  The aim of this paper is to present some results relating the properties of
stability, concentration and approximation to the identity of convolution
through not necessarily mollification type families of heavy tailed Markov
kernels. A particular case is provided by the kernels $K_t$ obtained as the $t$
mollification of $L^{\sigma(t)}$ selected from the family
$\mathcal{L}=\{L^{\sigma}:
\widehat{L^{\sigma}}(\xi)=e^{-|\xi|^\sigma},0<\sigma<2\}$, by a given function
$\sigma$ with values in the interval $(0,2)$. We show that a basic Harnack type
inequality, introduced by C.~Calder\'{o}n in the convolution case, becomes at
once natural to the setting and useful to connect the concepts of stability,
concentration and approximation of the identity. Some of the general results
are extended to spaces of homogeneous type since most of the concepts involved
in the theory are given in terms of metric and measure.
