A new framework in which to pose stochastic partial differential equations (SPDEs), with three distinct novelties in comparison to the traditional variational framework. The first is that Stratonovich SPDEs are explicitly considered. Stratonovich SPDEs are renowned for applications in physics, and are traditionally mathematically treated by a conversion to Ito form. Heuristically this conversion is known but a proper treatment in the infinite dimensional setting has been absent, perhaps due to the lack of rigorous results concerning martingale properties. This is compounded by the second novelty in that differential noise is accounted for; that is, one assumes the noise operator is bounded only from a smaller Hilbert Space into a larger one and not on the same space. This results in some additional regularity in the Itô form required to solve the original Stratonovich SPDE, which has certainly not been appreciated despite the area of gradient-dependent Stratonovich noise booming in popularity from physical principles in fluid dynamics and regularisation by noise results. The third is the loss of an explicit duality structure (Gelfand Triple), which is not expected in the study of analytically strong solutions. We note that this extends the classical variational framework presented by Röckner and Pardoux for example (see quoted books below) on all three accounts.

 

Book Information
ISBN 9783031695858
Author Dan Crisan
Format Paperback
Page Count 125
Imprint Springer International Publishing AG
Publisher Springer International Publishing AG