Perturbations of bi-continuous semigroups with applications to transition semigroups on \(C_b(H)\)
In the paper under review, the author shows an unbounded perturbation theorem of Miyadera-Voigt type for bi-continuous semigroups on the space \(C_b(\Omega)\) of all bounded and continuous functions on \(\Omega\) endowed with the compact open-topology \(\tau_c\), with \(\Omega\) a Polish space (the bi-continuous semigroups were introduced and studied by \textit{F. Kühnemund} [Bi-continuous semigroups on spaces with two topologies: theory and applications (Ph. D. Thesis, Tübingen) (2001)]. Then such a result is applied in order to prove the existence of the transition semigroup on the whole space \(C_b(H)\), endowed with the compact open-topology \(\tau_c\), for any bounded nonlinear drift term \(F\in C_b(H, H)\), associated to the stochastic differential equation \[ \begin{aligned} dX(t)&= (AX(t)+F(X(t)))dt+Q^{1/2}dW(t), \quad t\geq 0,\\ X(0)&= x , \quad x\in H,\end{aligned} \] where \((A,D(A))\) is the generator of a \(C_0\)-semigroup \((S(t))_{t\geq 0}\) on a separable Hilbert space \(H\), \(W\) is an \(H\)-valued cylindrical Wiener process, and \[ Q(t)=\int_0^t S(s)QS^*(s)\,ds, \quad t\geq 0, \] with \(Q\) a bounded, positive, self-adjoint operator on \(H\) for which \(\operatorname {Ker}Q=\{0\}\).
scientific article; zbMATH DE number 2077987