Abstract.
We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped with entropy-like ``distances''. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general.
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Received: 31 October 2000 / Revised version: 21 February 2003 / Published online: 12 May 2003
L. Miclo also thanks the hospitality and support of the Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil, where part of this work was done.
Mathematics Subject Classification (2000): 60J05, 60J22, 37A30, 37A25, 39A11, 39A12, 46E39, 28A33, 47D07
Key words or phrases: Lipschitz contraction – Generalized relative entropy – Markov kernel – Dobrushin's ergodic coefficient – Orlicz norm – Dirichlet form – Spectral gap – Modified logarithmic Sobolev inequality – Inhomogeneous Gaussian chains – Loose of memory property
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Del Moral, P., Ledoux, M. & Miclo, L. On contraction properties of Markov kernels. Probab. Theory Relat. Fields 126, 395–420 (2003). https://doi.org/10.1007/s00440-003-0270-6
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DOI: https://doi.org/10.1007/s00440-003-0270-6