arXiv subjects

Eero Saksman

Publications and source records attributed to Eero Saksman.

At least 19 recordsLinked to original sources

arXiv2026

Analysis of the Singular Spectrum for General Perturbations

We investigate the behavior of the singular spectrum of self-adjoint operators under families of Hermitian perturbations, even non-compact ones. Under mild assumptions we have the ``shift'' of the singular spectrum for almost all values of the parameter. Moreover, if we consider trace class perturbations, we observe the ``shift'' outside a countable set of the values of the parameter. While similar results were known for finite rank perturbations, the extensions to trace class perturbations are far from easy, and the proofs require significant new ideas. In addition, some of our results hold (surprisingly enough!) even for all positive and bounded perturbations. This is a consequence of our generalized Aleksandrov disintegration theorem established in this paper. We use some advanced techniques, involving Sz.-Nagy--Foia\c s theory and the operator ${\bf A}_2$ condition.

math.SP↗

arXiv2026

Integral Means Spectrum for the Random Riemann Zeta Function

We study the integral means spectrum associated with the analytic function whose derivative is the so-called randomized Riemann zeta-function, introduced some time ago by Bagchi. The randomized $ζ$-function, $ζ_{\mathrm{rand}}(σ+ih)$, is known to represent the asymptotic statistical behaviour of the random vertical shifts of the actual $ζ$-function in the critical strip, $1/2 <σ\leq 1, h\in \mathbb R$, and appears in a number of recent works on the asymptotic behavior of the moments and maxima of the $ζ$-function on short intervals along the critical axis $σ=1/2$. Using probability and basic analytic number theory, we show that the complex integral means spectrum of the primitive of $ζ_{\mathrm{rand}}$ is almost surely of the form conjectured 30 years ago by Kraetzer, for the so-called universal integral means spectrum of univalent functions in the disc. The Riemann $ζ$-function and its random version have recently been rigorously related to the so-called Gaussian multiplicative chaos (GMC), initiated by Kahane 40 years ago. In the case of the holomorphic multiplicative chaos on the unit disc -- an important stochastic object closely related to Liouville quantum gravity on the unit circle -- we prove that the integral means spectrum of the primitive is almost surely also of the same Kraetzer form. However, we establish that neither the primitive of the random function $ζ_{\mathrm{rand}}$, nor that of the holomorphic GMC are injective. Building on earlier work by one of the authors and Webb on the convergence of Riemann $ζ$-function on the critical line to a holomorphic GMC distribution, we finally provide an alternative derivation of the integral means spectrum for the random Riemann $ζ$-function.

math.CV↗

arXiv2026

Conformal welding of independent Gaussian multiplicative chaos measures

We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the boundary of the unit disk we show that there exist conformal maps to complementary domains on the Riemann sphere such that the pushforward of the normalised measures agree on their common boundary.

math.PR↗

arXiv2025

1D stochastic pressure equation with log-correlated Gaussian coefficients

We study unique solvability for one dimensional stochastic pressure equation with diffusion coefficient given by the Wick exponential of log-correlated Gaussian fields. We prove well-posedness for Dirichlet, Neumann and periodic boundary data, and the initial value problem, covering the cases of both the Wick renormalization of the diffusion and of point-wise multiplication. We provide explicit representations for the solutions in both cases, characterized by the $S$-transform and the Gaussian multiplicative chaos measure.

math.PR↗

arXiv2025

Fractional Heat Semigroup Characterization of Distances from Functions in Lipschitz Spaces to Their Subspaces

