Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method
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arXiv
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2026
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| _version_ | 1866908954650476544 |
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| author | Fan, Zexi Li, Bowen Lu, Jianfeng |
| author_facet | Fan, Zexi Li, Bowen Lu, Jianfeng |
| contents | In this note, we consider the underdamped Langevin dynamics with invariant measure $μ(\mathrm{d}x\,\mathrm{d}v) \propto e^{-U(x)-|v|^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $μ_x(\mathrm{d}x)\propto e^{-U(x)}\,\mathrm{d}x$ satisfies a Poincaré inequality with constant $m>0$, and that $\nabla^2 U\ge -K\,\mathrm{Id}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault--Mouhot--Schmeiser, employing a gap-shifted corrector \begin{equation*}
A_m=(m- L_{\mathrm{o}})^{-1}(L_aΠ_v)^*, \end{equation*} where $L_{\mathrm{o}}=Δ_x-\nabla U\cdot\nabla_x$ is the overdamped generator, $L_a$ is the generator of the Hamiltonian flow, and $Π_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate \begin{equation*}
Λ=\frac{1}{6\Bigl(\sqrt{2+\frac{K}{2m}}+\sqrt{4+\frac{K}{2m}}\Bigr)}\sqrt{m}. \end{equation*} In particular, for convex $U$, this recovers the optimal $O(\sqrt{m})$ rate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method Fan, Zexi Li, Bowen Lu, Jianfeng Analysis of PDEs In this note, we consider the underdamped Langevin dynamics with invariant measure $μ(\mathrm{d}x\,\mathrm{d}v) \propto e^{-U(x)-|v|^2/2}\,\mathrm{d}x\,\mathrm{d}v$. Assume that the position marginal $μ_x(\mathrm{d}x)\propto e^{-U(x)}\,\mathrm{d}x$ satisfies a Poincaré inequality with constant $m>0$, and that $\nabla^2 U\ge -K\,\mathrm{Id}$ for some $K\ge 0$. We revisit the modified $L^2$ method of Dolbeault--Mouhot--Schmeiser, employing a gap-shifted corrector \begin{equation*} A_m=(m- L_{\mathrm{o}})^{-1}(L_aΠ_v)^*, \end{equation*} where $L_{\mathrm{o}}=Δ_x-\nabla U\cdot\nabla_x$ is the overdamped generator, $L_a$ is the generator of the Hamiltonian flow, and $Π_v$ denotes averaging over the velocity variable. We establish an explicit hypocoercive $L^2$-convergence rate \begin{equation*} Λ=\frac{1}{6\Bigl(\sqrt{2+\frac{K}{2m}}+\sqrt{4+\frac{K}{2m}}\Bigr)}\sqrt{m}. \end{equation*} In particular, for convex $U$, this recovers the optimal $O(\sqrt{m})$ rate. |
| title | Sharp hypocoercive convergence estimates for underdamped Langevin dynamics via the modified $L^2$ method |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.10068 |