{"paper":{"title":"Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math-ph","math.MP","math.PR"],"primary_cat":"math.OC","authors_text":"Hilbert J. Kappen, Joris Bierkens, Michael Chertkov, Vladimir Y. Chernyak","submitted_at":"2013-03-01T09:59:31Z","abstract_excerpt":"We consider long term average or `ergodic' optimal control poblems with a special structure: Control is exerted in all directions and the control costs are proportional to the square of the norm of the control field with respect to the metric induced by the noise. The long term stochastic dynamics on the manifold will be completely characterized by the long term density $\\rho$ and the long term current density $J$. As such, control problems may be reformulated as variational problems over $\\rho$ and $J$. We discuss several optimization problems: the problem in which both $\\rho$ and $J$ are var"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.0126","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}