Random holomorphic functions

The talk is devoted to translation flows, random with respect to an invariant ergodic measure of the Teichmueller flow on the moduli space of Abelian differentials. The main results are an asymptotic expansion for the ergodic integrals of our flows and limit theorems for our translation flows. The main construction is that of finitely-additive transverse invariant measures on the vertical foliation of an abelian differential, invariant under the horizontal foliation.