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Lévy-type processes and pseudodifferential operators. (English) Zbl 0984.60054
Barndorff-Nielsen, Ole E. (ed.) et al., Lévy processes. Theory and applications. Boston: Birkhäuser. 139-168 (2001).
In this survey the authors show how pseudodifferential operators arise naturally in the theory of Markov processes and that this opens the way to use Fourier analytic techniques for general Markov processes. In Section 1, it is seen that one can identify the characteristic exponent $$\psi(\xi)$$ of a Lévy process with the symbol of its infinitesimal generator which is a pseudodifferential operator with constant coefficients. In the subsequent sections, they discuss sample path properties and global properties of Lévy-type processes, construction of Lévy-type processes and subordination of general Lévy-type processes and its relations to functional analysis. In the last section, they mention very briefly three other topics where pseudodifferential operators enter the theory of Markov processes.
For the entire collection see [Zbl 0961.00012].

##### MSC:
 60G51 Processes with independent increments; Lévy processes