Abstract: Loewner's differential equation, which describes the time-evolution of slit mappings, was initially introduced to attack an extremal problem for univalent functions (Bieberbach's conjecture) solved finally by de Branges in 1984. It was then used to indroduce the Schramm--Loewner evolution (SLE), which yielded a breakthrough in the analysis of two-dimensional critical models in statistical physics, in 2000. In this talk, I shall describe yet another, recent application of Loewner's method to one-dimensional, non-commutative stochastic processes. In non-commutative probability, we can consider several ``independences'' for algebraic random variables and corresponding stochastic processes with ``independent'' increments. Loewner chains are naturally associated with such processes if we consider ``monotone'' independence. If time permits, I shall also discuss how some function-theoretic (or potential-theoretic) properties are related to probabilistic ones. This talk is based on a joint work with Takahiro Hasebe (Hokkaido University) and Ikkei Hotta (Yamaguchi University).

Zoom Meeting ID: 181 155 584