Format:
Online-Ressource (digital)
ISBN:
9780387894881
Series Statement:
Universitext
Content:
Preface to the Second Edition -- Preface to the First Edition -- Introduction -- Framework -- Applications to stochastic ordinary differential equations -- Stochastic partial differential equations driven by Brownian white noise -- Stochastic partial differential equations driven by Lévy white noise -- Appendix A. The Bochner-Minlos theorem -- Appendix B. Stochastic calculus based on Brownian motion -- Appendix C. Properties of Hermite polynomials -- Appendix D. Independence of bases in Wick products -- Appendix E. Stochastic calculus based on Lévy processes- References -- List of frequently used notation and symbols -- Index.
Content:
The first edition of Stochastic Partial Differential Equations: A Modeling, White Noise Functional Approach, gave a comprehensive introduction to SPDEs driven by space-time Brownian motion noise. In this, the second edition, the authors extend the theory to include SPDEs driven by space-time Lévy process noise, and introduce new applications of the field. Because the authors allow the noise to be in both space and time, the solutions to SPDEs are usually of the distribution type, rather than a classical random field. To make this study rigorous and as general as possible, the discussion of SPDEs is therefore placed in the context of Hida white noise theory. The key connection between white noise theory and SPDEs is that integration with respect to Brownian random fields can be expressed as integration with respect to the Lebesgue measure of the Wick product of the integrand with Brownian white noise, and similarly with Lévy processes. The first part of the book deals with the classical Brownian motion case. The second extends it to the Lévy white noise case. For SPDEs of the Wick type, a general solution method is given by means of the Hermite transform, which turns a given SPDE into a parameterized family of deterministic PDEs. Applications of this theory are emphasized throughout. The stochastic pressure equation for fluid flow in porous media is treated, as are applications to finance. Graduate students in pure and applied mathematics as well as researchers in SPDEs, physics, and engineering will find this introduction indispensible. Useful exercises are collected at the end of each chapter. From the reviews of the first edition: "The authors have made significant contributions to each of the areas. As a whole, the book is well organized and very carefully written and the details of the proofs are basically spelled out... This is a rich and demanding book… It will be of great value for students of probability theory or SPDEs with an interest in the subject, and also for professional probabilists." —Mathematical Reviews "...a comprehensive introduction to stochastic partial differential equations." —Zentralblatt MATH.
Note:
Description based upon print version of record
,
""Preface to the Second Edition""; ""Preface to the First Edition""; ""Contents""; ""1 Introduction""; ""1.1 Modeling by Stochastic Differential Equations""; ""2 Framework""; ""2.1 White Noise""; ""2.1.1 The 1-Dimensional, d-Parameter Smoothed White Noise""; ""2.1.2 The (Smoothed) White Noise Vector""; ""2.2 The Wiener--ItÃ? Chaos Expansion""; ""2.2.1 Chaos Expansion in Terms of Hermite Polynomials""; ""2.2.2 Chaos Expansion in Terms of Multiple ItÃ? Integrals""; ""2.3 The Hida Stochastic Test Functions and Stochastic Distributions. The Kondratiev Spaces (S)m; N,(S)-m; N""
,
""2.3.1 The Hida Test Function Space (S) and the Hida Distribution Space (S)""""2.3.2 Singular White Noise""; ""2.4 The Wick Product""; ""2.4.1 Some Examples and Counterexamples""; ""2.5 Wick Multiplication and Hitsuda/Skorohod Integration""; ""2.6 The Hermite Transform""; ""2.7 The (S)N,r Spaces and the S-Transform""; ""2.8 The Topology of (S)-1N""; ""2.9 The F-Transform and the Wick Product on L1()""; ""2.10 The Wick Product and Translation""; ""2.11 Positivity""; ""3 Applications to Stochastic Ordinary Differential Equations""; ""3.1 Linear Equations""
,
""3.1.1 Linear 1-Dimensional Equations""""3.1.2 Some Remarks on Numerical Simulations""; ""3.1.3 Some Linear Multidimensional Equations""; ""3.2 A Model for Population Growth in a Crowded, Stochastic Environment""; ""3.2.1 The General (S)-1 Solution""; ""3.2.2 A Solution in L1()""; ""3.2.3 A Comparison of Model A and Model B""; ""3.3 A General Existence and Uniqueness Theorem""; ""3.4 The Stochastic Volterra Equation""; ""3.5 Wick Products Versus Ordinary Products: a Comparison Experiment""; ""3.5.1 Variance Properties""; ""3.6 Solution and Wick Approximation of Quasilinear SDE""
,
""3.7 Using White Noise Analysis to Solve General Nonlinear SDEs""""4 Stochastic Partial Differential Equations Driven by Brownian White Noise""; ""4.1 General Remarks""; ""4.2 The Stochastic Poisson Equation""; ""4.2.1 The Functional Process Approach""; ""4.3 The Stochastic Transport Equation""; ""4.3.1 Pollution in a Turbulent Medium""; ""4.3.2 The Heat Equation with a Stochastic Potential""; ""4.4 The Stochastic Schrödinger Equation""; ""4.4.1 L1()-Properties of the Solution""; ""4.5 The Viscous Burgers Equation with a Stochastic Source""; ""4.6 The Stochastic Pressure Equation""
,
""4.6.1 The Smoothed Positive Noise Case""""4.6.2 An Inductive Approximation Procedure""; ""4.6.3 The 1-Dimensional Case""; ""4.6.4 The Singular Positive Noise Case""; ""4.7 The Heat Equation in a Stochastic, Anisotropic Medium""; ""4.8 A Class of Quasilinear Parabolic SPDEs""; ""4.9 SPDEs Driven by Poissonian Noise""; ""5 Stochastic Partial Differential Equations Driven by Lévy Processes""; ""5.1 Introduction""; ""5.2 The White Noise Probability Space of a Lévy Process (d=1)""; ""5.3 White Noise Theory for a Lévy Process (d=1)""; ""5.3.1 Chaos Expansion Theorems""
,
""5.3.2 The Lévy--Hida--Kondratiev Spaces""
Additional Edition:
ISBN 9780387894874
Additional Edition:
Buchausg. u.d.T. Stochastic partial differential equations New York, NY : Springer, 2010 ISBN 9780387894874
Language:
English
Subjects:
Economics
,
Mathematics
Keywords:
Stochastische partielle Differentialgleichung
;
Stochastische partielle Differentialgleichung
DOI:
10.1007/978-0-387-89488-1
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Author information:
Øksendal, Bernt K. 1945-
Author information:
Holden, Helge 1956-
