, Volume 2, Issue 3, pp 239-256

Conservativeness and Extensions of Feller Semigroups

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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.