Conservativeness and extensions of Feller semigroups
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Submission date: 04. Dec. 1997
published in: Positivity, 2 (1998) 3, p. 239-256
DOI number (of the published article): 10.1023/A:1009748105208
MSC-Numbers: 47D07, 60J35, 47G30
Keywords and phrases: feller semigroup, feller process, conservativeness, positive maximum principle, pseudo-differential operator
Let denote a Feller semigroup on the space of functions vanishing at infinity, , and its (canonical) extension to the bounded measurable functions. We show that is necessary and sufficient for the invariance of under . If the generator of the semigroup is a pseudo-differential operator we can restate this condition in terms of the symbol. As a by-product, we obtain necessary and sufficient conditions for the conservativeness of the semigroup which are again expressed through the symbol.