An arXiv preprint proposes a mathematical way to measure the fine-grained shape of collisions between independent Markov processes—not only whether they meet, but how collision mass is distributed across time and space. Almost surely under its stated geometric and heat-kernel assumptions, the collision sets have precise fractal dimensions. The paper also shows that Hausdorff measures built for collision times, and for collision points in the supercritical spatial regime, stay within fixed positive multiples of the natural collision measures.
Two views of a collision
The work studies a family of independent process components on a common metric-measure space, with at least two components. In the main theorem, every component starts from the same point under a common-point starting law. The collision measure is then split into two marginal views: one records when collisions occur, and the other records where they occur. That turns time and location into separate geometric objects rather than two ways of describing the same summary.
The framework assumes Ahlfors regularity at small scales, meaning that local volume follows a controlled power law, together with lower and upper heat-kernel estimates for short times. Its proofs combine positive continuous additive functionals, generalized Kac moment estimates, volume-growth and heat-kernel bounds, strong-Markov multiscale trials, and probability-amplification arguments. The setup is therefore a theoretical population of process components, rather than an empirical sample.
What the dimensions say
One key result concerns local dimension: the exponent describing how a measure shrinks around a typical point or time. Almost surely under the common-point law, the time marginal has one local dimension at typical collision times, while the spatial marginal has another at typical collision points. The paper expresses the time-side exponent as 1 minus half of its relevant spectral-dimension term, a scaling quantity used to distinguish the theorem’s regimes. The point-side exponent is the smaller of a process-based term and the space’s fractal dimension. The corresponding collision-time and collision-point sets have Hausdorff dimensions—the assigned sizes of irregular sets—exactly equal to the dimensions of the supports of their collision marginals.
When collision mass gathers in time
The results go beyond a single dimension by specifying how collision mass changes as the scale becomes tiny. At typical collision times, local mass in a time window of scale delta follows a power law in delta with a slowly varying log-log correction. The gauge is delta raised to the time-side exponent, multiplied by a log-log factor raised to half of the relevant spectral-dimension term. The normalized mass is bounded between positive deterministic constants; those constants are not given numerically.
That same gauge produces a Hausdorff-measure comparison on the time side. For every Borel set of times—that is, a standard measurable collection of times—the Hausdorff measure of its collision-time portion is bounded above and below by positive deterministic multiples of the temporal collision measure. On each finite time horizon, the resulting Hausdorff measure is positive and finite. The comparison is a bounded one, not an equality with a specified universal constant.
The paper also identifies unusually heavy, or thick, collision times. There, the largest local temporal masses use a stronger logarithmic scale: delta raised to the time-side exponent multiplied by a power of log(1 divided by delta). The theorem gives positive lower and upper bounds for this behavior, with the upper bound allowed to depend on the time horizon.
Points tell a less complete story
The spatial statement depends more sharply on the regime. When the relevant spectral dimension is above 2, typical collision points follow a power-law gauge in their distance scale with a log-log correction: r raised to the point-side exponent, multiplied by log-log(1 divided by r). At the critical value 2, the paper obtains separate lower and upper logarithmic bounds instead of one exact common scale. The typical spatial fluctuation is therefore identified in the supercritical regime but remains bracketed in the critical one.
In the same supercritical regime, the Hausdorff measure built from the spatial gauge is comparable, on every Borel set of collision points, to the spatial collision measure, and the total measure on a finite horizon is positive and finite. The theorem does not make that exact-measure claim in the critical or subcritical spatial regimes. It also gives a separate description of thick collision points, where the largest local masses have enhanced scales: r raised to the point-side exponent times log(1 divided by r) above the threshold, and r raised to the space’s fractal-dimension exponent times the square of log(1 divided by r) at the critical value.
A conditional map
The framework includes several model families named in the preprint, including symmetric stable processes, canonical diffusions on affine nested fractals such as the Sierpinski gasket, and stable-like jump processes on d-sets. Its conclusions remain conditional on the regularity and heat-kernel assumptions built into the construction. The document is an arXiv version 1 preprint dated 26 August 2026.
Paper data and sources
Original title: Exact Hausdorff measures and fine geometry of collisions of independent Markov processes
Authors: Ryoichiro Noda
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text