Let $Λ_s$ denote the inhomogeneous Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$. This article characterizes the distance $d(f, V)_{Λ_s}: = \inf_{g\in V} \|f-g\|_{Λ_s}$ from a function $f\in Λ_s$ to a non-dense subspace $V\subset Λ_s$ via the fractional semigroup $\{T_{α, t}: =e^{-t (-Δ)^{α/2}}: t\in (0, \infty)\}$ for any $α\in(0,\infty)$. Given an integer $ r >s/α$, a uniformly bounded continuous function $f$ on $\mathbb{R}^n$ belongs to the space $Λ_s$ if and only if there exists a constant $λ\in(0,\infty)$ such that \begin{align*} \left|(-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|\leq λt^{s -rα}\ \ \text{for any $x\in\mathbb{R}^n$ and $t\in (0, 1]$}.\end{align*} The least such constant is denoted by $λ_{ α, r, s}(f)$. For each $f\in Λ_s$ and $0<\varepsilon< λ_{α,r, s}(f)$, let $$ D_{α, r}(s,f,\varepsilon):=\left\{ (x,t)\in \mathbb{R}^n\times (0,1]:\ \left| (-Δ)^{\frac {αr}2} (T_{α, t^α} f)(x) \right|> \varepsilon t^{s -r α}\right\}$$ be the set of ``bad'' points. To quantify its size, we introduce a class of extended nonnegative \emph{admissible set functions} $ν$ on the Borel $σ$-algebra $\mathcal{B}(\mathbb{R}^n\times [0, 1])$ and define, for any admissible function $ν$, the \emph{critical index} $ \varepsilon_{α, r, s,ν}(f):=\inf\{\varepsilon\in(0,\infty):\ ν(D_{α, r}(s,f,\varepsilon))<\infty\}.$ Our result shows that, for a broad class of subspaces $V\subset Λ_s$, including intersections of $Λ_s$ with Sobolev, Besov, Triebel--Lizorkin, and Besov-type spaces, there exists an admissible function $ν$ depending on $V$ such that $\varepsilon_{α, r, s,ν}(f)\sim \mathrm{dist}(f, V)_{Λ_s}.$

math.FA↗

arXiv2025

Difference and Wavelet Characterizations of Distances from Functions in Lipschitz Spaces to Their Subspaces

Let $Λ_s$ denote the Lipschitz space of order $s\in(0,\infty)$ on $\mathbb{R}^n$, which consists of all $f\in\mathfrak{C}\cap L^\infty$ such that, for some constant $L\in(0,\infty)$ and some integer $r\in(s,\infty)$, \begin{equation*} \label{0-1}Δ_r f(x,y): =\sup_{|h|\leq y} |Δ_h^r f(x)|\leq L y^s, \ x\in\mathbb{R}^n, \ y \in(0, 1]. \end{equation*} Here (and throughout the article) $\mathfrak{C}$ refers to continuous functions, and $Δ_h^r$ is the usual $r$-th order difference operator with step $h\in\mathbb{R}^n$. For each $f\in Λ_s$ and $\varepsilon\in(0,L)$, let $ S(f,\varepsilon):= \{ (x,y)\in\mathbb{R}^n\times [0,1]: \frac {Δ_r f(x,y)}{y^s}>\varepsilon\}$, and let $μ: \mathcal{B}(\mathbb{R}_+^{n+1})\to [0,\infty]$ be a suitably defined nonnegative extended real-valued function on the Borel $σ$-algebra of subsets of $\mathbb{R}_+^{n+1}$. Let $\varepsilon(f)$ be the infimum of all $\varepsilon\in(0,\infty)$ such that $μ(S(f,\varepsilon))<\infty$. The main target of this article is to characterize the distance from $f$ to a subspace $V\cap Λ_s$ of $Λ_s$ for various function spaces $V$ (including Sobolev, Besov--Triebel--Lizorkin, and Besov--Triebel--Lizorkin-type spaces) in terms of $\varepsilon(f)$, showing that \begin{equation*} \varepsilon(f)\sim \mathrm{dist} (f, V\cap Λ_s)_{Λ_s}: = \inf_{g\in Λ_s\cap V} \|f-g\|_{Λ_s}.\end{equation*} Moreover, we present our results in a general framework based on quasi-normed lattices of function sequences $X$ and Daubechies $s$-Lipschitz $X$-based spaces.

