Pseudo-Differential Operators and Markov Processes: Generators and Potential Theory

Niels Jacob9781860943249, 9781860949562, 1860943241

In this volume two topics are discussed: the construction of Feller and Lp-sub-Markovian semigroups by starting with a pseudo-differential operator, and the potential theory of these semigroups and their generators. The first part of the text essentially discusses the analysis of pseudo-differential operators with negative definite symbols and develops a symbolic calculus; in addition, it deals with special approaches, such as subordination in the sense of Bochner. The second part handles capacities, function spaces associated with continuous negative definite functions, Lp -sub-Markovian semigroups in their associated Bessel potential spaces, Stein’s Littlewood-Paley theory, global properties of Lp-sub-Markovian semigroups, and estimates for transition functions.

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Table of contents :
Contents……Page 6
Preface……Page 8
General Notation……Page 10
Measures and Integrals……Page 12
Spaces of Functions Measures and Distributions……Page 13
Norms Scalar Products and Seminorms……Page 15
Notation from Functional Analysis Operators……Page 16
Notations related to Potential Theory and Harmonic Analysis……Page 17
Introduction: Pseudo-Differential Operators and Markov Processes……Page 20
II Generators and Their Potential Theory……Page 24
1 Introduction……Page 26
2.1 Second Order Elliptic Differential Operators as Generators of Feller and Sub-Markovian Semigroups……Page 36
2.2 Some Second Order Differential Operators with Non-Negative Characteristic Form as Generators of Sub-Markovian Semigroups……Page 71
2.3 Some Properties of Pseudo-Differential Operators with Negative Definite Symbols……Page 89
2.4 Hoh’s Symbolic Calculus for Pseudo-Differential Operators with Negative Definite Symbols……Page 110
2.5 Estimates for Pseudo-Differential Operators with Negative Definite Symbols Using the Symbolic Calculus……Page 136
2.6 Feller Semigroups and Sub-Markovian Semigroups Generated by Pseudo-Differential Operators……Page 149
2.7 Further Analytic Approaches for Constructing Feller and Sub- Markovian Semigroups……Page 161
2.8 Some Perturbation Results……Page 174
2.9 On Semigroups Obtained by Subordination……Page 195
2.10 Pseudo-Differential Operators with Variable Order of Differentiation as Generators of Feller Semigroups……Page 227
3.1 Capacities and Abstract Bessel Potential Spaces……Page 236
3.2 First Results on Lp-Sub-Markovian Semigroups in their Associated Bessel Potential Spaces……Page 267
3.3 Bessel Potential Spaces Associated with a Continuous Negative Definite Function……Page 293
3.4 Stein’s Littlewood-Paley Theory for Sub-Markovian Semigroups……Page 323
3.5 Global Properties of Lp-Sub-Markovian Semigroups……Page 359
3.6 Nash-Type and Sobolev-Type Inequalities-a Short Outline……Page 432
Bibliography……Page 440
Author Index……Page 466
Subject Index……Page 470