  The \v{C}ech and Rips constructions of persistent homology are stable with
respect to perturbations of the input data. However, neither is robust to
outliers, and both can be insensitive to topological structure of high-density
regions of the data. A natural solution is to consider 2-parameter persistence.
This paper studies the stability of 2-parameter persistent homology: We show
that several related density-sensitive constructions of bifiltrations from data
satisfy stability properties accommodating the addition and removal of
outliers. Specifically, we consider the multicover bifiltration, Sheehy's
subdivision bifiltrations, and the degree bifiltrations. For the multicover and
subdivision bifiltrations, we get 1-Lipschitz stability results closely
analogous to the standard stability results for 1-parameter persistent
homology. Our results for the degree bifiltrations are weaker, but they are
tight, in a sense. As an application of our theory, we prove a law of large
numbers for subdivision bifiltrations of random data.
