Pseudo-differential operators and Markov processes.

*(English)*Zbl 0860.60002
Mathematical Research. 94. Berlin: Akademie Verlag. 207 p. (1996).

By a result of Ph. Courrège it is known that generators of Markov processes in \(\mathbb{R}^n\) are pseudo-differential operators with symbols which are continuous negative definite functions with respect to the covariable. In his monograph, the author gives an overview about this subject in general and in particular reports on recent results. The book has the character of a survey, so there is no attempt made to give complete proofs, but the general idea is to fill the gap between probability theory and analysis and to give a non-technical introduction for mathematicians of both fields.

In the beginning the basic concept of a pseudo-differential operator as a generator of a process is explained. To get a first impression of the connection between symbol and process the classical case of Lévy processes is investigated. Next, it is shown by what means a process can be constructed starting with a given symbol. This is done in different settings such as Dirichlet spaces, Feller semigroups and the martingale problem. The following chapters consider the problem how information about the process can be obtained from the symbol. So subordination in the sense of Bochner, global properties of the process like recurrence and conservativeness and path properties of the process are studied. The next chapters are devoted to the corresponding Dirichlet problem which is considered in the framework of balayage spaces, Dirichlet spaces and in a probabilistic framework. All chapters are complemented by appendices which deepen the topic or give background information concerning the applied tools for analysts and probabilists as well. In the next two chapters related topics in the literature are discussed, i.e. boundary value problems and Wentzell boundary conditions, relativistic Hamiltonians, spectral analysis of Feller semigroups, operators on locally compact Abelian groups and nilpotent Lie groups. The text closes with a chapter that explains in more detail the techniques used to treat the class of pseudo-differential operators under consideration.

In the beginning the basic concept of a pseudo-differential operator as a generator of a process is explained. To get a first impression of the connection between symbol and process the classical case of Lévy processes is investigated. Next, it is shown by what means a process can be constructed starting with a given symbol. This is done in different settings such as Dirichlet spaces, Feller semigroups and the martingale problem. The following chapters consider the problem how information about the process can be obtained from the symbol. So subordination in the sense of Bochner, global properties of the process like recurrence and conservativeness and path properties of the process are studied. The next chapters are devoted to the corresponding Dirichlet problem which is considered in the framework of balayage spaces, Dirichlet spaces and in a probabilistic framework. All chapters are complemented by appendices which deepen the topic or give background information concerning the applied tools for analysts and probabilists as well. In the next two chapters related topics in the literature are discussed, i.e. boundary value problems and Wentzell boundary conditions, relativistic Hamiltonians, spectral analysis of Feller semigroups, operators on locally compact Abelian groups and nilpotent Lie groups. The text closes with a chapter that explains in more detail the techniques used to treat the class of pseudo-differential operators under consideration.

Reviewer: W.Hoh (Bielefeld)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35S99 | Pseudodifferential operators and other generalizations of partial differential operators |

60J25 | Continuous-time Markov processes on general state spaces |

31C25 | Dirichlet forms |

60J75 | Jump processes (MSC2010) |