Collision Measures in Stochastic Processes

• Collision measures are mathematical constructs that quantify the spatiotemporal intersections of independent stochastic processes via space–time occupation measures.
• They employ positive continuous additive functionals (PCAFs) and the Revuz correspondence to rigorously analyze convergence, scaling limits, and dynamics on varying metric spaces.
• Applications include random walks on electrical networks and critical random graphs, providing insights into inhomogeneity, fluctuation analysis, and rare-event asymptotics.

A collision measure is a mathematical construct quantifying the joint spatial–temporal occurrence of "collisions" between trajectories of stochastic processes, or, more generally, the spatiotemporal overlap of independent random processes. Recent developments recast collision measures as special instances of "space–time occupation measures" (STOMs)—random measures associated with positive continuous additive functionals (PCAFs)—and provide a unified analytic and convergence framework for their paper. This approach enables rigorous analysis of scaling limits, convergence on varying spaces, and links to resistance network theory and random graphs.

1. Space–Time Occupation Measures and PCAFs

The central object is the positive continuous additive functional (PCAF) $A = (A_t)_{t \geq 0}$ of a Markov process $X$. The associated space–time occupation measure (STOM) is defined for any Borel set $E \subset S \times [0, \infty)$ as

$$\Pi(E) = \int_0^\infty 1_{(X_t, t) \in E} \, dA_t\qquad\text{(Eq. 1)}$$

This construction generalizes classical occupation times (where $A_t = t$ yields the total time spent in $E$) and enables the inclusion of more complex functionals like local times or additive functionals weighted by a smooth measure. The deep connection between PCAFs and "smooth measures" is made explicit via the Revuz correspondence: for non-negative bounded Borel $f$ and any $\alpha > 0$,

$$\mathbb{E}^x \left[ \int_{0}^\infty e^{-\alpha t} f(X_t) dA_t \right] = \int_S r^{(\alpha)}(x, y) f(y) \mu(dy)\qquad\text{(Eq. 2)}$$

Here $r^{(\alpha)}(x,y)$ is the $\alpha$-potential kernel and $\mu$ is the unique smooth measure associated with $A$. Thus, the STOM generalizes occupation measures and provides an analytic tool for tracking the "presence" of a process across space and time.

2. Gromov–Hausdorff–Type Convergence Theorems

The convergence theory of STOMs is developed under settings where both the underlying spaces and processes vary. Given a sequence of metric measure spaces, processes, smooth measures, and heat kernels $(S_n, d_{S_n}, \rho_n, \mu_n, a_n, p_n, L_{x_n})$ converging to $(S, d_S, \rho, \mu, a, p, L_x)$ in a Gromov–Hausdorff–type topology, the associated PCAFs and STOMs also converge, provided a uniform potential decay holds:

$$\lim_{\alpha \to \infty} \limsup_n \Vert R_{p_n}^{\alpha}(\mu_n^{(r)}) \Vert_\infty = 0 \qquad \forall r > 0\qquad\text{(Eq. 4)}$$

This technical condition ensures that "short-time" heat kernel integrals against $\mu_n$ vanish, preventing mass accumulation. The space of objects is enhanced to carry measures, heat kernels, and law maps, allowing joint convergence of processes, associated functionals, and measures even when the underlying geometry is random or changes with $n$ (e.g., in critical random graphs). Through mollification of the heat kernel and delicate use of the metric, convergence of PCAFs and thus STOMs is guaranteed under these hypotheses.

3. Collision Measure Construction

Within this framework, collision measures are constructed by considering two independent Markov processes $X^{1}, X^{2}$ on the same space $S$. The product process $\hat{X}_t = (X_t^1, X_t^2)$, under the law $$\mathbb{P}^{(x_1,x_2)} = \mathbb{P}^{x_1} \otimes \mathbb{P}^{x_2}$$, collides when $\hat{X}_t$ lies on the diagonal:

$$\operatorname{diag}: S \rightarrow S \times S,\quad x \mapsto (x, x)$$

The collision measure records the times and locations where collisions occur:

