Hyper-contractivity and entropy decay in discrete time
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918345265119232 |
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| author | Salez, Justin |
| author_facet | Salez, Justin |
| contents | Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,π)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with $p< q$ implies a one-step entropy contraction estimate of the form ${\mathrm H}(μT\,|\,π)\le θ\, {\mathrm H}(μ\,|\,π)$, with $θ=p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17305 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hyper-contractivity and entropy decay in discrete time Salez, Justin Probability 60j05, 60j25 Consider a measure-preserving transition kernel $T$ on an arbitrary probability space $(\mathbb X,\mathcal cA,π)$. In this level of generality, we prove that a one-step hyper-contractivity estimate of the form $\|T\|_{p\to q}\le 1$ with $p< q$ implies a one-step entropy contraction estimate of the form ${\mathrm H}(μT\,|\,π)\le θ\, {\mathrm H}(μ\,|\,π)$, with $θ=p/q$. Neither reversibility, nor any sort of regularity is required. This static implication is simultaneously simpler and stronger than the celebrated dynamic relation between exponential hyper-contractivity and exponential entropy decay along continuous-time Markov semi-groups. |
| title | Hyper-contractivity and entropy decay in discrete time |
| topic | Probability 60j05, 60j25 |
| url | https://arxiv.org/abs/2602.17305 |