Invariant probability measures on holomorphic functions on the unit disk

We study probability measures on the space of holomorphic functions on the unit disk that are invariant under pre-composition by conformal automorphisms and give no mass to constant functions. We determine the slowest possible growth of functions supported by such measures. Let Mf(R) denote the maximum modulus of f on the hyperbolic ball of radius R centered at the origin. We show that given a non-decreasing function G:[1,∞)→(0,∞), there exists an Aut(D)-invariant probability measure $μ$ on Hol(D, C) functions \ such that log⁺ Mf(R)=Of(G(R)) for $μ$ almost every f if and only if ∫₁^∞ (R/G(R)) \, dR <∞. For the lower bound, we use a mass transport argument to show that almost every trajectory of the gradient flow of log |f| has finite energy. To establish sharpness, we construct holomorphic functions of slow growth on infinite Riemann surfaces.