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