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Conservativeness and Extensions of Feller Semigroups

Authors:	Schilling René L

Institution:	(1) Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße 22–26 D-04103 Leipzig, Germany

Abstract:	Let $\{ T_t \} _{t \geqslant 0}$ denote a Feller semigroup on $C_\infty (\mathbb{R}^n )$ , and $\{ \tilde T_t \} _{t \geqslant 0}$ itsextension to the bounded measurable functions. We show that $T_t 1 \in C_b (\mathbb{R}^n )$ . 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.

Keywords:	Feller semigroup Feller process conservativeness positive maximum principle pseudo-differential operator
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