Stochastic Calculus in Infinite Dimensions and SPDEs
Springer International Publishing (Verlag)
978-3-031-69585-8 (ISBN)
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.
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 .
The monograph is a well readable overview on some key tools and results for studying linear and nonlinear SPDEs in the infinite dimensional context. I very well recommend it to readers in both probability theory and stochastic processes, as well as partial differential equations and functional analysis. (Andrej Srakar, zbMATH 1562.60001, 2025)
The book successfully addresses the key problems in the study of nonlinear stochastic evolution equations and introduces the reader to the modern research tools, all in under 150 pages. (Sergey V. Lototsky, Mathematical Reviews, July, 2025)
| Erscheinungsdatum | 03.09.2024 |
|---|---|
| Reihe/Serie | SpringerBriefs in Mathematics |
| Zusatzinfo | X, 136 p. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik |
| Schlagworte | Cylindrical Brownian Motion • Infinite Dimensional Setting • Stochastic Calculus • stochastic integration in Hilbert space • stochastic partial differential equations |
| ISBN-10 | 3-031-69585-2 / 3031695852 |
| ISBN-13 | 978-3-031-69585-8 / 9783031695858 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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