• First, push forward a "weighting measure" $\mu$ (often the invariant measure of the process) via the diagonal to $S \times S$.
• Use Revuz correspondence and the product process to define a PCAF and associated STOM for the product process.
• The collision measure $\Pi$ on $S \times [0,\infty)$ is then

$$\Pi = \hat{\Pi} \circ (\operatorname{diag} \times \mathrm{id}_{[0,\infty)})$$

In the discrete setting with canonical weighting,

$$\Pi(dx\, dt) = \sum_x 1\{ X_t^1 = X_t^2 = x \} \delta_x(dx)\, dt$$

The measure thus assigns unit mass at each $(x, t)$ where the two processes collide at site $x$ at time $t$.

4. Applications: Random Walks on Electrical Networks and Random Graphs

A significant application concerns random walks on electrical networks and resistance metric spaces, especially critical random graphs. In variable-speed random walks (VSRWs) on critical Galton–Watson trees, critical Erdős–Rényi graphs, and the uniform spanning tree, the canonical weighting measure is the ordinary counting measure (owing to uniform invariance), and scaling limits of the collision measure can be rigorously established. For constant-speed random walks (CSRWs), the squared conductance measure is relevant, and analogous convergence statements for collision measures hold under appropriate measure convergence. Heat kernel estimates (e.g., $$\lim_{\delta\to 0}\sup_x \int_0^\delta p(2t, x, y)dt = 0$$ for all bounded sets) can be verified via volume growth techniques when the spectral dimension is sufficiently low (ensuring recurrence and frequent collisions). This specialization provides concrete limiting descriptions for collision measures in high-profile models of random graphs and networks.

5. Significance of Collision Measures and STOMs

The developed theory has several important implications:

• Sensitive Probes of Inhomogeneity: Collision measures can capture residual inhomogeneities even when single-particle dynamics homogenize in the limit. For example, in the Bouchaud trap model, collisions occur more frequently near deep traps despite homogenized marginal motion, highlighting the sensitivity of collision measures.
• Unified Treatment of Scaling and Limits: The Gromov–Hausdorff-convergence approach facilitates the paper of processes, spaces, and measures that all change with $n$, as in the continuum limits of random walks on complex or random structures.
• Fluctuation Analysis and Rare-Event Asymptotics: The framework opens new avenues for analyzing rare event statistics, fluctuation theorems, and scaling exponents of collision phenomena in interacting and disordered media.
• Generalization Beyond Pairs: While the focus is on binary collisions, extension to multi-particle collisions, correlated processes, and systems with non-conservative Markovian behavior is natural within the STOM formalism.

6. Open Problems and Research Directions

Further research may include:

• Extension to Correlated Processes: Developing collision measure limits for non-independent processes or incorporating interaction-driven effects.
• Multi-Particle and Higher Moments: Analyzing joint collision measures for more than two processes, thus engaging higher-order intersection theory in stochastic processes.
• Rates of Convergence: Quantifying convergence rates and fluctuation scaling for collision measures under various scaling regimes.
• Disordered Systems and Non-Conservativeness: Characterizing how persistent microscopic irregularities manifest in collision measures, especially in systems featuring explosion or lack of conservativeness, where the flexible STOM approach provides a distinct advantage.

7. Summary Table: Main Components

Concept	Definition/Role	Example Setting
Space–Time Occupation Measure (STOM)	$\Pi(E) = \int_0^\infty 1_{(X_t, t) \in E} dA_t$	Occupation/local time, collision measures
Collision Measure	STOM on diagonal of product process; tracks where/when trajectories coincide	Random walks on $\mathbb{Z}$/critical branching graphs
Gromov–Hausdorff Convergence	Topology for joint convergence of spaces, measures, kernels, processes	Scaling limit of random walks on random graphs

Collision measures, approached via the STOM–PCAF–Revuz correspondence formalism, unify and extend prior work on occupation times, enabling rigorous quantification and convergence analyses of encounter structures in stochastic processes. These measures serve as central analytical tools in the evolution and scaling of collision phenomena, with broad applications in random graph theory, statistical mechanics, and the paper of disordered systems (Noda, 22 Oct 2025).