# Translation-invariant probability measures on entire functions

## TL;DR

This paper investigates translation-invariant probability measures on entire functions, establishing a lower bound on the growth of functions in their support and exploring related properties using tools from complex analysis and ergodic theory.

## Contribution

It provides a sharp lower bound on the growth of entire functions under such measures and constructs measures that achieve this bound, answering a question posed by Weiss.

## Key findings

- Established a sharp lower bound for the growth of entire functions in the support of translation-invariant measures.
- Constructed measures that attain the established growth bound.
- Analyzed decay of tails and growth of recurrent entire and meromorphic functions.

## Abstract

We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions and from the ergodic theory. We also prove several companion results, which concern the decay of the tails of non-trivial translation-invariant probability measures on the space of entire functions and the growth of locally uniformly recurrent entire and meromorphic functions.