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E-grāmata: Second Order Elliptic Integro-Differential Problems

(Universita di Roma, 'Lasapienza', Italy), (Wayne State University, Detroit, Michigan, USA)
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Second Order Elliptic Integro-Differential ProblemsPalielināt
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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.
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


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