  We define the resurgence and asymptotic resurgence numbers associated to a
pair of graded families of ideals in a Noetherian ring. These notions
generalize the well-studied resurgence and asymptotic resurgence of an ideal in
a polynomial ring. We examine when these invariant are finite and rational. We
investigate situations where these invariant can be computed via Rees
valuations or realized as actual limits of well-defined sequences. We study how
the asymptotic resurgence changes when a family is replaced by its integral
closure. Many examples are given to illustrate that whether or not known
properties of resurgence and asymptotic resurgence of an ideal would extend to
that of a pair of graded families of ideals generally depends on the Noetherian
property and finite generation of the Rees algebras of these families.
