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MiS Preprint

Conservativeness and extensions of Feller semigroups

Rene Schilling


Let $\{T_i\}_i \geq 0$ denote a Feller semigroup on the space of functions vanishing at infinity, $C_{\infty} (R^K)$, and $\{T_i\}_i \geq 0$ its (canonical) extension to the bounded measurable functions. We show that $T_i 1 \in C_b (\mathbb{R}^k)$ is necessary and sufficient for the invariance of $C_b (\mathbb{R}^k)$ under $\{T_i\}_i \geq 0$. 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.

MSC Codes:
47D07, 60J35, 47G30
feller semigroup, feller process, conservativeness, positive maximum principle, pseudo-differential operator

Related publications

1998 Repository Open Access
Rene L. Schilling

Conservativeness and extensions of Feller semigroups

In: Positivity, 2 (1998) 3, pp. 239-256