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Explore local times for Brownian motion indexed by the Brownian tree in this advanced mathematics lecture, focusing on the fundamental building blocks of scaling limits for random planar maps including the Brownian sphere and Brownian plane. Delve into the mathematical framework that connects stochastic processes with geometric structures, examining how local times behave in this specialized context. Learn about a significant application proving that volume processes of spheres in the Brownian plane possess continuous derivatives, and discover how the volume process paired with its derivative forms a Markovian system. Investigate the explicit stochastic differential equation governing this Markovian pair, with coefficients involving the classical Airy function from mathematical physics. Gain insights into collaborative research methodologies through discussion of joint work with Ed Perkins from the University of British Columbia, demonstrating how complex theoretical problems in probability theory and geometric analysis are approached through international mathematical collaboration.
