Model Examples

Abstract

AbstractThis chapter is devoted to some basic model examples, which will serve as a guide throughout the book. They are the opportunity to illustrate some of the ideas, definitions and properties of Markov semigroups, operators and processes and will help to set up the framework of the investigation of more general Markov semigroups and generators which will be achieved in the next chapter. The chapter starts with the three geometric models of the heat semigroup on the Euclidean space, the sphere and the hyperbolic space. Further sections discuss the heat semigroup with Neumann, Dirichlet or periodic conditions on \({\mathbb{R}}\) or on an interval of \({\mathbb{R}}\). More general Sturm–Liouville operators on an interval of the real line are examined next, illustrated by the examples of the Ornstein–Uhlenbeck or Hermite, Laguerre and Jacobi operators.KeywordsInvariant MeasureOrthogonal PolynomialHeat KernelInfinitesimal GeneratorJacobi OperatorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.