Characterize the limiting intensity of the maximum-particle thinning

Determine whether the limiting Poisson point process obtained from the maximum-particle thinning M(θ_t^λ) has intensity measure of the form A_∞e^{-μx}dx+B_∞dx, where μ=λ+√(λ²−2) and A_∞,B_∞ are possibly random non-negative constants.

Background

The paper studies the large-time behavior of a branching Brownian motion started from a point process θ. To analyze the leading particles, it introduces the non-Markovian thinning M(θ_t^λ), which retains the maximum descendant from each initial atom. The authors explain that any limiting object should be a Poisson point process, but unlike the uniformly tagged-particle thinning, M(θ_t^λ) is non-Markovian, so an invariant-measure or Choquet–Deny argument is unavailable to identify its intensity.

The proposed form includes both an exponentially decaying component and a constant component. The constant term is important because its positivity would correspond to a limiting Poisson point process without a finite top particle, while the later results show that fixed-point limits of the full BBM must nevertheless have finite top particles almost surely.