Markov Operators, Positive Semigroups and Approximation Processes

✍ Scribed by Francesco Altomare; Mirella Cappelletti; Vita Leonessa; Ioan Rasa

• Publisher: De Gruyter
• Year: 2014
• Tongue: English
• Leaves: 326
• Series: De Gruyter Studies in Mathematics; 61
• Category: Library

✦ Synopsis

This research monograph gives a detailed account of a theory which is mainly concerned with certain classes of degenerate differential operators, Markov semigroups and approximation processes. These mathematical objects are generated by arbitrary Markov operators acting on spaces of continuous functions defined on compact convex sets; the study of the interrelations between them constitutes one of the distinguishing features of the book.

Among other things, this theory provides useful tools for studying large classes of initial-boundary value evolution problems, the main aim being to obtain a constructive approximation to the associated positive C₀-semigroups by means of iterates of suitable positive approximating operators. As a consequence, a qualitative analysis of the solutions to the evolution problems can be efficiently developed.

The book is mainly addressed to research mathematicians interested in modern approximation theory by positive linear operators and/or in the theory of positive C₀-semigroups of operators and evolution equations. It could also serve as a textbook for a graduate level course.

✦ Table of Contents

Preface
Introduction
Guide to the reader and interdependence of sections
Notation
1 Positive linear operators and approximation problems
1.1 Positive linear functionals and operators
1.1.1 Positive Radon measures
1.1.2 Choquet boundaries
1.1.3 Bauer simplices
1.2 Korovkin-type approximation theorems
1.3 Further convergence criteria for nets of positive linear operators
1.4 Asymptotic behaviour of Lipschitz contracting Markov semigroups
1.5 Asymptotic formulae for positive linear operators
1.6 Moduli of smoothness and degree of approximation by positive linear operators
1.7 Notes and comments
2 C0-semigroups of operators and linear evolution equations
2.1 C0-semigroups of operators and abstract Cauchy problems
2.1.1 C0-semigroups and their generators
2.1.2 Generation theorems and abstract Cauchy problems
2.2 Approximation of C0-semigroups
2.3 Feller and Markov semigroups of operators
2.3.1 Basic properties
2.3.2 Markov Processes
2.3.3 Second-order differential operators on real intervals and Feller theory
2.3.4 Multidimensional second-order differential operators and Markov semigroups
2.4 Notes and comments
3 Bernstein-Schnabl operators associated with Markov operators
3.1 Generalities, definitions and examples
3.1.1 Bernstein-Schnabl operators on [0,1]
3.1.2 Bernstein-Schnabl operators on Bauer simplices
3.1.3 Bernstein operators on polytopes
3.1.4 Bernstein-Schnabl operators associated with strictly elliptic differential operators
3.1.5 Bernstein-Schnabl operators associated with tensor products of Markov operators
3.1.6 Bernstein-Schnabl operators associated with convex combinations of Markov operators
3.1.7 Bernstein-Schnabl operators associated with convex convolution products of Markov operators
3.2 Approximation properties and rate of convergence
3.3 Preservation of Hölder continuity
3.3.1 Smallest Lipschitz constants and triangles
3.3.2 Smallest Lipschitz constants and parallelograms
3.4 Bernstein-Schnabl operators and convexity
3.5 Monotonicity properties
3.6 Notes and comments
4 Differential operators and Markov semigroups associated with Markov operators
4.1 Asymptotic formulae for Bernstein-Schnabl operators
4.2 Differential operators associated with Markov operators
4.3 Markov semigroups generated by differential operators associated with Markov operators
4.4 Preservation properties and asymptotic behaviour
4.5 The special case of the unit interval
4.5.1 Degenerate differential operators on [0,1]
4.5.2 Approximation properties by means of Bernstein-Schnabl operators
4.5.3 Preservation properties and asymptotic behaviour
4.5.4 The saturation class of Bernstein-Schnabl operators and the Favard class of their limit semigroups
4.6 Notes and comments
5 Perturbed differential operators and modified Bernstein-Schnabl operators
5.1 Lototsky-Schnabl operators
5.2 A modification of Bernstein-Schnabl operators
5.3 Approximation properties
5.4 Preservation properties
5.5 Asymptotic formulae
5.6 Modified Bernstein-Schnabl operators and first-order perturbations
5.7 The unit interval
5.7.1 Complete degenerate second-order differential operators on [0, 1]
5.7.2 Approximation properties by means of modified Bernstein-Schnabl operators
5.8 The d-dimensional simplex and hypercube
5.9 Notes and comments
Appendices
A.1 A classification of Markov operators on two dimensional convex compact subsets
A.2 Rate of convergence for the limit semigroup of Bernstein operators
Bibliography
Symbol index
Index
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