We consider a family of fractional Brownian fields $\{B^{H}\}_{H\in (0,1)}$ on $\mathbb{R}^{d}$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $ψ$ such that the covariance of $$ X^{H}(x) = Γ(H)^{\frac{1}{2}} \left( B^{H}(x) - \int_{\mathbb{R}^{d}} B^{H}(u) ψ(u, x)du\right), $$ converges to the covariance of a log-correlated Gaussian field when $H \downarrow 0$.
We then use Berestycki's ``good points'' approach in order to derive the limiting measure of the so-called multiplicative chaos of the fractional Brownian field $$ M^{H}_γ(dx) = e^{γX^{H}(x) - \frac{γ^{2}}{2} E[X^{H}(x)^{2}] }dx, $$ as $H\downarrow 0$ for all $γ\in (0,γ^{*}(d)]$, where $γ^{*}(d)>\sqrt{\frac{7}{4}d}$. As a corollary we establish the $L^{2}$ convergence of $M^{H}_γ$ over the sets of ``good points'', where the field $X^H$ has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of ``good points'' with probability close to $1$. Moreover, on these sets the volatility converges in $L^2$ to the volatility of multifractal random walks.
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof is based on a study of the deterministic version of the problem and the excursion theory.
We prove that the marginal law $σ_{t}\boxtimesν$ of free positive multiplicative Brownian motion is log-unimodal for all $t>0$ if $ν$ is a multiplicatively symmetric log-unimodal distribution, and that $σ_{t}\boxtimesν$ is log-unimodal for sufficiently large $t$ if $ν$ is supported on a suitably chosen finite interval. Counterexamples are given when $ν$ is not assumed to be symmetric or having a bounded support.
This review-style article presents an overview of recent progress in constructing and studying critical Gaussian multiplicative chaos. A proof that the critical measure in any dimension can be obtained as a limit of subcritical measures is given.
We propose a new definition of the Gaussian multiplicative chaos (GMC) and an approach based on the relation of subcritical GMC to randomized shifts of a Gaussian measure. Using this relation we prove general uniqueness and convergence results for subcritical GMC that hold for Gaussian fields with arbitrary covariance kernels.
We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.
We study non-Gaussian log-correlated multiplicative chaos, where the random field is defined as a sum of independent fields that satisfy suitable moment and regularity conditions. The convergence, existence of moments and analyticity with respect to the inverse temperature are proven for the resulting chaos in the full subcritical range. These results are generalizations of the corresponding theorems for Gaussian multiplicative chaos. A basic example where our results apply is the non-Gaussian Fourier series \[\sum_{k=1}^\infty \frac{1}{\sqrt{k}}(A_k \cos(2πk x) + B_k \sin(2πk x)),\] where $A_…
In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in 1985. As main application, we give a rigorous mathematical meaning to the Kolmogorov-Obukhov model of energy dissipation in a turbulent flow.
In this article we study the tail probability of the mass of Gaussian multiplicative chaos. With the novel use of a Tauberian argument and Goldie's implicit renewal theorem, we provide a unified approach to general log-correlated Gaussian fields in arbitrary dimension and derive precise first order asymptotics of the tail probability, resolving a conjecture of Rhodes and Vargas. The leading order is described by a universal constant that captures the generic property of Gaussian multiplicative chaos, and may be seen as the analogue of the Liouville unit volume reflection coefficients in higher…
We present new, short and self-contained proofs of the convergence (with an adequate renormalization) of four different sequences to the critical Gaussian Multiplicative Chaos:(a) the derivative martingale (b) the critical martingale (c) the exponential of the mollified field (d) the subcritical Gaussian Multiplicative Chaos.
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire subcritical phase $(γ< \sqrt{2d})$ and that the limit is universal (i.e., the limiting measure is independent of the regularisation of the underlying field)
We construct the analogue of Gaussian multiplicative chaos measures for the local times of planar Brownian motion by exponentiating the square root of the local times of small circles. We also consider a flat measure supported on points whose local time is within a constant of the desired thickness level and show a simple relation between the two objects. Our results extend those of Bass, Burdzy and Khoshnevisan and in particular cover the entire $L^1$-phase or subcritical regime. These results allow us to obtain a nondegenerate limit for the appropriately rescaled size of thick points, thereb…
Brownian multiplicative chaos measures, introduced in [Jeg20, AHS20, BBK94], are random Borel measures that can be formally defined by exponentiating $γ$ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $γ$ is less than 2 were studied. This article considers the critical case where $γ=2$, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation…
Motivated by isotropic fully developed turbulence, we define a theory of symmetric matrix valued isotropic Gaussian multiplicative chaos. Our construction extends the scalar theory developed by J.P. Kahane in 1985.
