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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.

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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