TWO PROBLEMS ON RANDOM ANALYTIC FUNCTIONS IN FOCK SPACES

Let f(z) = Σ^∞_n=0a_nzⁿbe an entire function on the complex plane and let Rf(z) = Σ^∞_n=0a_nX_nzⁿbe its randomization induced by a standard sequence (X_n)_nof independent Bernoulli, Steinhaus or Gaussian random variables. In this paper, we characterize those functions f(z) such that Rf(z) is almost surely in the Fock space F^p_αfor any p; α ∈ (0, ∞). Then such a characterization, together with embedding theorems which are of independent interests, is used to obtain a Littlewood-type theorem, aka 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 F(∞; q; α), an endpoint version of Fock spaces, via entropy integrals; (b) a complete description of random lacunary elements in Fock spaces; (c) a complete description of random multipliers between different Fock spaces.