2025/05/08 by Samuel G. G. Johnston, Andreas E. Kyprianou, Johnston, Samuel G. G. +5
Arts and Humanities · #Architecture and Art History Studies
paper · pdf · doi:10.48550/arxiv.2505.05451
Let R:(0,∞) → [0,∞) be a measurable function. Consider coalescing Brownian motions started from every point in the subset \ (0,x) : x ∈ ℝ \ of [0,∞) × ℝ (with [0,∞) denoting time and ℝ denoting space) and proceeding according to the following rule: the interval \t\ × [Lt,Ut] between two consecutive Brownian motions instantaneously fragments' at rate R(Ut - Lt). At a fragmentation event at a time t, we initiate new coalescing Brownian motions from each of the points \ (t,x) : x ∈ [Lt,Ut]\. The resulting process, which we call the R-marble, is easily constructed when R is bounded, and may be considered a random subset of the Brownian web. Under mild conditions, we show that it is possible to construct the R-marble when R is unbounded as a limit as n → ∞ of Rn-marbles where Rn(g) = R(g) \wedge n. The behaviour of this limiting process is mainly determined by the shape of R near zero. The most interesting case occurs when the limit limg \downarrow 0 g2 R(g) = λ exists in (0,∞), in which case we find a phase transition. For λ≥ 6, the limiting object is indistinguishable from the Brownian web, whereas if λ< 6, then the limiting object is a nontrivial stochastic process with large gaps. When R(g) = λ/g2, the R-marble is a self-similar stochastic process which we refer to as the Brownian marble with parameter λ> 0. We give an explicit description of the spacetime correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-3 process.