We show that for $γ<\sqrt{4/3}$, it is possible to define the Levy area of a planar Brownian motion with the Liouville measure of intermittency parameter $γ$ as the underlying area measure. We also consider the case of smoother curves, and study some properties of the integration map thus defined.
We study the total mass of high points in a random model for the Riemann-Zeta function. We consider the same model as in [8], [2], and build on the convergence to 'Gaussian' multiplicative chaos proved in [14]. We show that the total mass of points which are a linear order below the maximum divided by their expectation converges almost surely to the Gaussian multiplicative chaos of the approximating Gaussian process times a random function. We use the second moment method together with a branching approximation to establish this convergence.
We consider Gaussian multiplicative chaos measures defined in a general setting of metric measure spaces. Uniqueness results are obtained, verifying that different sequences of approximating Gaussian fields lead to the same chaos measure. Specialized to Euclidean spaces, our setup covers both the subcritical chaos and the critical chaos, actually extending to all non-atomic Gaussian chaos measures.
We prove convergence of a stochastic approximation of powers of the Riemann $ζ$ function to a non-Gaussian multiplicative chaos measure, and prove that this measure is a non-trivial multifractal random measure. The results cover both the subcritical and critical chaos. A basic ingredient of the proof is a 'good' Gaussian approximation of the induced random fields that is potentially of independent interest.
In this article we study the tail probability of the mass of critical Gaussian multiplicative chaos (GMC) associated to a general class of log-correlated Gaussian fields in any dimension, including the Gaussian free field (GFF) in dimension two. More precisely, we derive a fully explicit formula for the leading order asymptotics for the tail probability and demonstrate a new universality phenomenon. Our analysis here shares similar philosophy with the subcritical case but requires a different approach due to complications in the analogous localisation step, and we also employ techniques from r…
We consider a pair of probability measures $μ,ν$ on the unit circle such that $Σ_λ(η_ν(z))=z/η_μ(z)$. We prove that the same type of equation holds for any $t\geq 0$ when we replace $ν$ by $ν\boxtimesλ_t$ and $μ$ by $\mathbb{M}_t(μ)$, where $λ_t$ is the free multiplicative analogue of the normal distribution on the unit circle of $\mathbb{C}$ and $\mathbb{M}_t$ is the map defined by Arizmendi and Hasebe. These equations are a multiplicative analogue of equations studied by Belinschi and Nica. In order to achieve this result, we study infinite divisibility of the measures associated with subord…
In this note we prove that suitable positive powers of the absolute value of the characteristic polynomial of a Haar distributed random unitary matrix converge in law, as the size of the matrix tends to infinity, to a Gaussian multiplicative chaos measure once correctly normalized. We prove this in the whole $L^1$- or subcritical phase of the chaos measure.
Bulk/boundary quotients of Gaussian multiplicative chaos measures II: Tail profile of bulk Gaussian multiplicative chaos measures in the exact scale-invariant case
本文研究精确尺度不变核下边界具有均匀奇异性的高斯乘性混沌测度的右尾轮廓，通过将局部化技巧应用于边界上的辅助测度，推广了 Rhodes-Vargas 和 Wong 的平坦背景几何结果，并用体/边界商的变体刻画尾部轮廓系数。
The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By considering the case of the 2D Gaussian free field, we review convergence, uniqueness and characterisations of the measures; revisit Kahane's convexity inequalities and existence and scaling of moments; discuss the measurability of the underlying field with respect to the GMC measure and present a KPZ relation for scaling exponents.
This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter $γ^2$ beyond the transition phase (i.e. $γ^2>2d$) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a simplified proof of the classical KPZ formula as well as the dual KPZ formula for atomic Gaussian multiplicative chaos. In particular, this framework allows to construct singular Liouville measures and …
The objective of this note is to study the probability that the total mass of a sub-critical Gaussian multiplicative chaos (GMC) with arbitrary base measure $σ$ is small. When $σ$ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty$. However, when $σ$ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $σ$, the total GMC mass has negative moments of all orders.
Let $M_γ$ be a subcritical Gaussian multiplicative chaos measure associated with a general log-correlated Gaussian field defined on a bounded domain $D \subset \mathbb{R}^d$, $d \geq 1$. We find an explicit formula for its singularity spectrum by showing that $M_γ$ satisfies almost surely the multifractal formalism, i.e., we prove that its singularity spectrum is almost surely equal to the Legendre-Fenchel transform of its $L^q$-spectrum. Then, applying this result, we compute the lower singularity spectrum of the multifractal random walk and of the Liouville Brownian motion.
