenger [18, §3] and Khalil-Luethi [50], who consider some classes of measures supported on compact-by-solvable groups, and work of the first-named author [74] on spread-out measures.…”

Section: Recurrence Equidistribution Topology Of Homogeneous Measuresmentioning

confidence: 99%

Expanding measures: Random walks and rigidity on homogeneous spaces

Prohaska¹,

Sert²,

Shi³

2021

Preprint

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Let G be a real Lie group, Λ < G a lattice and H G a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures µ on H and, applying recent work of Eskin-Lindenstrauss, prove that µ-stationary probability measures on G/Λ are homogeneous. Transferring a construction by Benoist-Quint and drawing on ideas of Eskin-Mirzakhani-Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on G/Λ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in G/Λ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons-Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a non-conformal and weighted setting. Contents1. Introduction 1 2. H-expansion: Definition and basic properties 11 3. Examples of H-expanding measures 14 4. Measure rigidity 22 5. Countability of homogeneous subspaces 28 6. Height functions with contraction properties 31 7. Recurrence, equidistribution, topology of homogeneous measures 41 8. Birkhoff genericity 45 9. Connections to Diophantine approximation on fractals 50 Appendix A. Epimorphic subgroups and subalgebras 55 References 57

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“…Note however that we could not use existing concepts, since they are only well-defined for so-called "ψ-irreducible" chains, which our random walks are generally not. We refer to Meyn-Tweedie [11] for further reading on general state space Markov chains, and to [12] for a treatment of random walks with "spread out" increment distribution µ, which are ψ-irreducible.…”

Section: Periodicitymentioning

confidence: 99%

“…Answers are available only in special cases: Breuillard [7] established (1.2) for certain measures µ supported on unipotent subgroups, Buenger [8] proved it for some sparse solvable measures, and in previous work the author dealt with the case of spread out measures [12].…”

Section: Introductionmentioning

confidence: 99%

Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

Prohaska

2019

Preprint

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The purpose of this note is to discuss three aspects of weak* convergence of the n-step distributions of random walks on finite volume homogeneous spaces G/Γ of semisimple real Lie groups. First, we investigate the obvious obstruction to the upgrade from Cesàro to non-averaged convergence: periodicity.We give examples where it occurs and conditions under which it does not. In a second part, we prove convergence towards Haar measure with exponential speed from almost every starting point. Finally, we establish strong uniformity for the Cesàro convergence towards Haar measure for uniquely ergodic random walks.

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Spread out random walks on homogeneous spaces

Cited by 4 publications

References 26 publications

Expanding measures: Random walks and rigidity on homogeneous spaces

Expanding measures: Random walks and rigidity on homogeneous spaces

Expanding measures: Random walks and rigidity on homogeneous spaces

Aspects of Convergence of Random Walks on Finite Volume Homogeneous Spaces

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