Heat-kernel condition for collision measures in one-dimensional Bouchaud trap models

Establish the small-time heat-kernel integrability condition needed for convergence of collision measures of two independent one-dimensional Bouchaud trap models in the regime where the trap distribution has finite mean but infinite second moment. Concretely, for the sequence of Bouchaud trap model environments with state spaces S_n, heat kernels p_n(t,x,y), and canonical weighting measures μ_n equal to the squared invariant measures, show that for every r>0, lim_{δ→0} limsup_{n→∞} sup_{x1,x2∈S_n^{(r)}} ∫_0^δ ∫_{S_n^{(r)}} p_n(t,x1,y) p_n(t,x2,y) μ_n(dy) dt = 0, where S_n^{(r)}={x∈S_n : d_{S_n}(ρ_n,x)≤r}. Verifying this heat-kernel condition will validate the STOM-based convergence of collision measures in this heavy-tailed Bouchaud trap model setting, in which the invariant measures homogenize to Lebesgue while the squared invariant measures converge to the invariant measure of the Fontes–Isopi–Newman diffusion.

Background

The paper develops a general STOM framework to prove convergence of collision measures for pairs of independent stochastic processes. A key technical ingredient for collision-measure convergence is a short-time heat-kernel integrability condition involving the product of two heat kernels and the weighting measure (the squared invariant measure in the canonical uniform-collision setting).

In the one-dimensional Bouchaud trap model with trap distribution in the regime of finite mean but infinite second moment, the invariant measures of the walks converge to Lebesgue, whereas their squared invariant measures converge to a singular limit (the invariant measure of the Fontes–Isopi–Newman diffusion). The authors note that verifying the required heat-kernel condition in this specific heavy-tailed setting is not yet achieved; recent work provides partial evidence, but a full proof is missing.

Establishing this condition would enable the application of the paper’s STOM-based convergence theorems to collision measures in Bouchaud trap models, illuminating how collisions reflect persistent microscopic inhomogeneities even when the individual processes homogenize.