  In recent years, a number of functional inequalities have been derived for
Poisson random measures, with a wide range of applications. In this paper, we
prove that such inequalities can be extended to the setting of marked temporal
point processes, under mild assumptions on their Papangelou conditional
intensity. First, we derive a Poincar\'e inequality. Second, we prove two
transportation cost inequalities. The first one refers to functionals of marked
point processes with a Papangelou conditional intensity and is new even in the
setting of Poisson random measures. The second one refers to the law of marked
temporal point processes with a Papangelou conditional intensity, and extends a
related inequality which is known to hold on a general Poisson space. Finally,
we provide a variational representation of the Laplace transform of functionals
of marked point processes with a Papangelou conditional intensity. The proofs
make use of an extension of the Clark-Ocone formula to marked temporal point
processes. Our results are shown to apply to classes of renewal, nonlinear
Hawkes and Cox point processes.
