  We develop a canonical framework for the study of the problem of registration
of multiple point processes subjected to warping, known as the problem of
separation of amplitude and phase variation. The amplitude variation of a real
random function $\{Y(x):x\in[0,1]\}$ corresponds to its random oscillations in
the $y$-axis, typically encapsulated by its (co)variation around a mean level.
In contrast, its phase variation refers to fluctuations in the $x$-axis, often
caused by random time changes. We formalise similar notions for a point
process, and nonparametrically separate them based on realisations of i.i.d.
copies $\{\Pi_i\}$ of the phase-varying point process. A key element in our
approach is to demonstrate that when the classical phase variation assumptions
of Functional Data Analysis (FDA) are applied to the point process case, they
become equivalent to conditions interpretable through the prism of the theory
of optimal transportation of measure. We demonstrate that these induce a
natural Wasserstein geometry tailored to the warping problem, including a
formal notion of bias expressing over-registration. Within this framework, we
construct nonparametric estimators that tend to avoid over-registration in
finite samples. We show that they consistently estimate the warp maps,
consistently estimate the structural mean, and consistently register the warped
point processes, even in a sparse sampling regime. We also establish
convergence rates, and derive $\sqrt{n}$-consistency and a central limit
theorem in the Cox process case under dense sampling, showing rate optimality
of our structural mean estimator in that case.
