Convergence of space-time occupation measures of stochastic processes and its application to collisions

Abstract: We introduce a new perspective on positive continuous additive functionals (PCAFs) of Markov processes, which we call space--time occupation measures (STOMs). This notion provides a natural generalization of classical occupation times and occupation measures, and offers a unified framework for studying their convergence. We analyze STOMs via so-called smooth measures associated with PCAFs through the Revuz correspondence. We establish that if the underlying spaces, the processes living on them, their heat kernels, and the associated smooth measures converge, and if the corresponding potentials of these measures satisfy a uniform decay condition, then the associated PCAFs and STOMs also converge in suitable Gromov--Hausdorff-type topologies. We then apply this framework to the analysis of collisions of independent stochastic processes. Specifically, by exploiting the STOM formulation, we introduce the notion of collision measures, which record both the collision sites and times of two processes, and prove general convergence theorems for these measures. The abstract results are further specialized to random walks on electrical networks via the theory of resistance metric spaces, leading to concrete scaling limits for collision measures of random walks on critical random graphs, such as critical Galton--Watson trees, critical Erd\H{o}s--R\'enyi random graphs, and the uniform spanning tree.