Markov Semigroups
This opening chapter introduces, somewhat informally, some of the basic ideas and concepts in the investigation of Markov semigroups, operators and processes, at the interface between analysis, partial differential equations, probability theory and geomet
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Markov Semigroups
This chapter is introductory and descriptive. It aims to introduce, somewhat informally, some of the basic ideas and concepts in the investigation of Markov semigroups, operators and processes, at the interface between analysis, partial differential equations, probability theory and geometry, loosely jumping from one area to another. Readers only familiar with one, or two, or even none of these fields, will find the necessary background in the various appendices. The chapter is not intended to be read linearly, but should be used as a vade mecum of ideas and tools on the topic of Markov semigroups. Each section introduces and sheds light on one or several aspects of the investigation of Markov semigroups, aiming at this point to develop intuition rather than giving precise definitions and hypotheses. Pointers to more precise descriptions and examples appearing in subsequent chapters of the book are provided. In particular, some of the main features of the analysis will be illustrated by model examples in Chap. 2, while Chap. 3 will develop the complete formalism describing the suitable environment in which the full theory may easily be developed. For the reader’s convenience, a first set of precise assumptions concerning state spaces, measures, semigroups etc., is nevertheless put forward here in Sect. 1.14. There are many ways to address Markov semigroups and generators. In this chapter, we investigate their elementary properties starting with the assumption that we already know the semigroup itself which then leads to the definition of the so-called infinitesimal generator of the semigroup, which in return entirely describes the latter. In Chap. 3, conversely, we begin with the generator, or rather its carré du champ operator, given on some suitable class of functions, and use it to determine properties of the semigroup. This setting then describes the formalism in which the monograph will evolve. Generally speaking, a semigroup P = (Pt )t≥0 is a family of operators acting on some suitable function space with the semigroup property Pt ◦ Ps = Pt+s , t, s ≥ 0, P0 = Id. Such families naturally arise in numerous settings, and describe evolution equations which may be investigated from various viewpoints. In particular, these semigroups appear in the probabilistic context describing the family of laws of Markov processes (Xt )t≥0 living on a measurable space E, the fundamental relaD. Bakry et al., Analysis and Geometry of Markov Diffusion Operators, Grundlehren der mathematischen Wissenschaften 348, DOI 10.1007/978-3-319-00227-9_1, © Springer International Publishing Switzerland 2014
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Markov Semigroups
tion being given by the conditional expectation representation Pt f (x) = E f (Xt ) | X0 = x for t ≥ 0, x ∈ E and f : E → R a suitable measurable function. In order to illustrate, and introduce the reader to, these investigations, let us describe the simplest example, namely the heat or Brownian semigroup on the Euclidean space Rn . This example may be introduced in many ways, depending
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