Frequent Hypercyclicity of Random Entire Functions for the Differentiation Operator |
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Authors: | Miika Nikula |
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Institution: | 1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014?, Helsinki, Finland
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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. |
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