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(More customer reviews)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.
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Stochastic Partial Differential Equations : A Modeling, White Noise Functional Approach (Probability and Its Applications)The main emphasis of this work is on stochastic partial differential equations. First the stochastic Poisson equation and the stochastic transport equation are discussed; then the authors go on to deal with the Schrodinger equation, the heat equation, the nonlinear Burgers' equation with a stochastic source, and the pressure equation. The white noise approach often allows for solutions given by explicit formulas in terms of expectations of certain auxiliary processes. The noise in the above examples are all of a Gaussian white noise type. In the end, the authors also show how to adapt the analysis to SPDEs involving noise of Poissonian type.
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