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

Ryoichiro Noda

TL;DR

The paper develops space--time occupation measures (STOMs) as a unified framework for PCAFs of Markov processes and derives convergence results when the underlying spaces, heat kernels, and smooth measures converge under a uniform potential-decay condition. It introduces collision measures within the STOM framework to capture both collision sites and times for independent processes, proving general convergence theorems and applying them to scaling limits on resistance metric spaces and critical random graphs. The methodology relies on a robust Gromov--Hausdorff-type topology for spaces with additional structures, heat-kernel estimates, and the Revuz correspondence linking PCAFs to smooth measures. This framework enables precise analysis of collisions in disordered media and yields concrete scaling limits for random walks on complex networks, providing new tools for studying multi-particle interactions in heterogeneous environments.

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 Erdős--Rényi random graphs, and the uniform spanning tree.

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

TL;DR

The paper develops space--time occupation measures (STOMs) as a unified framework for PCAFs of Markov processes and derives convergence results when the underlying spaces, heat kernels, and smooth measures converge under a uniform potential-decay condition. It introduces collision measures within the STOM framework to capture both collision sites and times for independent processes, proving general convergence theorems and applying them to scaling limits on resistance metric spaces and critical random graphs. The methodology relies on a robust Gromov--Hausdorff-type topology for spaces with additional structures, heat-kernel estimates, and the Revuz correspondence linking PCAFs to smooth measures. This framework enables precise analysis of collisions in disordered media and yields concrete scaling limits for random walks on complex networks, providing new tools for studying multi-particle interactions in heterogeneous environments.

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 Erdős--Rényi random graphs, and the uniform spanning tree.
Paper Structure (48 sections, 115 theorems, 519 equations)

This paper contains 48 sections, 115 theorems, 519 equations.

Table of Contents

  1. Introduction
  2. An overview of the framework
  3. Main results
  4. Application to the study of collisions
  5. Organization of the paper and notational conventions
  6. Preliminaries on topologies and metrizations
  7. The Hausdorff and Fell topologies
  8. The weak and vague topologies
  9. The compact-convergence topology with variable domains
  10. Topologies on the space of càdlàg functions
  11. The spaces for PCAFs and STOMs
  12. Topology on the space of metric spaces equipped with additional structures
  13. General framework
  14. Product and composition
  15. Structures used in the present paper
  16. ...and 33 more sections

Key Result

Theorem 1.3

Assume that the following conditions are satisfied. Then in $\mathfrak{M}_\bullet\bigl(\tau_{\mathcal{M}} \times \tau_{\mathrm{HK}} \times \tau_{\mathrm{SP, STOM}}\bigr)$.

Theorems & Definitions (281)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Noda_pre_Aging
  • Lemma 2.4
  • proof
  • ...and 271 more