{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:51:43Z","timestamp":1760057503718,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,8]],"date-time":"2025-02-08T00:00:00Z","timestamp":1738972800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Magnus Ehrnrooth Foundation"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study the joint moments of occupation times on the legs of a diffusion spider. Specifically, we give a recursive formula for the Laplace transform of the joint moments, which extends earlier results for a one-dimensional diffusion. For a Bessel spider, of which the Brownian spider is a special case, our approach yields an explicit formula for the joint moments of the occupation times.<\/jats:p>","DOI":"10.3390\/e27020179","type":"journal-article","created":{"date-parts":[[2025,2,10]],"date-time":"2025-02-10T05:53:22Z","timestamp":1739166802000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Occupation Times on the Legs of a Diffusion Spider"],"prefix":"10.3390","volume":"27","author":[{"given":"Paavo","family":"Salminen","sequence":"first","affiliation":[{"name":"Faculty of Science and Engineering, \u00c5bo Akademi University, 20500 \u00c5bo, Finland"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4697-8245","authenticated-orcid":false,"given":"David","family":"Stenlund","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,8]]},"reference":[{"key":"ref_1","first-page":"37","article-title":"A diffusion with a discontinuous local time","volume":"Volume 52\u201353","author":"Walsh","year":"1978","journal-title":"Temps locaux"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/BFb0083979","article-title":"On Walsh\u2019s Brownian motions","volume":"Volume 1372","author":"Barlow","year":"1989","journal-title":"S\u00e9minaire de Probabilit\u00e9s, XXIII"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF00776238","article-title":"Construction of right processes from excursions","volume":"73","author":"Salisbury","year":"1986","journal-title":"Probab. Theory Related Fields"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"490","DOI":"10.1007\/s10959-015-0654-4","article-title":"On the joint law of the occupation times for a diffusion process on multiray","volume":"30","author":"Yano","year":"2017","journal-title":"J. Theoret. Probab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2215","DOI":"10.1214\/aop\/1176989018","article-title":"Diffusion processes on graphs and the averaging principle","volume":"21","author":"Freidlin","year":"1993","journal-title":"Ann. Probab."},{"key":"ref_6","first-page":"544","article-title":"On occupation time functionals for diffusion processes and birth-and-death processes on graphs","volume":"11","author":"Weber","year":"2001","journal-title":"Ann. Appl. Probab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1007\/BFb0083980","article-title":"Une extension multidimensionnelle de la loi de l\u2019arc sinus","volume":"Volume 1372","author":"Barlow","year":"1989","journal-title":"S\u00e9minaire de Probabilit\u00e9s, XXIII"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s11009-010-9203-x","article-title":"Random motion on simple graphs","volume":"14","author":"Papanicolaou","year":"2012","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1214\/ECP.v17-2319","article-title":"A scaling proof for Walsh\u2019s Brownian motion extended arc-sine law","volume":"17","author":"Vakeroudis","year":"2012","journal-title":"Electron. Commun. Probab."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2228","DOI":"10.1016\/j.spa.2014.02.005","article-title":"Harmonic functions on Walsh\u2019s Brownian motion","volume":"124","author":"Fitzsimmons","year":"2014","journal-title":"Stoch. Process. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1685","DOI":"10.1007\/s10959-015-0626-8","article-title":"Some limit theorems for heights of random walks on a spider","volume":"29","year":"2016","journal-title":"J. Theoret. Probab."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"330","DOI":"10.1007\/s10959-017-0788-7","article-title":"Limit theorems for local and occupation times of random walks and Brownian motion on a spider","volume":"32","year":"2019","journal-title":"J. Theoret. Probab."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1016\/j.orl.2016.05.001","article-title":"Exercising control when confronted by a (Brownian) spider","volume":"44","author":"Ernst","year":"2016","journal-title":"Oper. Res. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1921","DOI":"10.1016\/j.spa.2018.06.012","article-title":"Semimartingales on rays, Walsh diffusions, and related problems of control and stopping","volume":"129","author":"Karatzas","year":"2019","journal-title":"Stoch. Process. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.spa.2020.10.010","article-title":"Embedding of Walsh Brownian motion","volume":"134","author":"Bayraktar","year":"2021","journal-title":"Stoch. Process. