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The presented diffusions are time reversible and can be constructed for any prespecified stationary distribution on the torus, including highly multimodal mixtures. We give results on asymptotic likelihood theory, allowing one-sample inference and tests of linear hypotheses for $ k $ groups of diffusions, including homogeneity. We show that exact and direct diffusion bridge simulation is possible too. A class of circular jump processes with similar properties is also proposed. Several numerical experiments illustrate the methodology for the circular and two-dimensional torus cases. The new family of diffusions is applied (i) to test several homogeneity hypotheses on the movement of ants and (ii) to simulate bridges between the three-dimensional backbones of two related proteins.<\/jats:p>","DOI":"10.1093\/biomet\/asaf050","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T07:52:00Z","timestamp":1751615520000},"source":"Crossref","is-referenced-by-count":2,"title":["A family of toroidal diffusions with exact likelihood inference"],"prefix":"10.1093","volume":"113","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9224-4111","authenticated-orcid":false,"given":"E","family":"Garc\u00eda-Portugu\u00e9s","sequence":"first","affiliation":[{"name":"Department of Statistics, Universidad Carlos de Madrid , Av. de la Universidad 30 , 28911 Legan\u00e9s,","place":["Spain"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7233-5377","authenticated-orcid":false,"given":"M","family":"S\u00f8rensen","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, University of Copenhagen , Universitetsparken 5 , DK-2100 Copenhagen,","place":["Denmark"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2025,7,9]]},"reference":[{"key":"2026032607093867700_asaf050-B1","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511809781","volume-title":"L\u00e9vy Processes and Stochastic Calculus","author":"Applebaum","year":"2009","edition":"2nd ed"},{"key":"2026032607093867700_asaf050-B2","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1214\/aos\/1013699998","article-title":"The control of the false discovery rate in multiple testing under dependency","volume":"29","author":"Benjamini","year":"2001","journal-title":"Ann. 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