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Coalescing systems of non-Brownian particles

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A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the starting set is compact. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We show that Arratia’s conclusion is valid for Brownian motions on the Sierpinski gasket and for stable processes on the real line with stable index greater than one. Joint work with Steve Evans and Ben Morris.

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