论文标题
通过malliavin conculus的假想乘法混乱的密度
Density of imaginary multiplicative chaos via Malliavin calculus
论文作者
论文摘要
我们考虑假想的高斯乘法混乱，即复杂的wick指数$μ_β：=：e^{iβγ（x）}：$ in $ d \ geq 1 $ dimensions中的对数相关的高斯字段$γ$。我们证明了一个基本的密度结果，表明对于任何非零连续测试函数$ f $，复杂值随机变量$μ_β（f）$具有光滑的密度W.R.T. Lebesgue以$ \ mathbb {c} $进行测量。作为推论，我们推断出单位圆上虚构混乱的负面力矩也不对应于Fyodorov-Bouchaud公式的分析延续，即使定义明确。 令人惊讶的是，对于虚构的混乱而言，基本密度结果并不容易证明，本文的主要贡献之一是引入了Malliavin conculus对（复杂）乘法混乱的研究。为了将Malliavin微积分应用于虚构的混乱，我们通过小型绕道到操作员理论开发了一种新的分解定理，用于非排分对数相关的磁场，并为假想混乱的Sobolev规范获得小球概率。
We consider the imaginary Gaussian multiplicative chaos, i.e. the complex Wick exponential $μ_β:= :e^{iβΓ(x)}:$ for a log-correlated Gaussian field $Γ$ in $d \geq 1$ dimensions. We prove a basic density result, showing that for any nonzero continuous test function $f$, the complex-valued random variable $μ_β(f)$ has a smooth density w.r.t. the Lebesgue measure on $\mathbb{C}$. 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.
