The boundary of the zero set and boundary local time of one-dimensional super-Brownian motion
Super-Brownian motion is a measure-valued Markov process which arises as the scaling limit of several discrete models, including branching random walk. In dimension one, it has a continuous density. In this talk I will discuss the construction of a boundary local time for the density, which is a random measure supported on the boundary of its zero set. I will then show how a close analysis of the right endpoint of the density's support is used to prove that the local time is positive almost surely (when the process itself is non-zero). An application energy method using the local time then gives an almost sure characterization of the Hausdorff dimension of the boundary of the zero set, completing a result which was previously only known to hold with positive probability.
This talk includes joint work with Ed Perkins.