Translation-invariant probability measures on entire functions

Lev Buhovsky, Adi Glücksam*, Alexander Logunov, Mikhail Sodin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We study non-trivial translation-invariant probability measures on the space of entire functions of one complex variable. The existence (and even an abundance) of such measures was proven by Benjamin Weiss. Answering Weiss’ question, we find a relatively sharp lower bound for the growth of entire functions in the support of such measures. The proof of this result consists of two independent parts: the proof of the lower bound and the construction, which yields its sharpness. Each of these parts combines various tools (both classical and new) from the theory of entire and subharmonic functions and from the ergodic theory. We also prove several companion results, which concern the decay of the tails of non-trivial translation-invariant probability measures on the space of entire functions and the growth of locally uniformly recurrent entire and meromorphic functions.

Original languageEnglish (US)
Pages (from-to)307-339
Number of pages33
JournalJournal d'Analyse Mathematique
Volume139
Issue number1
DOIs
StatePublished - Oct 1 2019
Externally publishedYes

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)

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