{"paper":{"title":"Free Brownian motion and free convolution semigroups: multiplicative case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.FA","authors_text":"Ping Zhong","submitted_at":"2012-10-23T00:08:18Z","abstract_excerpt":"We consider a pair of probability measures $\\mu,\\nu$ on the unit circle such that $\\Sigma_{\\lambda}(\\eta_{\\nu}(z))=z/\\eta_{\\mu}(z)$. We prove that the same type of equation holds for any $t\\geq 0$ when we replace $\\nu$ by $\\nu\\boxtimes\\lambda_t$ and $\\mu$ by $\\mathbb{M}_t(\\mu)$, where $\\lambda_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\\mathbb{C}$ and $\\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.6090","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":""},"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"}