A Fixed Point Formula of a Random Dynamical System

Abstract

I will discuss a particular dynamical system on the space of probability measures on the line and its relation to electrical networks. This system has powerful contractive properties that ensure it converges almost surely to a delta mass, with very few assumptions on the choice of the initial probability measure. However this is guaranteed by non-constructive arguments, and very little is known about how the location of the limiting delta mass depends on the initial condition. I will describe some recent solutions to this problem, for certain types of initial conditions, based on the connection with electrical network theory.