[Paper Review] On the regularity of complex multiplicative chaos

Janne Junnila, Eero Saksman+1 more|arXiv (Cornell University)|May 28, 2019

Stochastic processes and statistical mechanics19 references4 citations

TL;DR

This paper establishes the moment finiteness and Besov regularity of complex multiplicative chaos measures μβ for complex β in the subcritical eye-shaped domain Ea. Using Gaussian multiplicative chaos theory and moment estimates via wavelet analysis and scaling, it proves that μβ has finite p-th moments and belongs to local Besov spaces Bp,q^s(U) up to a sharp regularity threshold depending on Re(β) and Im(β), extending prior results on imaginary chaos to general complex β.

ABSTRACT

Denote by $μ_β="\exp(βX)"$ the Gaussian multiplicative chaos which is defined using a log-correlated Gaussian field $X$ on a domain $U\subset\mathbb{R}^d$. The case $β\in\mathbb{R}$ has been studied quite intensively, and then $μ_β$ is a random measure on $U$. It is known that $μ_β$ can also be defined for complex values $β$ lying in certain subdomain of $\mathbb{C}$, and then the realizations of $μ_β$ are random generalized functions on $U$. In this note we complement the results of Junnila et al. (where the case of purely imaginary $β$ was considered) by studying the Besov-regularity of $μ_β$ and the finiteness of moments for general complex values of $β$.

Motivation & Objective

• To extend the regularity and moment finiteness results of real and imaginary β to general complex β in Gaussian multiplicative chaos theory.
• To determine the precise range of complex β for which the p-th moments of complex multiplicative chaos measures are finite.
• To characterize the optimal local Besov regularity of complex multiplicative chaos measures μβ in terms of β's real and imaginary parts.
• To provide sharp regularity thresholds for μβ in Besov spaces Bp,q^s(U), generalizing known results for real and purely imaginary β.
• To establish the optimality of the derived bounds under natural smoothness assumptions on the covariance kernel g.

Proposed method

• The paper uses wavelet-based Besov space characterizations to analyze the regularity of complex multiplicative chaos μβ.
• It applies moment estimates via the scaling properties of the log-correlated field X and the associated chaos measure μβ.
• The method involves decomposing test functions using wavelets and estimating the L^p norms of wavelet coefficients of μβ.
• It leverages the analyticity of μβ in β on the eye-shaped domain Ea, established in prior work, to extend regularity results to complex β.
• The analysis relies on the local translational invariance of the chaos measure and scaling relations derived from Gaussian chaos theory.
• A key technical step involves bounding the moments of wavelet coefficients using the moment condition on the field's covariance kernel g ∈ H^{d+ε}_{loc}(U×U).

Experimental results

Research questions

• RQ1 For which complex β ∈ C is the p-th moment of the complex multiplicative chaos measure μβ finite?
• RQ2 What is the optimal local Besov regularity s such that μβ ∈ Bp,q,loc^s(U) for complex β?
• RQ3 How does the regularity threshold depend on both the real and imaginary parts of β?
• RQ4 Can the moment and regularity bounds be extended beyond the purely imaginary β case studied in [9]?
• RQ5 Are the derived bounds on moments and regularity optimal under the given assumptions on the covariance kernel g?

Key findings

• The p-th moment of μβ is finite for all β in the region Ea_p = Ea ∩ ({|Re(β)| < √(2d)/p} ∪ {(p−1)Re(β)^2 + Im(β)^2 < 2d(p−1)/p}), provided g ∈ H^{d+ε}_{loc}(U×U).
• For β ∈ Ea with p ≥ 1, μβ belongs to the local Besov space Bp,q,loc^s(U) for all s < −[(p−1)Re(β)^2 + Im(β)^2]/2 when |Re(β)| ≤ √(2d)/p.
• When |Re(β)| ≥ √(2d)/p, the regularity threshold is s < d/p − √(2d)|Re(β)| + (Re(β)^2 − Im(β)^2)/2, and μβ ∈ Bp,q,loc^s(U) for any q ≥ 1.
• The derived bounds on moments and regularity are conjectured to be optimal, based on the structure of the moment estimates and scaling behavior.
• The results generalize the moment and regularity theory of complex multiplicative chaos from purely imaginary β to the full subcritical complex domain Ea.
• The analysis confirms that the regularity threshold depends non-trivially on both Re(β) and Im(β), reflecting the interplay between the real and imaginary parts in the complex chaos construction.