The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of perturbations. This has two main implications: it makes the space of persistence diagrams rather pathological and it is often provides very pessimistic bounds with respect to outliers. In this paper, we provide new stability results with respect to the $p$-Wasserstein distance between persistence diagrams. This includes an elementary proof for the setting of functions on sufficiently finite spaces in terms of the $p$-norm of the perturbations, along with an algebraic framework for $p$-Wasserstein distance which extends the results to wider class of modules. We also provide apply the results to a wide range of applications in topological data analysis (TDA) including topological summaries, persistence transforms and the special but important case of Vietoris-Rips complexes.

This paper now only the cellular stability and applications. For the algebraic part see previous version as it will be made into a standalone paper shortly