Stochastic partial differential equations: an introduction.

*(English)*Zbl 1361.60002
Universitext. Cham: Springer (ISBN 978-3-319-22353-7/pbk; 978-3-319-22354-4/ebook). vi, 266 p. (2015).

The three main approaches for analyzing stochastic partial differential equations (SPDEs) are the “martingale approach”, the “semigroup approach” and the “variational approach”.

The present book provides an introduction to the “variational approach” and to the “semigroup approach”. First, the authors present several examples of SPDEs – such as the stochastic generalized Burgers equation, stochastic Navier-Stokes equations and the stochastic porous media equation – which are all of the general form \[ \begin{split} \text{d}X(t)(\xi) = A (t, X(t)(\xi), D_{\xi} X(t)(\xi), D_{\xi}^2 X(t)(\xi)) \text{d}t \\ + B (t, X(t)(\xi), D_{\xi} X(t)(\xi), D_{\xi}^2 X(t)(\xi)) \text{d}W(t),\end{split} \] driven by a cylindrical Wiener process \(W\). These types of SPDEs are covered by the theory developed in this book.

After preparing the relevant foundations, such as infinite dimensional Wiener processes and stochastic integration in infinite dimension, the authors present two chapters with an introduction to the “variational approach” and one chapter with an introduction to the “semigroup approach”. Concerning the “variational approach”, in the first chapter the coefficients of the SPDE have to satisfy monotonicity and coercivity conditions, and – more generally – in the second chapter, they just have to fulfill local monotonicity and generalized coercivity conditions.

The book contains several appendices including the required prerequisites from functional analysis and probability theory.

The present book provides an introduction to the “variational approach” and to the “semigroup approach”. First, the authors present several examples of SPDEs – such as the stochastic generalized Burgers equation, stochastic Navier-Stokes equations and the stochastic porous media equation – which are all of the general form \[ \begin{split} \text{d}X(t)(\xi) = A (t, X(t)(\xi), D_{\xi} X(t)(\xi), D_{\xi}^2 X(t)(\xi)) \text{d}t \\ + B (t, X(t)(\xi), D_{\xi} X(t)(\xi), D_{\xi}^2 X(t)(\xi)) \text{d}W(t),\end{split} \] driven by a cylindrical Wiener process \(W\). These types of SPDEs are covered by the theory developed in this book.

After preparing the relevant foundations, such as infinite dimensional Wiener processes and stochastic integration in infinite dimension, the authors present two chapters with an introduction to the “variational approach” and one chapter with an introduction to the “semigroup approach”. Concerning the “variational approach”, in the first chapter the coefficients of the SPDE have to satisfy monotonicity and coercivity conditions, and – more generally – in the second chapter, they just have to fulfill local monotonicity and generalized coercivity conditions.

The book contains several appendices including the required prerequisites from functional analysis and probability theory.

Reviewer: Stefan Tappe (Hannover)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

60H05 | Stochastic integrals |