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Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach (Universitext)

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SUMMARY: This book presents a new approach to stochastic partial differential equations based on white noise analysis. The framework makes heavy use of functional analysis and its main starting point is the Wiener chaos expansion and analogous expansions on different functional spaces (Schwartz spaces).

A stochastic PDE is a PDE containing a random noise term, which may be additive or multiplicative. One of the problems when working with Stochastic PDEs is to define a notion of solution which is meaningfully extendable to the nonlinear case. Problems arises because the noise term is highly irregular: for each sample of the noise, one has a (nonlinear) PDE with a very irregular term in it. In physical terms, one may encounter "ultraviolet" divergences. So, one is first faced with an existence/ unicity problem for such equations. Additionally, one would like to describe probabilitic properties of such solutions.

The method proposed by the authors can be described as follows: first, one expands the noise term in the PDE using a Wiener chaos expansion. Truncating the expansion at a certain order n yields a "regularized" equation in which the noise is smoothened. This can be roughly described as an ultraviolet cutoff. The equation then has a unique solution in an appropriate functional space. The solution of the original SPDE is then defined as the sequence of truncated solutions. In some cases, this sequence may converge in some classical sense in an appropriate function space to a weak or strong solution defined in the usual sense. But, in general, this is not the case and the notion of solution defined by the authors may be different from classical notions.

Although the title contains the word 'modeling', it may look as the abstract definition of solution proposed by the authors may have little to do with the physical notion of solution. One feels a need for a justification why this definition of a solution is physically relevant at all, which I feel is lacking. The authors give some examples, such as the noisy Burgers equation and the Kardar-Parisi-Zhang equation, but the results predicted for the solutions seem to be different than the ones predicted for example by renormalization group analysis for example regarding the scaling exponents for KPZ. Also, it would be interesting to compare this notion of solution with more classical ones for example using the semigroup/ Green function approach.

The approach proposed bears a strong resemblance to ultraviolet regularization schemes used in renormalization group theory. In fact, this framework may be seenas a probabilistic setting for renormalization methods.Unfortunately there is little discussion of this point in the book.

The first chapters contain an interesting review of white noise expansions and chaos expansions, useful in their own interest.

Overall I recommend this book as interesting for researchers in mathematical and theoretical physics.

The authors have prepared a very accessible introduction for elements of the Malliavin calculus, analysis on the Hida space, and the Wick product with applications to stochastic PDEs. This material is also a prerequisite for some of the new modeling theories which extend the classical SPDE models based on semimartingale diffusions to a more general setting. As an example of these extensions, see Mishura's Stochastic Calculus for Fractional Brownian Motion and Related Processes or the work of Biagini, Hu, Oksendal, and Zhang in Stochastic Calculus for Fractional Brownian Motion and Applications.

The reader will need some prerequisites to get into the text. For probability theory, I recommend Chung's A Course in Probability Theory Revised. For classical Itô calculus, I recommend Rogers & Williams Diffusions, Markov Processes and Martingales: Volume 2, Itô Calculus . You'll need a good background in functional analysis, and I recommend Rudin's Functional Analysis.

In addition to the prerequisites, there are several corequisites that the reader will want to have handy. I note that the authors do appeal to specific results from Reed & Simon's Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Another often cited reference is the text by Hida, et.al White Noise: An Infinite Dimensional Calculus. Finally, I have found that Nualart's The Malliavin Calculus and Related Topics provides more complete understanding of white noise calculus, particularly the Skorohod integral.

In chapter 1, the authors provide a wonderfully intuitive motivation for the theory by considering the moving boundary problem involving a stochastic model of rates of absorption of moisture in a porous rock. Using charts and photographs, the authors demonstrate how traditional methods to solve the SPDE fail to capture the true fractal nature of the solution. The authors then go on to explain why it is necessary to generalize from real-valued stochastic processes to processes taking values in certain spaces of distributions. These are the celebrated spaces of Kondratiev and Hida. The inability of ordinary multiplication to apply in such spaces of distributions leads to the introduction of the Wick Product.

The aim of the next chapter, Chapter 2, is two-fold. First, the authors introduce the slogan:
"Itô Calculus with ordinary multiplication is Ordinary Calculus with Wick multiplication"
and then go on to make this slogan rigous be laying the foundations for elements of Malliavin calculus, including the Wiener Chaos expansion and the Skorohod integral. Along the way, they show that, for adapted processes, the Itô and Skorohod integrals coincide. Second, the chapter wraps up by introducing the Hermite transform, along with the related s-transform and f-transform. These transforms, combined with Wick calculus provide an extremely useful suite of tools for analyzing and solving SPDEs.

The final two chapters now apply these white noise space techniques, first to stochastic ordinary differential equations in Chapter 3 and then to multivariate stochastic partial differential equations in Chapter 4. Each of the chapters (from chapter 2 through chapter 4) contains an exercise section with a number of problems to help solidify the material in the mind of the reader.

The book contains several appendices. In Appendix A, a proof of a special version of the Bochner-Milnos Theorem is provided, which gives the existence of the all-important white noise probability measure on the space of tempered distributions. Appendix B is a brief review of Itô calculus, while Appendix C is a nice summary on the Hermite polynomials. The final Appendix is a technical section proving that the Wick product is well-defined.