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<p>We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley–Stroock and Aida–Shigekawa perturbation arguments (Holley and Stroock, J. Stat. Phys. <b>46</b>(5–6), 1159–1194 <CitationRef CitationID="CR16">1987</CitationRef>; Aida and Shigekawa, J. Funct. Anal. <b>126</b>(2), 448–475 <CitationRef CitationID="CR1">1994</CitationRef>), the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti (<CitationRef CitationID="CR7">2023</CitationRef>). The second method combines the Wasserstein-2 contraction method, used in Monmarché (Ann. Henri Lebesgue <b>6</b>, 941–973 <CitationRef CitationID="CR26">2023</CitationRef>) to prove a Poincaré inequality in some non-equilibrium cases, with Wang’s hypercontractivity results (Wang, Ann. Probab. <b>37</b>(4), 1587–1604 <CitationRef CitationID="CR32">2009</CitationRef>).</p>

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Logarithmic Sobolev Inequalities for Non-equilibrium Steady States

  • 期刊论文

  • 发表日期：

    2025年3月24日

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  • Pierre Monmarché,
  • Songbo Wang

摘要

We consider two methods to establish log-Sobolev inequalities for the invariant measure of a diffusion process when its density is not explicit and the curvature is not positive everywhere. In the first approach, based on the Holley–Stroock and Aida–Shigekawa perturbation arguments (Holley and Stroock, J. Stat. Phys. 46(5–6), 1159–1194 1987; Aida and Shigekawa, J. Funct. Anal. 126(2), 448–475 1994), the control on the (non-explicit) perturbation is obtained by stochastic control methods, following the comparison technique introduced by Conforti (2023). The second method combines the Wasserstein-2 contraction method, used in Monmarché (Ann. Henri Lebesgue 6, 941–973 2023) to prove a Poincaré inequality in some non-equilibrium cases, with Wang’s hypercontractivity results (Wang, Ann. Probab. 37(4), 1587–1604 2009).