In this talk I will discuss some of the most basic questions one can ask about Brownian motions on infinite dimensional compact groups such as the product of countably many torii.
On a compact group G, at a given time t > 0, the law of Brownian motion may or may not be singular with respect to Haar measure. Suppose there is a finite positive time t at which the law is not singular w.r.t Haar measure. Does it imply that there is a finite time s > t at which the law is absolutely continuous w.r.t Haar measure?