{"paper":{"title":"Nash inequality for Diffusion Processes Associated with Dirichlet Distributions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Feng-Yu Wang, Weiwei Zhang","submitted_at":"2018-01-28T11:13:00Z","abstract_excerpt":"For any $N\\ge 2$ and $\\alpha=(\\alpha_1,\\cdots, \\alpha_{N+1})\\in (0,\\infty)^{N+1}$, let $\\mu^{(N)}_{\\alpha}$ be the Dirichlet distribution with parameter $\\alpha$ on the set $\\Delta^{ (N)}:= \\{ x \\in [0,1]^N:\\ \\sum_{1\\le i\\le N}x_i \\le 1 \\}.$ The multivariate Dirichlet diffusion is associated with the Dirichlet form\n  $${\\scr E}_\\alpha^{(N)}(f,f):= \\sum_{n=1}^N \\int_{ \\Delta^{(N)}} \\bigg(1-\\sum_{1\\le i\\le N}x_i\\bigg) x_n(\\partial_n f)^2(x)\\,\\mu^{(N)}_\\alpha(d x)$$ with Domain ${\\scr D}({\\scr E}_\\alpha^{(N)})$ being the closure of $C^1(\\Delta^{(N)})$. We prove the Nash inequality\n  $$\\mu_\\alpha^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09209","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"}