{"title":"SU(2) 上布朗运动的耦合和 <mml:math xmlns:m","authors":"Magalie Bénéfice","doi":"10.1016/j.spa.2024.104434","DOIUrl":null,"url":null,"abstract":"<div><p>The Lie groups <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>) consists in simultaneously coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. The strategy is to alternate between reflection and synchronous (with noise) couplings on the sphere. We also describe some more general constructions of co-adapted couplings on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and on <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":51160,"journal":{"name":"Stochastic Processes and their Applications","volume":"176 ","pages":"Article 104434"},"PeriodicalIF":1.1000,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Couplings of Brownian motions on SU(2) and SL(2,R)\",\"authors\":\"Magalie Bénéfice\",\"doi\":\"10.1016/j.spa.2024.104434\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The Lie groups <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> (resp. <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>) consists in simultaneously coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span>. The strategy is to alternate between reflection and synchronous (with noise) couplings on the sphere. We also describe some more general constructions of co-adapted couplings on <span><math><mrow><mi>S</mi><mi>U</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></mrow></math></span> and on <span><math><mrow><mi>S</mi><mi>L</mi><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span>.</p></div>\",\"PeriodicalId\":51160,\"journal\":{\"name\":\"Stochastic Processes and their Applications\",\"volume\":\"176 \",\"pages\":\"Article 104434\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2024-07-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Stochastic Processes and their Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0304414924001406\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastic Processes and their Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304414924001406","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Couplings of Brownian motions on SU(2) and SL(2,R)
The Lie groups and can be viewed as model spaces in subRiemannian geometry. Coupling two subelliptic Brownian motions on (resp. ) consists in simultaneously coupling two Brownian motions on the sphere (resp. the hyperbolic plane) and their swept areas. Using this approach we propose an explicit construction of a co-adapted successful coupling on . The strategy is to alternate between reflection and synchronous (with noise) couplings on the sphere. We also describe some more general constructions of co-adapted couplings on and on .
期刊介绍：
Stochastic Processes and their Applications publishes papers on the theory and applications of stochastic processes. It is concerned with concepts and techniques, and is oriented towards a broad spectrum of mathematical, scientific and engineering interests.
Characterization, structural properties, inference and control of stochastic processes are covered. The journal is exacting and scholarly in its standards. Every effort is made to promote innovation, vitality, and communication between disciplines. All papers are refereed.
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