Density of imaginary multiplicative chaos via Malliavin calculus

• 成果类型: Article
• 署名作者: Aru, Juhan; Jego, Antoine; Junnila, Janne
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-022-01135-y
• 发表日期: 2022
• 页码: 749-803
• 关键词: convergence; FIELDS

摘要

We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential mu(beta ):=: e(i beta Gamma(x)) : for a log-correlated Gaussian field Gamma in d >= 1 dimensions. We prove a basic density result, showing that for any nonzero continuous test function f, the complex-valued random variable mu(beta)(f) has a smooth density w.r.t. the Lebesgue measure on C. As a corollary, we deduce that the negative moments of imaginary chaos on the unit circle do not correspond to the analytic continuation of the Fyodorov-Bouchaud formula, even when well-defined. Somewhat surprisingly, basic density results are not easy to prove for imaginary chaos and one of the main contributions of the article is introducing Malliavin calculus to the study of (complex) multiplicative chaos. To apply Malliavin calculus to imaginary chaos, we develop a new decomposition theorem for non-degenerate log-correlated fields via a small detour to operator theory, and obtain small ball probabilities for Sobolev norms of imaginary chaos.