Two problems on random analytic functions in Fock spaces

Abstract Let $f(z)=\sum _{n=0}^\infty a_n z^n$ be an entire function on the complex plane, and let ${\mathcal R} f(z) = \sum _{n=0}^\infty a_n X_n z^n$ be its randomization induced by a standard sequence $(X_n)_n$ of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions $f(z)$ such that ${\mathcal R} f(z)$ is almost surely in the Fock space ${\mathcal F}_{\alpha }^p$ for any $p, \alpha \in (0,\infty )$ . Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, also known as regularity improvement under randomization within the scale of Fock spaces. Other results obtained in this paper include: (a) a characterization of random analytic functions in the mixed-norm space ${\mathcal F}(\infty , q, \alpha )$ , an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; and (c) a complete description of random multipliers between different Fock spaces.