Abstract

Consider a Brownian motion constrained to remain within a cone in theplane, and conditioned to exit it at its apex. As it explores thisspace, its path can be divided into sections living within smallersubcones with random apexes: cone excursions. Cutting out theseexcursions produces a process with jumps, and the procedure can beiterated indefinitely within the cut-out sections. What emerges is agrowth-fragmentation, a type of branching process with infiniteactivity. We demonstrate this and characterise the law of thegrowth-fragmentation for a particular choice of apex angle. Theresulting process can be seen as describing the boundary lengths ofcertain SLE curves drawn on a quantum disc, and mirrors paralleldevelopments in the field of random planar maps. A key element in thework is an interesting pathwise construction of the 3/2-stable processconditioned to stay positive. This is joint work with Ellen Powell (Durham) and William Da Silva (Vienna).