Liouville equation and the q-statistical formalism

We revisit some aspects of the foundations of the q-non-extensive thermostatistical formalism, which are particularly relevant to its astronomical applications. We analyse some important features of escort mean values and escort distributions, in connection with the dynamical evolution of ensemble probability distributions associated with the statistical description of general classical dynamical systems. Furthermore, we compare the roles played by escort mean values and escort distributions when studying classical N-body systems at the levels of (i) the Liouville equation and (ii) the Vlasov equation.