Growth and asymptotics of perturbed recurrent semigroups
Given a strongly continuous semigroup \((e^{tA})_{t\geq 0}\) on \(L^2(\Omega ,m)\), where \((\Omega ,m)\) is any \(\sigma\)-finite measure space. It is assumed that this semigroup and its adjoint \((e^{tA^*})_{t\geq 0}\) are sub-Markovian. Therefore, by interpolation, \((e^{tA})_{t\geq 0}\) induces positive strongly continuous semigroups on \(L^p(\Omega ,m)\) \((1\leq p<\infty)\). In this paper the author proves that if \((e^{tA})_{t\geq 0}\) is recurrent, i.e., \((e^{tA})_{t\geq 0}\) is irreducible and \(\sup_{n\geq 0}\int_0^ne^{tA}f(x) dt=\infty \) (a.e.), for all \(f\in L^1(\Omega ,m), f\not \equiv 0\), then \((e^{tA})_{t\geq 0}\) is not \(L^1\)-asymptotically stable but every positive semigroup dominated strictly by \((e^{tA})_{t\geq 0}\) is \(L^1\)-asymptotically stable. In the case where \((e^{tA})_{t\geq 0}\) is self-adjoint and recurrent, the author proves that the growth bound \(\omega(-B)>0\) for any self-adjoint strongly continuous semigroup with \(e^{-tA} <e^{-tB}, t\geq 0\). The results are applied to study the asymptotic behavior of perturbations of second order elliptic operators by measures.
perturbations of second-order elliptic operators by measures
scientific article; zbMATH DE number 1057421