Second Order Elliptic Integro-Differential Problems: 1st Edition (Paperback) book cover

Second Order Elliptic Integro-Differential Problems

1st Edition

By Maria Giovanna Garroni, Jose Luis Menaldi

Chapman and Hall/CRC

240 pages

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pub: 2002-02-20
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Description

The Green function has played a key role in the analytical approach that in recent years has led to important developments in the study of stochastic processes with jumps. In this Research Note, the authors-both regarded as leading experts in the field- collect several useful results derived from the construction of the Green function and its estimates.

The first three chapters form the foundation for the rest of the book, presenting key results and background in integro-differential operators, and integro-differential equations. After a summary of the properties relative to the Green function for second-order parabolic integro-differential operators, the authors explore important applications, paying particular attention to integro-differential problems with oblique boundary conditions. They show the existence and uniqueness of the invariant measure by means of the Green function, which then allows a detailed study of ergodic stopping time and control problems.

Table of Contents

PREFACE

GLOSSARY OF BASIC NOTATIONS

ELLIPTIC EQUATIONS

Background

Problems Not in Divergence Form

Problems in Divergence Form

Markov-Feller Processes

INTEGRO-DIFFERENTIAL OPERATORS

Discussion

The Whole Space

Bounded Domains

Adjoint Operators

Unbounded Functions and Commutator

Relation with Jump Processes

INTEGRO-DIFFERENTIAL EQUATIONS

Problems Not in Divergence Form

Problems in Divergence Form

GREEN FUNCTION ESTIMATES

Discussion

Basic Properties

Green Spaces

Dirichlet Boundary Conditions

INVARIANT DENSITY MEASURE

Discussion

Ergodicity

Asymptotic Behavior

Boundary Singularity

STOPPING TIME PROBLEMS

Discussion

Setting of the Problem

Variational Inequality

Asymptotic Behavior

ERGODIC CONTROL PROBLEMS

Stochastic Control

Hamilton-Jacobi-Bellman Equation

BIBLIOGRAPHY

INDEX

About the Series

Chapman & Hall/CRC Research Notes in Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics