Govind Menon (1.1) Stochastic Loewner evolution with branching and the Dyson superprocess

The purpose of this talk is to advertize a new stochastic Loewner evolution that was developed in my student Vivian Healey's 2017 thesis. This work was motivated by the desire to better understand the Brownian map by constructing explicit conformal maps that embed random trees, especially Aldous' continuum random tree, into the upper half plane. The main results are (i) conformal maps that embed finite Galton-Watson trees into the upper half plane via a stochastic Loewner evolution that includes branching; and (ii) a description of the scaling limit. Part (i) is completely rigorous, while part (ii) includes both formal calculations and rigorous results. At least formally, the scaling limit is a stochastic Loewner evolution driven by a measure valued diffusion that is the scaling limit of Dyson Brownian motion plus branching. We call this diffusion the Dyson superprocess. It can be seen as the free analogue of the Dawson Watanabe superprocess and it has several interesting connections with stochastic PDE.

Presented at the 27th Annual PCMI Summer Session, Random Matrices, held June 25 – July 15, 2017. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI).

The Institute for Advanced Study / IAS / Park City Mathematics Institute (PCMI) is designed for mathematics educators at the secondary and post-secondary level, as well as mathematics researchers and students at the post-secondary level. These groups find at PCMI an intensive mathematical experience geared to their individual needs. Moreover, the interaction among groups with different backgrounds and professional needs increases each participant’s appreciation of the mathematical community as a whole as well as the work of participants in different areas.