We give a general coalgebraic account of the notions of infinite trace and infinite execution in state-based, dynamical systems, by extending the generic theory of finite traces and executions developed by Hasuo and coauthors \cite{jacobs-journal}. The systems we consider are modelled as coalgebras of endofunctors obtained as the composition of a computational type (e.g.~nondeterministic or stochastic) with a general transition type. This generalises existing work by Jacobs \cite{jacobs-04} that only accounts for a nondeterministic computational type. We subsequently introduce path-based temporal (including fixpoint) logics for coalgebras of such endofunctors, whose semantics is based upon the notion of infinite execution. Our approach instantiates to both nondeterministic and stochastic computations, yielding, in particular, path-based fixpoint logics in the style of CTL* for nondeterministic systems, as well as generalisations of the logic PCTL for probabilistic systems.

\begin{thebibliography}{1}

\bibitem{jacobs-journal}
I.~Hasuo, B.~Jacobs, and A.~Sokolova.
\newblock Generic trace semantics via coinduction.
\newblock {\em Logical Methods in Computer Science}, 3:1--36, 2007.

\bibitem{jacobs-04}
B.~Jacobs.
\newblock Trace semantics for coalgebras.
\newblock In {\em Proc.\,\,CMCS~2004}, volume 106 of {\em ENTCS}, 2004.

\end{thebibliography}