# Conservativeness and Extensions of Feller Semigroups

Conservativeness and Extensions of Feller Semigroups 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Conservativeness and Extensions of Feller Semigroups

, Volume 2 (3) – Oct 7, 2004
18 pages

/lp/springer_journal/conservativeness-and-extensions-of-feller-semigroups-d0SzH6kz3a
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1023/A:1009748105208
Publisher site
See Article on Publisher Site

### 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.

### Journal

PositivitySpringer Journals

Published: Oct 7, 2004

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