Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator

Abstract

We study the random entire functions defined as power series $$f(z) = \sum _{n=0}^\infty (X_n/n!) z^n$$ with independent and identically distributed coefficients $$(X_n)$$ and show that, under very weak assumptions, they are frequently hypercyclic for the differentiation operator $$D: H({\mathbb {C}}) \rightarrow H({\mathbb {C}}),\,f \mapsto Df = f'$$ . This gives a very simple probabilistic construction of $$D$$ -frequently hypercyclic functions in $$H({\mathbb {C}})$$ . 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.