Two problems on random analytic functions in Fock spaces

Xiang Fang, Pham Trong Tien

Research output: Contribution to journalArticlepeer-review


Let (Formula Presented) be an entire function on the complex plane, and let (Formula Presented) be its randomization induced by a standard sequence of independent Bernoulli, Steinhaus, or Gaussian random variables. In this paper, we characterize those functions such that is almost surely in the Fock space for any. 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, 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.

Original languageEnglish
Pages (from-to)1176-1198
Number of pages23
JournalCanadian Journal of Mathematics
Issue number4
StatePublished - 8 Aug 2023


  • Fock spaces
  • Random analytic functions
  • mixed norm space


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