We determine the rate of decay of the expectation Z(t) of some multiplicative functional related to Brownian motion up to time t. This permits to prove that the Wiener measure, penalized by this multiplicative functional, converges as t goes to infinity to a probability measure (p.m.) . We obtain the law of the canonical process under this new p.m.
In 2001, Knight constructed a stochastic process modeling the one dimensional interaction of two particles, one being Newtonian in the sense that it obeys Newton's laws of motion, and the other particle being Brownian. We construct a multi-particle analog, using Skorohod map estimates in proving a propagation of chaos and characterizing the hydrodynamic limit as the solution to a PDE with free boundary condition. Stochastic methods are used to show existence and uniqueness for the free boundary problem, and also present an algorithm of approximating the solution.
We prove that when suitably normalized, small enough powers of the absolute value of the characteristic polynomial of random Hermitian matrices, drawn from one-cut regular unitary invariant ensembles, converge in law to Gaussian multiplicative chaos measures. We prove this in the so-called $L^2$-phase of multiplicative chaos. Our main tools are asymptotics of Hankel determinants with Fisher-Hartwig singularities. Using Riemann-Hilbert methods, we prove a rather general Fisher-Hartwig formula for one-cut regular unitary invariant ensembles.
The random trigonometric series $\sum_{n=1}^\infty ρ_n \cos (nt +ω_n)$ on the circle $\mathbb{T}$ are studied under the conditions $\sum |ρ_n|^2=\infty$ and $ρ_n\to 0$, where $\{ω_n\}$ are iid and uniformly distributed on $\mathbb{T}$. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which are the limits of the exponentiations of partials sums. which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on…
We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.
In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purp…
We provide a deep connection between elastic drifted Brownian motions and inverses to tempered subordinators. Based on this connection, we establish a link between multiplicative functionals and dynamical boundary conditions given in terms of non-local equations in time. Indeed, we show that the multiplicative functional associated to the elastic Brownian motion with drift is equivalent to a multiplicative functional associated with fractional boundary conditions of tempered type. By exploiting such connection we write some functionals in terms of a simple (positive and non-decreasing) process…
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…
We show that the imaginary multiplicative chaos $\exp(iβΓ)$ determines the gradient of the underlying field $Γ$ for all log-correlated Gaussian fields with covariance of the form $-\log |x-y| + g(x,y)$ with mild regularity conditions on $g$, for all $d \geq 2$ and for all $β\in (0,\sqrt{d})$. In particular, we show that the 2D continuum zero boundary Gaussian free field is measurable w.r.t. its imaginary chaos.
The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question raised in work of Soundararajan and Zaman, who proved moment bounds for the same quantity in analogy to those of Harper in the random multiplicative setting. Having a simpler quantity allows us to make the proof close to self-contained, and perhaps somewhat more accessible.
In this paper, we study Gaussian multiplicative chaos in the critical case. We show that the so-called derivative martingale, introduced in the context of branching Brownian motions and branching random walks, converges almost surely (in all dimensions) to a random measure with full support. We also show that the limiting measure has no atom. In connection with the derivative martingale, we write explicit conjectures about the glassy phase of log-correlated Gaussian potentials and the relation with the asymptotic expansion of the maximum of log-correlated Gaussian random variables.
The purpose of these notes, based on a course given by the second author at Les Houches summer school, is to explain the probabilistic construction of Polyakov's Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, these notes contain a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. These notes are rather short and require no prior knowledge on the topic.
We study the characteristic polynomials of both the Gaussian Orthogonal and Symplectic Ensembles. We show that for both ensembles, powers of the absolute value of the characteristic polynomials converge in law to Gaussian multiplicative chaos measures after normalization for sufficiently small real powers. The main tool is a new asymptotic relation between the fractional moments of the absolute characteristic polynomials of Gaussian Orthogonal, Unitary, and Symplectic Ensembles.
We investigate a special sequence of random variables $A(N)$ defined by an exponential power series with independent standard complex Gaussians $(X(k))_{k \geq 1}$. Introduced by Hughes, Keating, and O'Connell in the study of random matrix theory, this sequence relates to Gaussian multiplicative chaos (in particular "holomorphic multiplicative chaos'' per Najnudel, Paquette, and Simm) and random multiplicative functions. Soundararajan and Zaman recently determined the order of $\mathbb{E}[|A(N)|]$. By constructing an algorithm to calculate $A(N)$ in $O(N^2 \log N)$ steps, we produce computatio…
