# Reference complementing Hairer's “Introduction to Stochastic PDEs”

I want to study stochastic partial differential equations, and the standard reference seems to be Hairer's notes - after all, he got the first Fields Medal for work on this topic.

However, I also noticed that his notes are rather terse, and hence I wanted to ask if anyone could suggest resources that complement his notes. Could you name some references, and comment on strengths/weaknesses/applications?

I like Hairer's notes, but of course these are lecture notes. And some introductory books can be easier to read since there are more explanations.

There are basically three approaches to analysing SPDEs: the “martingale (or martingale measure) approach”, the “semigroup (or mild solution) approach” and the “variational approach”. Hairer is using the semigroup approach. Now it depends which perspective you want. The same as in Hairer and therefore using the reference to understand Hairer's note more, or if you want to learn SPDEs from another perspective.

Stochastic Equations in Infinite Dimensions by Da Prato & Zabczyk

If you are looking for something which complements Hairer's notes, this book is the way to go. It is self-contained and introduces every concept carefully. It is using the semigroup theory, first establishing existence and uniqueness of evolutionary SPDEs. Afterwards, they prove several regularity results.

Stochastic Partial Differential Equations: An Introduction by Liu & Röckner

A nice short introduction to SPDEs. I like it and it has a lot of similarities with Hairer's notes, also introducing the semigroup theory as an auxiliary tool, but just as a side note; the focus is on the variational approach. It presents all the existence and uniqueness results for monotone coefficients. It is also looking at the stochastic Navier-Stokes equation and many more (Cahn-Hilliard, ...).

PDE and Martingale Methods in Option Pricing by Pasucci

A huge book covering a lot of topics and not only SPDEs. It introduces the techniques used for SPDES quite nicely and I look to look these up in this book. Nice existence results on parabolic PDEs with variable coefficients. As written in the title, it uses mainly the martingale approach.

Stochastic Partial Differential Equations by Lototsky & Rozovsky

If you are looking for something on SPDEs it is mostly in here, quite recent book from 2017. The authors take the previous books on SPDEs into account and put it all into this book, character of a compendium. Nonetheless the proofs are really complete and steps are written out in a thorough way.

Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach by Holden & Øksendal & Ubøe & Zhang

I like their approach and it became a classic. Using white noise analysis and heavy functional analysis machinery they present a different perspective. Not sure if it is really complimentary to Hairer but it is worth a read.

A Concise Course on Stochastic Partial Differential Equations by Prevot & Röckner

Similar to Hairer's notes this book also more has the structure of lecture notes, but using the variational approach. A nice short read.

• What about Walsh for the martingale approach. – Shalop Aug 7 '18 at 19:55
• @Shalop True, when speaking of theory of SPDEs one should definitely point out Walsh's work. To be honest I've never really read through it throughly because of the old $LaTeX$ typeset. – Fritz Aug 7 '18 at 20:03