{"paper":{"title":"Free positive multiplicative Brownian motion and the free additive convolution of semicircle and uniform distribution","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CO","math.FA","math.OA"],"primary_cat":"math.PR","authors_text":"Martin Auer","submitted_at":"2025-05-09T12:16:29Z","abstract_excerpt":"The free positive multiplicative Brownian motion $(h_t)_{t\\geq0}$ is the large $N$ limit in non-commutative distribution of matrix geometric Brownian motion. It can be constructed by setting $h_t:=g_{t/2}g_{t/2}^*$, where $(g_t)_{t\\geq0}$ is a free multiplicative Brownian motion, which is the large $N$ limit in non-commutative distribution of the Brownian motion in $\\operatorname{Gl}(N,\\mathbb{C})$. One key property of $(h_t)_{t\\geq0}$ is the fact that the corresponding spectral distributions $(\\nu_t)_{t\\geq0}\\subset M^1((0,\\infty))$ form a semigroup w.r.t. free multiplicative convolution. In "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.05984","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2505.05984/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"}