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Pseudo differential operators with negative definite symbols of variable order. (English) Zbl 0977.35151
In the remarkable paper [A symbolic calculus for pseudodifferential operators generating Feller semigroups, Osaka J. Math. 35, No. 4, 789-820 (1998; Zbl 0922.47045)] a deep inside of the analysis of pseudodifferential operators generating Markov processes is achieved. The author developed a symbolic calculus for (certain) pseudodifferential operators with negative definite symbols, i.e., pseudodifferential operators having a symbol \(p(x,\xi)\) such that \(\xi\mapsto p(x,\xi)\) admits a Lévy-Khinchin representation. In the paper under review this calculus is extended to operators of variable order, i.e., pseudo-differential operators with symbol \((p(x,\xi))^{\alpha(x)}\) where \(p(x,\xi)\) is as above. Such operators occur naturally in the probabilistic context of stable-like processes and sub-ordination in the sense of Bochner. The principal results of the paper are first of all the completely established symbolic calculus allowing a parametrics construction for “elliptic” elements and then its application to the construction of a Feller semigroup generated by \(-(p(x,0))^{\alpha(x)}\) which yields the existence of a corresponding Feller process.

35S05 Pseudodifferential operators as generalizations of partial differential operators
47D07 Markov semigroups and applications to diffusion processes
60J35 Transition functions, generators and resolvents
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