Stochastic differential equations in infinite dimensions with applications to stochastic partial differential equations.

*(English)*Zbl 1228.60002
Probability and Its Applications. Berlin: Springer (ISBN 978-3-642-16193-3/hbk; 978-3-642-16194-0/ebook). xvi, 291 p. (2011).

This volume is a systematic study of existence, uniqueness and fine properties of solutions to stochastic differential equations (SDEs) in infinite dimensions.

The authors commence with an excursion to the theory of finite-dimensional partial differential equations (PDEs) and illustrate their reformulation as a system of infinite-dimensional equations. A solid working knowledge of semigroup theory is also developed. Then, substantial clarity and time is devoted to the presentation of stochastic calculus in infinite dimensions, including the treatment of cylindrical and Q-Wiener processes and the construction of their respective Itô integrals and Itô formulas. The exposition of the central topic, SDEs in infinite dimensions, is launched with proofs of the existence of (strong, mild and weak) solutions and uniqueness and fine properties within the classical framework of regularity of coefficients and Lipschitz continuity assumptions, using both semigroup and variational methods.

A brief excursion to SDEs with discontinuous drifts is spent with a study of spin and interacting particle systems. The authors then conclude with a lucid account on the stability theory for SDEs in infinite dimensions, on the issue of ultimate boundedness and the topic of invariant measures.

The authors commence with an excursion to the theory of finite-dimensional partial differential equations (PDEs) and illustrate their reformulation as a system of infinite-dimensional equations. A solid working knowledge of semigroup theory is also developed. Then, substantial clarity and time is devoted to the presentation of stochastic calculus in infinite dimensions, including the treatment of cylindrical and Q-Wiener processes and the construction of their respective Itô integrals and Itô formulas. The exposition of the central topic, SDEs in infinite dimensions, is launched with proofs of the existence of (strong, mild and weak) solutions and uniqueness and fine properties within the classical framework of regularity of coefficients and Lipschitz continuity assumptions, using both semigroup and variational methods.

A brief excursion to SDEs with discontinuous drifts is spent with a study of spin and interacting particle systems. The authors then conclude with a lucid account on the stability theory for SDEs in infinite dimensions, on the issue of ultimate boundedness and the topic of invariant measures.

Reviewer: Jianing Zhang (Berlin)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

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