{"paper":{"title":"Analysis of the archetypal functional equation in the non-critical case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Gregory Derfel, Leonid V. Bogachev, Stanislav A. Molchanov","submitted_at":"2014-09-22T09:20:45Z","abstract_excerpt":"We study the archetypal functional equation of the form $y(x)=\\iint_{\\mathbb{R}^2} y(a(x-b))\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)$ ($x\\in\\mathbb{R}$), where $\\mu$ is a probability measure on $\\mathbb{R}^2$; equivalently, $y(x)=\\mathbb{E}\\{y(\\alpha(x-\\beta))\\}$, where $\\mathbb{E}$ is expectation with respect to the distribution $\\mu$ of random coefficients $(\\alpha,\\beta)$. Existence of non-trivial (i.e., non-constant) bounded continuous solutions is governed by the value $K:=\\iint_{\\mathbb{R}^2}\\ln|a|\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)=\\mathbb{E}\\{\\ln|\\alpha|\\}$; namely, under mild technical conditions no"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6126","kind":"arxiv","version":2},"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"}