Frequent hypercyclicity of random entire functions for the differentiation operator

TL;DR

This paper demonstrates that certain random entire functions are frequently hypercyclic for the differentiation operator, providing a simple probabilistic method to construct such functions with controlled growth rates.

Contribution

It introduces a new probabilistic construction of frequently hypercyclic entire functions for the differentiation operator using random power series.

Findings

Random entire functions are frequently hypercyclic for D under weak assumptions.

These functions can be constructed with near minimal growth rates.

The growth rate differs from the slowest possible by at most a logarithmic factor.

Abstract

In this note we study the random entire functions defined as power series $f(z) = \sum_{n=0}^\infty \frac{X_n}{n!} z^n$ with independent and identically distributed coefficients $(X_n)$ and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator $D: H(\C) \to H(\C)$, $f \mapsto Df = f'$. This gives a very simple probabilistic construction of $D$-frequently hypercyclic functions in $H(\C)$. Moreover we show that, under more restrictive assumptions on the distribution of the $(X_n)$, these random entire functions have a growth rate that differs from the slowest growth rate possible for $D$-frequently hypercyclic entire functions at most by a factor of a power of a logarithm.