Summary: The main goal of this paper is to establish some equivalence results on stability, recurrence, and ergodicity between a piecewise deterministic Markov process (PDMP) \(\{X(t)\}\) and an embedded discrete-time Markov chain \(\{\Theta_{n}\}\) generated by a Markov kernel \(G\) that can be explicitly characterized in terms of the three local characteristics of the PDMP, leading to tractable criterion results. First we establish some important results characterizing \(\{\Theta_{n}\}\) as a sampling of the PDMP \(\{X(t)\}\) and deriving a connection between the probability of the first return time to a set for the discrete-time Markov chains generated by \(G\) and the resolvent kernel \(R\) of the PDMP. From these results we obtain equivalence results regarding irreducibility, existence of \(\sigma\)-finite invariant measures, and (positive) recurrence and (positive) Harris recurrence between \(\{X(t)\}\) and \(\{\Theta_{n}\}\), generalizing the results of F. Dufour and O. L. V. Costa [SIAM J. Control Optim. 37, 1483–1502 (1999; Zbl 0936.60062)] in several directions. Sufficient conditions in terms of a modified Foster-Lyapunov criterion are also presented to ensure positive Harris recurrence and ergodicity of the PDMP. We illustrate the use of these conditions by showing the ergodicity of a capacity expansion model.