Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary
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2024
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| author | Eberle, Andreas Lörler, Francis |
| author_facet | Eberle, Andreas Lörler, Francis |
| contents | Non-reversible lifts reduce the relaxation time of reversible diffusions at most by a square root. For reversible diffusions on domains in Euclidean space, or, more generally, on a Riemannian manifold with boundary, non-reversible lifts are in particular given by the Hamiltonian flow on the tangent bundle, interspersed with random velocity refreshments, or perturbed by Ornstein-Uhlenbeck noise, and reflected at the boundary. In order to prove that for certain choices of parameters, these lifts achieve the optimal square-root reduction up to a constant factor, precise upper bounds on relaxation times are required. A key tool for deriving such bounds by space-time Poincaré inequalities is a quantitative space-time divergence lemma. Extending previous work of Cao, Lu and Wang, we establish such a divergence lemma with explicit constants for general locally convex domains with smooth boundary in Riemannian manifolds satisfying a lower, not necessarily positive, curvature bound. As a consequence, we prove optimality of the lifts described above up to a constant factor, provided the deterministic transport part of the dynamics and the noise are adequately balanced. Our results show for example that an integrated Ornstein-Uhlenbeck process on a locally convex domain with diameter $d$ achieves a relaxation time of the order $d$, whereas, in general, the Poincaré constant of the domain is of the order $d^2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_16710 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary Eberle, Andreas Lörler, Francis Probability Analysis of PDEs Functional Analysis 60J25, 60J35, 58J65, 58J05, 35H10 Non-reversible lifts reduce the relaxation time of reversible diffusions at most by a square root. For reversible diffusions on domains in Euclidean space, or, more generally, on a Riemannian manifold with boundary, non-reversible lifts are in particular given by the Hamiltonian flow on the tangent bundle, interspersed with random velocity refreshments, or perturbed by Ornstein-Uhlenbeck noise, and reflected at the boundary. In order to prove that for certain choices of parameters, these lifts achieve the optimal square-root reduction up to a constant factor, precise upper bounds on relaxation times are required. A key tool for deriving such bounds by space-time Poincaré inequalities is a quantitative space-time divergence lemma. Extending previous work of Cao, Lu and Wang, we establish such a divergence lemma with explicit constants for general locally convex domains with smooth boundary in Riemannian manifolds satisfying a lower, not necessarily positive, curvature bound. As a consequence, we prove optimality of the lifts described above up to a constant factor, provided the deterministic transport part of the dynamics and the noise are adequately balanced. Our results show for example that an integrated Ornstein-Uhlenbeck process on a locally convex domain with diameter $d$ achieves a relaxation time of the order $d$, whereas, in general, the Poincaré constant of the domain is of the order $d^2$. |
| title | Space-time divergence lemmas and optimal non-reversible lifts of diffusions on Riemannian manifolds with boundary |
| topic | Probability Analysis of PDEs Functional Analysis 60J25, 60J35, 58J65, 58J05, 35H10 |
| url | https://arxiv.org/abs/2412.16710 |