Local times for Brownian motion indexed by the Brownian tree
We discuss local times for the process called Brownian motion indexed by the Brownian tree, which arises in scaling limits of several models of combinatorics, statistical physics and interacting particle systems. We show that the pair consisting of the local time and its derivative is a Markov process and satisfies an explicit stochastic differential equation whose coefficients involve the classical Airy function. As an application, we obtain a similar stochastic differential equation for the process of volumes of spheres in the Brownian plane. This is based in part on a joint work with Ed Perkins (UBC).