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Mon, 19 Jan 2026

15:30 - 16:30
L3

The Brownian marble

Prof. Andreas Kyprianou
(Dept of Mathematics University of Warwick)
Abstract

Fundamentally motivated by the two opposing phenomena of fragmentation and coalescence, we introduce a new stochastic object which is both a process and a geometry. The Brownian marble is built from coalescing Brownian motions on the real line, with further coalescing Brownian motions introduced through time in the gaps between yet to coalesce Brownian paths. The instantaneous rate at which we introduce more Brownian paths is given by λ/g2\lambda/g^2 where gg is the gap between two adjacent existing Brownian paths. We show that the process "comes down from infinity" when 0<λ<60<\lambda<6 and the resulting space-time graph of the process is a strict subset of the Brownian Web on R×[0,∞)\mathbb R \times [0,\infty). When λ≥6\lambda \geq 6, the resulting process "does not come down from infinity" and the resulting range of the process agrees with the Brownian Web.

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