Centered Sobolev inequality and exponential convergence in Φ-entropy

Lingyan Cheng and Liming Wu

Statistics & Probability Letters, 2019, vol. 148, issue C, 101-111

Abstract: In this short paper we find that the Sobolev inequality 1p−2∫fpdμ2p−∫f2dμ≤C∫|∇f|2dμ(p≥0) is equivalent to the exponential convergence of the Markov diffusion semigroup (Pt) to the invariant measure μ, in some Φ-entropy. We provide the estimate of the exponential convergence in total variation and a bounded perturbation result under the Sobolev inequality. Finally in the one-dimensional case we get some two-sided estimates of the Sobolev constant by means of the generalized Hardy inequality.

Keywords: Sobolev inequality; Diffusion process; ϕ-entropy; Exponential convergence (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1016/j.spl.2019.01.002

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