{"paper":{"title":"On bounded continuous solutions of the archetypal equation with rescaling","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":"2014-09-19T13:24:06Z","abstract_excerpt":"The `archetypal' equation with rescaling is given by $y(x)=\\iint_{\\mathbb{R}^2} y(a(x-b))\\,\\mu(\\mathrm{d}a,\\mathrm{d}b)$ ($x\\in\\mathbb{R}$), where $\\mu$ is a probability measure; equivalently, $y(x)=\\mathbb{E}\\{y(\\alpha(x-\\beta))\\}$, with random $\\alpha,\\beta$ and $\\mathbb{E}$ denoting expectation. Examples include: (i) functional equation $y(x)=\\sum_{i} p_{i} y(a_i(x-b_i))$; (ii) functional-differential (`pantograph') equation $y'(x)+y(x)=\\sum_{i} p_{i} y(a_i(x-c_i))$ ($p_{i}>0$, $\\sum_{i} p_{i}=1$). Interpreting solutions $y(x)$ as harmonic functions of the associated Markov chain $(X_n)$, w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5648","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"}