Chapman and Hall/CRC
281 pages | 30 B/W Illus.
As a relatively new area in mathematics, stochastic partial differential equations (PDEs) are still at a tender age and have not yet received much attention in the mathematical community. Filling the void of an introductory text in the field, Stochastic Partial Differential Equations introduces PDEs to students familiar with basic probability theory and Itô's equations, highlighting several computational and analytical techniques.
Without assuming specific knowledge of PDEs, the text includes many challenging problems in stochastic analysis and treats stochastic PDEs in a practical way. The author first brings the subject back to its root in classical concrete problems. He then discusses a unified theory of stochastic evolution equations and describes a few applied problems, including the random vibration of a nonlinear elastic beam and invariant measures for stochastic Navier-Stokes equations. The book concludes by pointing out the connection of stochastic PDEs to infinite-dimensional stochastic analysis.
By thoroughly covering the concepts and applications of stochastic PDEs at an introductory level, this text provides a guide to current research topics and lays the groundwork for further study.
"The book provides an excellent introduction to the theory of Stochastic Partial Differential Equations . . . It provides a well written and timely contribution to the literature."
– Evelyn Buckwar, in Zentralblatt Math, 2009
"The text may be characterized as an excellent guide to current research topics that opens possibilities for further developments in the field."
– In EMS Newsletter, 2008
"This introductory book fills a gap in the field."
– Nikita Y. Ratanov, in Mathematical Reviews, 2008d
". . . very well written introductory book . . . Overall I thoroughly recommend this book and believe that it will be a useful textbook with which to introduce students and young scientists to computational and analytical techniques for stochastic differential equations. This book is of great interest to applied mathematicians, theoretical physicists, naturalists, and all interested in statistical formulation of scientific problems."
– Andrzej Icha, Institute of Mathematics, in Pure and Applied Geophysics, June 2005
Brownian Motions and Martingales
Stochastic Differential Equations
SCALAR EQUATIONS OF FIRST ORDER
Generalized Itô's Formula
Linear Stochastic Equations
STOCHASTIC PARABOLIC EQUATIONS
Solution of Random Heat Equation
Linear Equations with Additive Noise
Some Regularity Properties
Random Reaction-Diffusion Equations
Parabolic Equations with Gradient-Dependent Noise
STOCHASTIC PARABOLIC EQUATIONS IN THE WHOLE SPACE
Linear and Similinear Equations
Positivity of Solutions
Correlation Functions of Solutions
STOCHASTIC HYPERBOLIC EQUATIONS
Wave Equation with Additive Noise
Semilinear Wave Equations
Wave Equations in Unbounded Domain
Randomly Perturbed Hyperbolic Systems
STOCHASTIC EVOLUTION EQUATIONS IN HILBERT SPACES
Hilbert Space-Valued Martingales
Stochastic Integrals in Hilbert Spaces
Stochastic Evolution Equations
Stochastic Evolution Equations of Second Order
ASYMPTOTIC BEHAVIOR OF SOLUTIONS
Itô's Formula and Lyapunov Functionals
Boundedness of Solutions
Stability of Null Solution
Small Random Perturbation Problems
Large Deviations Problems
Stochastic Burgers and Related Equations
Random Schrödinger Equation
Nonlinear Stochastic Beam Equations
Stochastic Stability of Cahn-Hilliard Equation
Invariant Measures for Stochastic Navier-Stokes Equations
DIFFUSION EQUATIONS IN INFINITE DIMENSIONS
Diffusion Processes and Kolmogorov Equations
Parabolic Equations and Related Elliptic Problems
Characteristic Functionals and Hopf Equations