On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Saloff‐Coste, Laurent

doi:10.1007/bf02922178

Cited by 10 publications

(12 citation statements)

References 18 publications

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“…Indeed, the spectral gap stays bounded and has a non-negative limit (which we shall compute later), whereas t (n) was shown by Saloff-Coste to be a O(log n). Similar results are presented in [SC04] in the broader setting of simple compact Lie groups or compact symmetric spaces, but without a proof of the cut-off phenomenon (Saloff-Coste gave a window for t (n) for every p ∈ [1, +∞]). The main result of our paper is that a cut-off indeed occurs for every p ∈ [1, +∞(, for every classical simple compact Lie group or classical simple compact symmetric space, and with a cut-off time equal to log n or 2 log n depending on the type of the space considered.…”

supporting

confidence: 60%

“…It has also been investigated by Chen and Saloff-Coste for Markov processes on continuous spaces, e.g. spheres and Lie groups; see in particular [SC94, SC04,CSC08] and the discussion of §1.4. However, in this case, cut-offs are easier to prove for the L p>1 -norm of p t (x) − 1, where p t (x) is the density of the process at time t and point x with respect to the equilibrium measure.…”

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confidence: 99%

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The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Méliot

2013

Potential Anal

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In this paper, we prove the cut-off phenomenon in total variation distance for the Brownian motions traced on the classical symmetric spaces of compact type, that is to say:(1) the classical simple compact Lie groups: special orthogonal groups, special unitary groups and compact symplectic groups;(2) the real, complex and quaternionic Grassmannian varieties (including the real spheres, and the complex or quaternionic projective spaces);(3) the spaces of real, complex and quaternionic structures.Denoting µt the law of the Brownian motion at time t, we give explicit lower bounds for d TV (µt, Haar) if t < t cut-off = α log n, and explicit upper bounds if t > t cut-off . This provides in particular an answer to some questions raised in recent papers by Chen and Saloff-Coste. Our proofs are inspired by those given by Rosenthal and Porod for products of random rotations in SO(n), and by Diaconis and Shahshahani for products of random transpositions in Sn.

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confidence: 60%

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confidence: 99%

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Méliot

2013

Potential Anal

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“…The study of the cutoff phenomenon for Markov diffusion processes goes back at least to the works of Saloff-Coste [52,53] in relation notably with Nash-Sobolev type functional inequalities, heat kernel analysis, and Diaconis-Wilson probabilistic techniques. We also refer to the more recent work [46] for the case of diffusion processes on compact groups and symmetric spaces, in relation with group invariance and representation theory, a point of view inspired by the early works of Diaconis on Markov chains and of Saloff-Coste on diffusion processes.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Universal cutoff for Dyson Ornstein Uhlenbeck process

Boursier¹,

Chafäı²,

Labbé³

2021

Preprint

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We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, entropy and Wasserstein. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions. Contents 1. Introduction and main results 1 2. Additional comments and open problems 9 3. Cutoff phenomenon for the OU 14 4. General exactly solvable aspects 15 5. The random matrix cases 18 6. Cutoff phenomenon for the DOU in TV and Hellinger 21 7. Cutoff phenomenon for the DOU in Wasserstein 26 Appendix A. Distances and divergences 27 Appendix B. Convexity and its dynamical consequences 30 Acknowledgements 33 References 33

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“…In the end, we see that L (4) is nothing else than It is well known that the Casimir operator of any compact semi simple Lie group has a constant Ricci curvature (see for example [11], prop. 3.17, or [25]). However, the explicit constant is not straightforward to compute, and for the sake of completeness, we provide for this an easy way through the use of the Γ 2 operator.…”

Section: Diffusion Processes On the Interior Of The Deltoid Curvementioning

confidence: 99%

Curvature dimension bounds on the deltoid model

Bakry¹,

Zribi²

2016

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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The deltoid curve in R 2 is the boundary of a domain on which there exist probability measures and orthogonal polynomials for theses measures which are eigenvectors of diffusion operators. As such, they may be considered as a two dimensional extension of the classical Jacobi operators. They belong to one of the 11 families of such bounded domains in R 2 . We study the curvature-dimension inequalities associated to these operators, and deduce various bounds on the associated polynomials, together with Sobolev inequalities related to the associated Dirichlet forms

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On the convergence to equilibrium of Brownian motion on compact simple Lie groups

Cited by 10 publications

References 18 publications

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

The Cut-off Phenomenon for Brownian Motions on Compact Symmetric Spaces

Universal cutoff for Dyson Ornstein Uhlenbeck process

Curvature dimension bounds on the deltoid model

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