Regularity theory and stochastic flows for parabolic SPDEs

Couverture
Introduction to the Series
Preface
1 Introduction
2 Stochastic Flows: Preliminary Comments
2.1 Regular Versions of Random Fields
2.2 Comments on the Literature
3 Preliminaries on Well-Posedness and Function Spaces
3 .1 Notations
3.2 Basic Abstract Well-Posedness Theorem
3.3 Basic Particular Case
3.4 An Extension
3.5 Hilbert Scales and Jnterpolation of Random Field Spaces
4 Regularity in Bounded Domains: Some Counterexamples
5 Regularity Theory for Homogeneous Problems
5.1 Abstract Regularity Theory
5.2 Application to Equations in Rd
5.3 Application to Second Order Equations in Bounded Domains: Dirichlet Boundary Condition
5.3.1 L²(D)-Solutions
5.3.2 H¹(D)-Solutions
5.3.3 H²(D)-Solutions
5.3.4 H^n(D)-Solutions, n>=3
5.3.5 H^{-1}(D)-Solutions
5.3.6 H^{-n}(D)-Solutions, n>=2
5.3.7 Summary of Regularity Results for Dirichlet Boundary Value Problem. H^alpha(D)-Solutions, alpha in R
5.3.8 Regularity for the Adjoint Equation
5.4 Application to Equations of Order 2m in Bounded Domains: Dirichlet Boundary Conditions
5.5 Application to Second Order Equations in Bounded Domains: Neumann Boundary Condition
5.5.1 Regularity Results for the Neumann Boundary Value Problem
5.5.2 Regularity for the Adjoint Equation
6 Non-Homogeneous Boundary Value Problems
6.1 General Framework, Standing Assumptions, and Applications
6.2 Preliminaries on Deterministic Boundary Value Problems
6.3 Orientation
6.4 Elementary Regularity
6.5 More Refined Abstract Regularity Results
7 Existence and Regularity of Stochastic Flows
7.1 First Abstract Theorems of Existence and Regularity
7.2 A Pathwise Green Formula
7.3 Transpositions of the Adjoint Flow
7.4 Applications
7.4.1 Dirichlet Boundary Value Problem
7.4.2 Equations of Order 2m with Dirichlet Boundary Conditions
7.4.3 Neumann Boundary Condition
8 An Alternative Approach
Acknowledgements
References
Index