Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincaré--Lions inequalities
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912434311135232 |
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| author | Brigati, Giovanni Stoltz, Gabriel Wang, Andi Q. Wang, Lihan |
| author_facet | Brigati, Giovanni Stoltz, Gabriel Wang, Andi Q. Wang, Lihan |
| contents | We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called \emph{weak confinement}. Indeed, any $\mathrm{L}^\infty$ distribution (in position and velocity) relaxes to equilibrium over time, and we quantify the convergence rate. In our situation, the spatial equilibrium distribution does not satisfy a Poincaré inequality. Instead, we assume a weighted Poincaré inequality, which allows for fat-tail or sub-exponential potential energies. We provide constructive and fully explicit estimates in $\mathrm{L}^2$-norm for $\mathrm{L}^\infty$ initial data. A key-ingredient is a new space-time weighted Poincaré--Lions inequality, entailing, in turn, a weak Poincaré--Lions inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincaré--Lions inequalities Brigati, Giovanni Stoltz, Gabriel Wang, Andi Q. Wang, Lihan Probability Analysis of PDEs Computation We study the long-time behavior of the underdamped Langevin dynamics, in the case of so-called \emph{weak confinement}. Indeed, any $\mathrm{L}^\infty$ distribution (in position and velocity) relaxes to equilibrium over time, and we quantify the convergence rate. In our situation, the spatial equilibrium distribution does not satisfy a Poincaré inequality. Instead, we assume a weighted Poincaré inequality, which allows for fat-tail or sub-exponential potential energies. We provide constructive and fully explicit estimates in $\mathrm{L}^2$-norm for $\mathrm{L}^\infty$ initial data. A key-ingredient is a new space-time weighted Poincaré--Lions inequality, entailing, in turn, a weak Poincaré--Lions inequality. |
| title | Explicit convergence rates of underdamped Langevin dynamics under weighted and weak Poincaré--Lions inequalities |
| topic | Probability Analysis of PDEs Computation |
| url | https://arxiv.org/abs/2407.16033 |