Second Order Elliptic Integro-Differential Problems

by Garroni; Maria Giovanna

• ISBN13: 9781584882008
• ISBN10: 158488200X
• Format: Nonspecific Binding
• Copyright: 2002-02-20
• Publisher: Chapman & Hall/

Summary

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	xi
Glossary of Basic Notations	xiii
Elliptic Equations	1	(42)
Backgrounds	1	(13)
Basic Function Spaces	1	(7)
Some Relations	8	(1)
Domain Conditions	9	(2)
Interpolation Inequalities	11	(1)
Extension and Trace Properties	12	(1)
Imbedding and Density Theorems	13	(1)
Problems Not in Divergence Form	14	(10)
Maximum Principles for Classic Solutions	16	(2)
A Priori Estimates for Classic Solutions	18	(2)
Existence and Uniqueness of Classic Solutions	20	(1)
Maximum Principales for Strong Solutions	20	(2)
A Priori Estimates for Strong Solutions	22	(1)
Existence and Uniqueness of Strong Solutions	23	(1)
Problems in Divergence Form	24	(11)
Interpretation of Weak Solutions	26	(3)
Maximum Principles for Weak Solutions	29	(2)
A Priori Estimates for Weak Solutions	31	(2)
Existence and Uniqueness of Weak Solutions	33	(2)
Markov-Feller Processes	35	(8)
Markov-Filler Semigroups	35	(3)
Wiener and Poisson Processes	38	(2)
Representation	40	(1)
Invariant Measure	41	(2)
Integro-Differential Operators	43	(40)
Discussion	43	(4)
The Whole Space	47	(8)
Bounded Domains	55	(9)
Adjoint Operators	64	(8)
Unbounded Functions and Commutator	72	(5)
Relation with Jump Processes	77	(6)
Integro-Differential Equations	83	(40)
Probelems Not in Divergence Form	83	(27)
Preliminaries and Comments	83	(5)
Maximum Principales for Classic Solutions	88	(6)
A Priori Estimates for Classic Solutions	94	(5)
Existence and Uniqueness of Classic Solutions	99	(2)
Maximum Principale for Strong Solutions	101	(1)
A Priori Estimates for Strong Solutions	102	(3)
Existence and Uniqueness of Strong Solutions	105	(5)
Problems in Divergence Form	110	(13)
Maximum Principles for Weak Solutions	114	(4)
Existence and Uniqueness of Weak Solutions	118	(5)
Green Function Estimates	123	(36)
Discussion	123	(5)
Parabolic Green Function	125	(2)
Elliptic Green Function	127	(1)
Basic Properties	128	(9)
Differential Part	129	(3)
Positive Lower Bound	132	(3)
Transition Density	135	(2)
Green Spaces	137	(21)
Bounded Time Interval	137	(17)
Unbounded Time Interval	154	(2)
Unbounded Domains	156	(2)
Dirichlet Boundary Conditions	158	(1)
Invariant Density Measure	159	(18)
Discussion	159	(6)
Ergodicity	165	(5)
Asymptotic Behavior	170	(3)
Boundary Singularity	173	(4)
Stopping Time Problems	177	(16)
Discussion	177	(1)
Setting of the Problem	178	(3)
Variational Inequality	181	(7)
Asymptotic Behavior	188	(5)
Ergodic Control Problems	193	(18)
Stochastic Control	193	(10)
Discussion	194	(1)
Reflected Diffusion with Jumps	194	(3)
Control Processes	197	(3)
Ergodic Optimal Control	200	(1)
Doeblin Condition	201	(2)
Hamilton-Jacobi-Bellman Equation	203	(8)
Bibliography	211	(8)
Index	219