{"title":"A solution theory for a general class of SPDEs.","authors":"André Süß, Marcus Waurick","doi":"10.1007/s40072-016-0088-8","DOIUrl":"https://doi.org/10.1007/s40072-016-0088-8","url":null,"abstract":"<p><p>In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of <math> <msup><mrow><mi>R</mi></mrow> <mi>d</mi></msup> </math> . The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green's function is not required.</p>","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"5 2","pages":"278-318"},"PeriodicalIF":0.0,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40072-016-0088-8","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"37105746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Measure valued solutions to the stochastic Euler equations in $$mathbb {R}^d$$Rd","authors":"J. U. Kim","doi":"10.1007/S40072-015-0060-Z","DOIUrl":"https://doi.org/10.1007/S40072-015-0060-Z","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"3 1","pages":"531-569"},"PeriodicalIF":0.0,"publicationDate":"2015-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73768016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Homogenization of Brinkman flows in heterogeneous dynamic media","authors":"H. Bessaih, Y. Efendiev, F. Maris","doi":"10.1007/s40072-015-0058-6","DOIUrl":"https://doi.org/10.1007/s40072-015-0058-6","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"41 1","pages":"479 - 505"},"PeriodicalIF":0.0,"publicationDate":"2015-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78123172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On some properties of space inverses of stochastic flows","authors":"J. Leahy, R. Mikulevičius","doi":"10.1007/s40072-015-0056-8","DOIUrl":"https://doi.org/10.1007/s40072-015-0056-8","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"1 1","pages":"445 - 478"},"PeriodicalIF":0.0,"publicationDate":"2015-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79516540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Multi Level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients","authors":"F. Nobile, Francesco Tesei","doi":"10.1007/s40072-015-0055-9","DOIUrl":"https://doi.org/10.1007/s40072-015-0055-9","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"6 1","pages":"398 - 444"},"PeriodicalIF":0.0,"publicationDate":"2015-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75857158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Moments, intermittency and growth indices for the nonlinear fractional stochastic heat equation","authors":"Le Chen, R. Dalang","doi":"10.1007/s40072-015-0054-x","DOIUrl":"https://doi.org/10.1007/s40072-015-0054-x","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"341 1","pages":"360 - 397"},"PeriodicalIF":0.0,"publicationDate":"2015-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82959802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularity of the solutions to SPDEs in metric measure spaces","authors":"Elena Issoglio, M. Zähle","doi":"10.1007/s40072-015-0048-8","DOIUrl":"https://doi.org/10.1007/s40072-015-0048-8","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"59 1","pages":"272 - 289"},"PeriodicalIF":0.0,"publicationDate":"2015-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73350983","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Controllability and qualitative properties of the solutions to SPDEs driven by boundary Lévy noise","authors":"E. Hausenblas, P. Razafimandimby","doi":"10.1007/s40072-015-0047-9","DOIUrl":"https://doi.org/10.1007/s40072-015-0047-9","url":null,"abstract":"","PeriodicalId":74872,"journal":{"name":"Stochastic partial differential equations : analysis and computations","volume":"29 1","pages":"221 - 271"},"PeriodicalIF":0.0,"publicationDate":"2015-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79349325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}