Stationary random entire functions and related questions

Let T be the action of the complex plane on the space of entire functions defined by translations, i.e T_w takes the entire function f(z) to the entire function f(z+w). B.Weiss showed in `97 that there exists a probability measure defined on the space of entire functions, which is invariant under this action. In this talk I will present optimal bounds on the minimal possible growth of functions in the support of such measures, and discuss other growth related problems inspired by this work. In particular, I will focus on the question of minimal possible growth of frequently oscillating subharmonic functions

The talk is partly based on a joint work with L. Buhovsky, A. Logunov, and M. Sodin.