Logarithmic Sobolev inequalities for non-equilibrium steady states
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917965337722880 |
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| author | Monmarché, Pierre Wang, Songbo |
| author_facet | Monmarché, Pierre Wang, Songbo |
| contents | 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 [J. Stat. Phys., 46(5-6):1159-1194, 1987, 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 [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_07858 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Logarithmic Sobolev inequalities for non-equilibrium steady states Monmarché, Pierre Wang, Songbo Probability Functional Analysis 26D10, 60J60, 35B40, 93E20 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 [J. Stat. Phys., 46(5-6):1159-1194, 1987, 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 [Ann. Appl. Probab., 33(6A):4608-4644, 2023]. The second method combines the Wasserstein-2 contraction method, used in [Ann. Henri Lebesgue, 6:941-973, 2023] to prove a Poincaré inequality in some non-equilibrium cases, with Wang's hypercontractivity results [Potential Anal., 53(3):1123-1144, 2020]. |
| title | Logarithmic Sobolev inequalities for non-equilibrium steady states |
| topic | Probability Functional Analysis 26D10, 60J60, 35B40, 93E20 |
| url | https://arxiv.org/abs/2309.07858 |