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"104229","DOI":"10.1016\/j.spa.2023.104229","article-title":"Diffusion spiders: Green kernel, excessive functions and optimal stopping","volume":"167","author":"Lempa","year":"2024","journal-title":"Stoch. Process. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1287\/moor.2023.1359","article-title":"On the diameter of the stopped spider process","volume":"49","author":"Bednarz","year":"2024","journal-title":"Math. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1007\/s10998-024-00580-7","article-title":"In memoriam P\u00e1l R\u00e9v\u00e9sz (1934\u20132022)","volume":"89","year":"2024","journal-title":"Period. Math. Hungar."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"413","DOI":"10.1214\/154957807000000013","article-title":"On the constructions of the skew Brownian motion","volume":"3","author":"Lejay","year":"2006","journal-title":"Probab. Surv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"487","DOI":"10.1214\/13-STS442","article-title":"Advection-dispersion across interfaces","volume":"28","author":"Ramirez","year":"2013","journal-title":"Statist. Sci."},{"key":"ref_21","first-page":"183","article-title":"Occupation and local times for skew Brownian motion with applications to dispersion across an interface","volume":"21","author":"Appuhamillage","year":"2011","journal-title":"Ann. Appl. Probab."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"7299","DOI":"10.1016\/j.jcp.2012.07.011","article-title":"Simulating diffusion processes in discontinuous media: Numerical scheme with constant time step","volume":"231","author":"Lejay","year":"2012","journal-title":"J. Comput. Phys."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"346","DOI":"10.1111\/sjos.12033","article-title":"Is a Brownian motion skew?","volume":"41","author":"Lejay","year":"2014","journal-title":"Scand. J. Statist."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/s00186-017-0602-4","article-title":"Timing in the presence of directional predictability: Optimal stopping of skew Brownian motion","volume":"86","author":"Alvarez","year":"2017","journal-title":"Math. Meth. Oper. Res."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.insmatheco.2011.10.004","article-title":"Arbitrage in skew Brownian motion models","volume":"50","author":"Rossello","year":"2012","journal-title":"Insur. Math. Econom."},{"key":"ref_26","first-page":"100732","article-title":"A study of maximizing skew Brownian motion with applications to option pricing","volume":"17","author":"Hussain","year":"2024","journal-title":"J. Radiat. Res. Appl. Sci."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1805","DOI":"10.1007\/s11009-021-09884-4","article-title":"First Hitting Time of Brownian Motion on Simple Graph with Skew Semiaxes","volume":"24","author":"Dassios","year":"2022","journal-title":"Methodol. Comput. Appl. Probab."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"613","DOI":"10.1214\/18-AAP1432","article-title":"Serve the shortest queue and Walsh Brownian motion","volume":"29","author":"Atar","year":"2019","journal-title":"Ann. Appl. Probab."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1007\/s10959-020-00993-3","article-title":"On occupation times of one-dimensional diffusions","volume":"34","author":"Salminen","year":"2021","journal-title":"J. Theoret. Probab."},{"key":"ref_30","unstructured":"It\u00f4, K., and McKean, H.P. (1974). Diffusion Processes and Their Sample Paths, Springer."},{"key":"ref_31","unstructured":"Borodin, A.N., and Salminen, P. (2015). Handbook of Brownian Motion\u2014Facts and Formulae, Birkh\u00e4user Verlag. [2nd ed.]. Probability and Its Applications."},{"key":"ref_32","unstructured":"Kac, M. (August, January 31). On some connections between probability theory and differential and integral equations. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Berkeley\/Los Angeles, CA, USA."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"P1.40","DOI":"10.37236\/9843","article-title":"On the connection between Stirling numbers and Bessel numbers","volume":"29","author":"Stenlund","year":"2022","journal-title":"Electron. J. Combin."},{"key":"ref_34","first-page":"627","article-title":"The Bessel numbers and Bessel matrices","volume":"31","author":"Yang","year":"2011","journal-title":"J. Math. Res. Expo."},{"key":"ref_35","unstructured":"Graham, R.L., Knuth, D.E., and Patashnik, O. (1994). Concrete Mathematics, Addison-Wesley Publishing Company. [2nd ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/179\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:29:18Z","timestamp":1760027358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/2\/179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,8]]},"references-count":35,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["e27020179"],"URL":"https:\/\/doi.org\/10.3390\/e27020179","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2025,2,8]]}}}