Minima of Independent Bessel Processes and of Distances Between Brownian Particles

Abstract

Consider a large, finite collection of particles performing Brownian motion independently in space. We examine the process obtained by taking the minimum, at each time, of the distances of the particles, either (a) from the origin, or (b) from each other. In both cases, when time and space are suitably renormalized, we obtain a narrow convergence result. We also consider the number of pairs of particles which approach each other closely, over a unit time interval.

Extremes of independent Gaussian processes

“…It follows from (51) that in order that Condition 1b of Theorem This, in particular, recovers the results of [11,12], where minima of independent Bessel processes were studied. Note that the proofs of [11,12] are based on the Markov property of Bessel processes.…”

“…}} be a zero-mean, unit-variance Gaussian vector with covariance ρ n := 1 − λ/ log n. Recalling (12) and taking into account the fact that Γ(t 1 , t 2 ) = ∞, we obtain that r n (t 1 , t 2 ) < ρ n for sufficiently large n. By Slepian's comparison lemma, see [10,Corollary 4…”

“…Then the process (2π) −1/2 nL n converges as n → ∞ to the process [11,12], where minima of independent Bessel processes were studied. Note that the proofs of [11,12] are based on the Markov property of Bessel processes. Global minima of stationary χ 2 -processes were studied by [1].…”

Minima and maxima of elliptical arrays and spherical processes

“…This reprint differs from the original in pagination and typographic detail. aforementioned paper investigates the asymptotics of the minimum of the absolute value of independent Gaussian processes extending some previous results of Penrose [26].…”

“…(b) The process {ζ Γ,S (t), t ∈ R} is defined by Γ and {S(t), t ∈ R} but does not depend on the variance function σ 2 (·). The processes ζ Γ,1 appears first in Penrose [26] and recently in Kabluchko [22]. We refer to {η Γ,S (t), t ∈ R} as Penrose-Kabluchko process.…”