  Piecewise Deterministic Markov Processes (PDMPs) are studied in a general
framework. First, different constructions are proven to be equivalent. Second,
we introduce a coupling between two PDMPs following the same differential flow
which implies quantitative bounds on the total variation between the marginal
distributions of the two processes. Finally two results are established
regarding the invariant measures of PDMPs. A practical condition to show that a
probability measure is invariant for the associated PDMP semi-group is
presented. In a second time, a bound on the invariant probability measures in
$V$-norm of two PDMPs following the same differential flow is established. This
last result is then applied to study the asymptotic bias of some non-exact PDMP
MCMC methods.
