On non-singular renewal kernels with an application to a semigroup of transition kernels

Abstract

We study the non-singularity and limit properties of the renewal kernel $\text{R=∑K}{}^{\mathrm{\ast n}}$ associated with a positive convolution kernel K(x,dy×dt) defined on a general measurable space (E, $\text{E}$). The principal tool is the use of embedded renewal measures. As an application we consider continuous parameter semigroups (R_t(x,dy);t⩾0) of transition kernels on (E, $\text{E}$).