  We develop a theory of insertion and deletion tolerance for point processes.
A process is insertion-tolerant if adding a suitably chosen random point
results in a point process that is absolutely continuous in law with respect to
the original process. This condition and the related notion of
deletion-tolerance are extensions of the so-called finite energy condition for
discrete random processes. We prove several equivalent formulations of each
condition, including versions involving Palm processes. Certain other seemingly
natural variants of the conditions turn out not to be equivalent. We illustrate
the concepts in the context of a number of examples, including Gaussian zero
processes and randomly perturbed lattices, and we provide applications to
continuum percolation and stable matching.
