{"paper":{"title":"Quadratic variation and local times of the horizontal component of the Peano curve (square filling curve)","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Darlington Hove, Farai J. Mhlanga, Phumlani L. Zondi, Rafa{\\l} M. {\\L}ochowski","submitted_at":"2025-01-14T09:36:10Z","abstract_excerpt":"We show that the horizontal component of the Peano curve has quadratic variation equal the limit of quadratic variations along the Lebesgue partitions for grids of the form $3^{-n}p\\mathbb{Z}+3^{-n}r$, $n=1,2,\\ldots$, where $p$ is a rational number, while $r$ is irrational number, but the value of such quadratic variation depends on $p$.\n  This also yields that the horizontal component of the Peano curve is an example of a deterministic function possessing local time (density of the occupation measure) with respect to the Lebesgue measure, whose local time can be expressed as the limit of norm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.07966","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.07966/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}