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Conservativeness of semigroups generated by pseudo differential operators. (English) Zbl 0917.60066
Let \(q(x,D)\) be a pseudo-differential operator with a symbol \(q(x,\xi)\). In a series of papers by Jacob and Hoh sufficient conditions were found for \(q(x,D)\) to be a generator of a Feller semigroup; see in particular N. Jacob [Math. Z. 215, No. 1, 151-166 (1994; Zbl 0795.35154)]. The author assumes that \(q(x,D)\) is a Feller generator and that an essential part of the above conditions holds (so that \(q(x,D)\) is a sum of a constant coefficient Lévy generator and a certain perturbation). It is shown that \(q(x,D)\) is conservative if and only if \(q(x,0)\equiv 0\). The sufficiency of this condition was already known [see W. Hoh, Stochastics Stochastics Rep. 55, No. 3/4, 225-252 (1995; Zbl 0880.47029)], but the author gives a new proof. The necessity was established by N. Jacob [Potential Anal. 8, No. 1, 61-68 (1998; Zbl 0908.60041)] only under strong additional assumptions.

MSC:
60J35 Transition functions, generators and resolvents
47D07 Markov semigroups and applications to diffusion processes
47G30 Pseudodifferential operators
35S05 Pseudodifferential operators as generalizations of partial differential operators
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