Functional inequalities for marked point processes

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