03739nam a22005415i 4500 978-3-031-69586-5 DE-He213 20251030112921.0 cr nn 008mamaa 240830s2024 sz | s |||| 0|eng d 9783031695865 978-3-031-69586-5 10.1007/978-3-031-69586-5 doi QA274-274.9 PBT bicssc PBWL bicssc MAT029000 bisacsh PBT thema PBWL thema Goodair, Daniel. author. aut http://id.loc.gov/vocabulary/relators/aut 83215 Stochastic Calculus in Infinite Dimensions and SPDEs [electronic resource] / by Daniel Goodair, Dan Crisan. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. X, 136 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda SpringerBriefs in Mathematics, 2191-8201 1 Introduction -- 2 Stochastic Calculus in Infinite Dimensions -- 3 Stochastic Differential Equations in Infinite Dimensions -- 4 A Toolbox for Nonlinear SPDEs -- 5 Existence Theory for Nonlinear SPDEs and the Stochastic Navier-Stokes Equations -- A Appendix -- References -- Index . Introducing a groundbreaking framework for stochastic partial differential equations (SPDEs), this work presents three significant advancements over the traditional variational approach. Firstly, Stratonovich SPDEs are explicitly addressed. Widely used in physics, Stratonovich SPDEs have typically been converted to Ito form for mathematical treatment. While this conversion is understood heuristically, a comprehensive treatment in infinite dimensions has been lacking, primarily due to insufficient rigorous results on martingale properties. Secondly, the framework incorporates differential noise, assuming the noise operator is only bounded from a smaller Hilbert space into a larger one, rather than within the same space. This necessitates additional regularity in the Ito form to solve the original Stratonovich SPDE. This aspect has been largely overlooked, despite the increasing popularity of gradient-dependent Stratonovich noise in fluid dynamics and regularisation by noise studies. Lastly, the framework departs from the explicit duality structure (Gelfand Triple), which is typically expected in the study of analytically strong solutions. This extension builds on the classical variational framework established by Röckner and Pardoux, advancing it in all three key aspects. Explore this innovative approach that not only addresses existing challenges but also opens new avenues for research and application in SPDEs. . Stochastic processes. 76770 Calculus. 83216 Stochastic Calculus. 83217 Crisan, Dan. author. (orcid)0000-0003-1486-1517 https://orcid.org/0000-0003-1486-1517 aut http://id.loc.gov/vocabulary/relators/aut 83218 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031695858 Printed edition: 9783031695872 Printed edition: 9783031695889 SpringerBriefs in Mathematics, 2191-8201 70788 https://doi.org/10.1007/978-3-031-69586-5 ZDB-2-SMA ZDB-2-SXMS 62876 62876