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- English
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Stochastic Partial Differential Equations
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About This Book
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 theor
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Chapter 1
Preliminaries
1.1 Introduction
The theory of stochastic ordinary differential equations has been well developed since K. Itô introduced the stochastic integral and the stochastic integral equation in the mid 1940s [40, 41]. Therefore such an equation is also known as an Itô equation in honor of its originator. Due to its diverse applications ranging from biology and physics to finance, the subject has become increasingly important and popular. For an introduction to the theory and many references, one is referred to the books [2, 32, 39, 55, 61] among many others.
Up to the early 1960s, most works on stochastic differential equations had been confined to ordinary differential equations. Since then, spurred by the demand from modern applications, partial differential equations with random parameters, such as the coefficients or the forcing term, have begun to attract the attention of many researchers. Most of them were motivated by applications to physical and biological problems. Notable examples are turbulent flow in fluid dynamics, diffusion and waves in random media [5, 8]. In general the random parameters or random fields involved need not be of white-noise type, but, in many applications, models with white noises provide reasonable approximations. Besides, as a mathematical subject, they pose many interesting and challenging problems in stochastic analysis. By a generalization of the Itô equations in Rd, it seems natural to consider stochastic partial differential equations of Itô type as a stochastic evolution equation in some Hilbert or Banach space. This book will be exclusively devoted to stochastic partial differential equations of Itô type.
The study of stochastic partial differential equations in a Hilbert space goes back ...
Table of contents
- Cover
- Halftitle Page
- Series Page
- Title Page
- Copyright Page
- Preface
- Table of Contents
- 1 Preliminaries
- 2 Scalar Equations of First Order
- 3 Stochastic Parabolic Equations
- 4 Stochastic Parabolic Equations in the Whole Space
- 5 Stochastic Hyperbolic Equations
- 6 Stochastic Evolution Equations in Hilbert Spaces
- 7 Asymptotic Behavior of Solutions
- 8 Further Applications
- 9 Diffusion Equations in Infinite Dimensions
- References
- Index