{"paper":{"title":"The orthogonal complements of $H^1(\\mathbb{R})$ in its regular Dirichlet extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jiangang Ying, Liping Li, Yuncong Shen","submitted_at":"2016-11-21T13:38:43Z","abstract_excerpt":"Consider the regular Dirichlet extension $(\\mathcal{E},\\mathcal{F})$ for one-dimensional Brownian motion, that $H^1(\\mathbb{R})$ is a subspace of $\\mathcal{F}$ and $\\mathcal{E}(f,g)=\\frac12\\mathbf{D}(f,g)$ for $f,g\\in H^1(\\mathbb{R})$. Both $H^1(\\mathbb{R})$ and $\\mathcal{F}$ are Hilbert spaces under $\\mathcal{E}_\\alpha$ and hence there is $\\alpha$-orthogonal compliment $\\mathcal{G}_\\alpha$. We give the explicit expression for functions in $\\mathcal{G}_\\alpha$ which then can be described by another two spaces. On the two spaces, there is a natural Dirichlet form in the wide sense and by the da"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06782","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":""},"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"}