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Consider a Brownian motion constrained to remain within a cone in the plane, and conditioned to exit it at its apex. As it explores this space, its path can be divided into sections living within smaller subcones with random apexes: cone excursions. Cutting out these excursions produces a process with jumps, and the procedure can be iterated indefinitely within the cut-out sections. What emerges is a growth-fragmentation, a type of branching process with infinite activity. We demonstrate this and characterise the law of the growth-fragmentation for a particular choice of apex angle. The resulting process can be seen as describing the boundary lengths of certain SLE curves drawn on a quantum disc, and mirrors parallel developments in the field of random planar maps. A key element in the work is an interesting pathwise construction of the 3/2-stable process conditioned to stay positive.
This is joint work with Ellen Powell (Durham) and William Da Silva (Vienna).
