On contraction properties of Markov kernels

• Published in: Probability theory and related fields Vol. 126; no. 3; pp. 395 - 420
• Main Authors: MICLO, L, LEDOUX, M, DEL MORAL, P
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer 01-06-2003; Berlin Springer Nature B.V; New York, NY
• Subjects: Dirichlet problem; Entropy; Ergodic processes; Exact sciences and technology; Finite differences and functional equations; Functional analysis; Global analysis, analysis on manifolds; Kernels; Markov processes; Mathematical analysis; Mathematics; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Markov process; Sobolev constant; Lipschitz contraction; Gaussian process; Dobrushin ergodic coefficient; Invariant measure; Markov kernel; Probability theory; Dirichlet form; Orlicz space; Probability measure
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-003-0270-6

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.