Collision and non-collision for diffusions on configuration space

2026/07/24 by Theodoros Assiotis, Kohei Suzuki
#math.PR #math-ph #math.MP

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Abstract

We develop criteria for collision and non-collision of reversible infinitely many interacting diffusion processes in the real line. The approach is potential-theoretic and is based on capacity estimates for symmetric Dirichlet forms on the configuration space. Our main results are model-independent in the sense that no determinantal or Pfaffian structure, prescribed interaction potential, or explicit labelled stochastic differential equation is required. The non-collision criterion involves only the second and third correlation functions of the reversible measure, whereas the collision criterion is based on a local lower bound for a finite-volume conditional density of the reversible measure. As an application, we identify the sharp collision threshold for the diffusion associated with the Sine_β-symmetric Dirichlet form: the collision set is polar if and only if β≥ 1. This provides an infinite-particle counterpart of the classical collision threshold for the finite-particle Dyson Brownian motion with inverse temperature β.

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