This book can be recommended to everybody interested in an advanced theory of stochastic differential equations and, in particular, in the stability problem. Reading it will be easier with some preliminary knowledge of infinite dimensional stochastic differential equations (SDE), nevertheless all necessary probability background is briefly reviewed in the book. The book is readable and systematically written. It starts with a chapter devoted to the theory of SDEs in infinite dimensions. Chapters 2-4 constitute the essential part of the monograph, with a detailed study of stability properties. Different notions of stability are introduced in chapter 1 (stability in probability, stability in p-th moment, asymptotical stability, almost sure stability, exponential stability, and many others). In chapter 2, stability is studied for stochastic linear evolution equations. The nonlinear equations are studied in chapter 3 and this part is the core of the book. Stochastic functional differential equations and their stability is treated as a more specific topic in chapter 4. The last part (chapter 5) contains applications and some related topics of stability. The book can be recommended to specialists in stochastic analysis but it can also be useful for researchers in the area of deterministic differential equations as a stochastic counterpart to this branch of mathematics. Advanced students of probability, researchers applying the stochastic systems in their work, and many others can also profit from this monograph.

Reviewer:

dh