Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 131-147.

On honesty of perturbed substochastic $C_0$-semigroups in $L^1$-spaces

Authors: Mustapha Mokhtar-Kharroubi (1) and Juergen Voigt (2)
Author institution: (1) Departement de Mathematiques, Universite de Franche-Comte, 16 Route de Gray, F-25030 Besancon, France
(2) Fachrichtung Mathematik, Technische Universit\"at Dresden, D-01062 Dresden, Germany

Summary: Let $T$ be the generator of a positive contraction semigroup on $L^{1}(\Omega ,\mu )$, and let $B\colon D(T)\rightarrow L^{1}(\Omega ,\mu )$ be a positive linear operator such that $\int(Tf+Bf)\leq 0$ for all $f\in D(T)_{+}$. It is known that there exists a minimal positive contraction semigroup generated by some operator $K\supseteq T+B.$ This paper deals mainly with the total mass carried by trajectories $ ( \eul^{tK}f;t\geq 0 )$ with non-negative initial data $f$. In particular, our analysis covers the problem of whether $K=\overline{T+B}$ or $K\supsetneq \overline{T+B}$ \ and the related problem of the stochasticity or lack of stochasticity of `formally conservative'' perturbed positive semigroups in $L^{1}$-spaces.

Keywords: $C_0$-semigroup, perturbation, substochastic, honest trajectory.

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