Pseudo Differential Operators and Markov Processes

In this volume two topics are discussed: the construction of Feller and L^p–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, L^p –sub-Markovian semigroups in their associated Bessel potential spaces, Stein's Littlewood–Paley theory, global properties of L^p–sub-Markovian semigroups, and estimates for transition functions.

• Generators of Feller and Sub-Markovian Semigroups:
  • Second Order Elliptic Differential Operators as Generators of Feller and Sub-Markovian Semigroups
  • Some Second Order Differential Operators with Non-Negative Characteristic Form as Generators of Sub-Markovian Semigroups
  • Some Properties of Pseudo-Differential Operators with Negative Definite Symbols
  • Hoh's Symbolic Calculus for Pseudo-Differential Operators with Negative Definite Symbols
  • Estimates for Pseudo-Differential Operators with Negative Definite Symbols Using the Symbolic Calculus
  • Feller Semigroups and Sub-Markovian Semigroups Generated by Pseudo-Differential Operators
  • Further Analytic Approaches for Constructing Feller and Sub-Markovian Semigroups
  • Some Perturbation Results
  • On Semigroups Obtained by Subordination
  • Pseudo-Differential Operators with Variable Order of Differentiation as Generators of Feller Semigroups
• Potential Theory of Semigroups and Generators:
  • Capacities and Abstract Bessel Potential Spaces
  • First Results on L^p-Sub-Markovian Semigroups in their Associated Bessel Potential Spaces
  • Bessel Potential Spaces Assocated with a Continuous Negative Definite Function
  • Stein's Littlewood–Paley Theory for Sub-Markovian Semigroups
  • Global Properties of L^p-Sub-Markovian Semigroups
  • Nash-Type and Sobolev-Type Inequalities — a Short Outline