Intro
Preface
Acknowledgement
Contents
List of Symbols
List of Abbreviations
1 Introduction
1.1 An Example
1.2 Contributions
1.3 Scope of Results
1.4 Structure and Content of This Text
1.5 Notation and Conventions
2 Preliminaries
2.1 Finite Dimensional Gaussian Measures
2.1.1 Univariate Gaussian Measures
2.1.2 Multivariate Gaussian Measures
2.1.3 Hermite Polynomials
2.2 Gaussian Measures on Separable Locally Convex Spaces
2.2.1 Cylindrical Sets
2.2.2 Definition and Basic Properties of Gaussian Measures
2.3 Cameron-Martin Space

2.4 Gaussian Product Measures
2.5 Gaussian Series
2.5.1 Some Abstract Results
2.5.2 Karhunen-Loève Expansion
2.5.3 Multiresolution Representations of GRFs
2.5.4 Periodic Continuation of a Stationary GRF
2.5.5 Sampling Stationary GRFs
2.6 Finite Element Discretization
2.6.1 Function Spaces
2.6.2 Finite Element Interpolation
3 Elliptic Divergence-Form PDEs with Log-Gaussian Coefficient
3.1 Statement of the Problem and Well-Posedness
3.2 Lipschitz Continuous Dependence
3.3 Regularity of the Solution
3.4 Random Input Data

3.5 Parametric Deterministic Coefficient
3.5.1 Deterministic Countably Parametric Elliptic PDEs
3.5.2 Probabilistic Setting
3.5.3 Deterministic Complex-Parametric Elliptic PDEs
3.6 Analyticity and Sparsity
3.6.1 Parametric Holomorphy
3.6.2 Sparsity of Wiener-Hermite PC Expansion Coefficients
3.7 Parametric Hs(D)-Analyticity and Sparsity
3.7.1 Hs(D)-Analyticity
3.7.2 Sparsity of Wiener-Hermite PC Expansion Coefficients
3.8 Parametric Kondrat'ev Analyticity and Sparsity
3.8.1 Parametric Ks(D)-Analyticity