I. Background. I.1. Introductory Material. I.2. Existence and Uniqueness Results. I.3. Classic Maximum Principles. I.4. Markov-Feller Processes -- II. Integro-Differential Parabolic Equations. II.1. Definition of the Integro-Differential Operator. II.2. Maximum Principles. II.3. Existence and Uniqueness Results. II.4. Stochastic Interpretation -- III. A Simple Cauchy Problem. III.1. Wiener and Poisson Processes. III.2. Essential Properties. III.3. Dirichlet & Neumann Problems in Half-space. III.4. Stochastic Representation -- IV. Green and Poisson Functions. IV.1. Definition of a Fundamental Solution. IV.2. Definition of a Green Function. IV.3. Definition of a Poisson Function -- V. Fundamental Solutions for Differential Equations. V.1. Estimates for the Heat Potentials. V.2. Operators with Constant Coefficients. V.3. Variable Coefficients. V.4. The Cauchy Problem. V.5. Layer Potentials -- VI. Classic Green and Poisson Functions. VI.1. Problems in Half-Space
VI.2. First and Second Boundary Value Problems. VI.3. Oblique Derivative Boundary Conditions -- VII. Green Spaces. VII.1. Definition of the Function Spaces. VII.2. An Integral Transformation. VII.3. Properties of the Integro-differential Operator. VII.4. Commutative Property -- VIII. The Construction of the Green Function. VIII.1. Volterra Equations. VIII.2. Existence of the Green Function. VIII.3. Reflected Diffusion Processes with Jumps -- IX. Estimates on the Green Function. IX.1. First Order Estimates. IX.2. Singular Integral Estimates. IX.3. Integro-Differential Operator Estimates. IX.4. Second Order Estimates
Analysis
Differential equations
Bibliography
Includes bibliographical references (pages 409-417)
