{"paper":{"title":"Weighted Helmholtz--Hodge decompositions, Lyapunov functions, and invariant measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerald Trutnau, Haesung Lee","submitted_at":"2026-05-25T11:16:37Z","abstract_excerpt":"We study weighted Helmholtz--Hodge decompositions of drift vector fields associated with second-order diffusion operators on $\\mathbb{R}^d$, $d\\ge 2$. Given a decomposition of the form \\[\n  \\mathbf{G}=A\\nabla\\Phi+\\mathbf{B}, \\] we relate the weighted divergence-free condition $\\mathrm{div}_{\\mu}(\\mathbf{B})=0$, where $\\mu=e^{2\\Phi}dx$, to infinitesimal invariance of $\\mu$ for the operator \\[\n  \\frac12 \\mathrm{trace}(A\\nabla^2)+\\langle \\mathbf{G},\\nabla\\cdot\\rangle. \\] We compare weighted, orthogonal, and strictly orthogonal Helmholtz--Hodge decompositions and show that uniqueness of the infini"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25715","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.25715/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}