math.FA↗

arXiv2025

Renormalized stochastic pressure equation with log-correlated Gaussian coefficients

We study periodic solutions to the following divergence-form stochastic partial differential equation with Wick-renormalized gradient on the $d$-dimensional flat torus $\mathbb{T}^d$, \[ -\nabla\cdot\left(e^{\diamond (- βX) }\diamond\nabla U\right)=\nabla \cdot (e^{\diamond (- βX)} \diamond \mathbf{F}), \] where $X$ is the log-correlated Gaussian field, $\mathbf{F}$ is a random vector field representing the flux, the in/out-flow of fluid per unit area per unit time, and $\diamond$ denotes the Wick product. The problem is a variant of the stochastic pressure equation, in which $U$ is modeling the pressure of a creeping water-flow in crustal rock that occurs in enhanced geothermal heating. In the original model, the Wick exponential term $e^{\diamond(-βX)}$ is modeling the random permeability of the rock. The porosity field is given by a log-correlated Gaussian random field $βX$, where $β<\sqrt{d}$. We use elliptic regularity theory in order to define a notion of a solution to this (a priori very ill-posed) problem, via modifying the $S$-transform from Gaussian white noise analysis, and then establish the existence and uniqueness of solutions. Moreover, we show that the solution to the problem can be expressed in terms of the Gaussian multiplicative chaos measure.

math.PR↗

arXiv2024

On the convergence of dynamic implementations of Hamiltonian Monte Carlo and No U-Turn Samplers

There is substantial empirical evidence about the success of dynamic implementations of Hamiltonian Monte Carlo (HMC), such as the No U-Turn Sampler (NUTS), in many challenging inference problems but theoretical results about their behavior are scarce. The aim of this paper is to fill this gap. More precisely, we consider a general class of MCMC algorithms we call dynamic HMC. We show that this general framework encompasses NUTS as a particular case, implying the invariance of the target distribution as a by-product. Second, we establish conditions under which NUTS is irreducible and aperiodic and as a corrolary ergodic. Under conditions similar to the ones existing for HMC, we also show that NUTS is geometrically ergodic. Finally, we improve existing convergence results for HMC showing that this method is ergodic without any boundedness condition on the stepsize and the number of leapfrog steps, in the case where the target is a perturbation of a Gaussian distribution.

stat.CO↗

arXiv2024

Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We study quasiconformal mappings in planar domains $Ω$ and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary $\partial Ω$ and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below $1$, the same conditions give global regularity in $Ω$ for the principal solutions with Beltrami coefficient supported in $Ω$.

math.AP↗

arXiv2023

Random analytic functions via Gaussian multiplicative chaos

We define a random analytic function $φ$ on the unit disc by letting a Gaussian multiplicative measure to be one of its Clark measures. We show that $φ$ is almost surely a Blaschke product and we provide rather sharp estimates for the density of its zeroes.

math.PR↗

arXiv2022

On the structure of Nevanlinna measures

In this paper, we study the structural properties of Nevanlinna measures, i.e. Borel measures that arise in the integral representation of Herglotz-Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually perfomed via the polydisc.

math.CV↗

arXiv2021

Approximation in the Zygmund and Hölder classes on $\mathbb{R}^n$

We determine the distance (up to a multiplicative constant) in the Zygmund class $Λ_{\ast}(\mathbb{R}^n)$ to the subspace $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image under the Bessel potential $J := (1-Δ)^{-1/2}$ of the space $\mathbf{bmo}(\mathbb{R}^n),$ which is a non-homogeneous version of the classical $\mathrm{BMO}.$ Locally, $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n)$ consists of functions that together with their first derivatives are in $\mathbf{bmo}(\mathbb{R}^n).$ More generally, we consider the same question when the Zygmund class is replaced by the Hölder space $Λ_{s}(\mathbb{R}^n),$ with $0 < s \leq 1$ and the corresponding subspace is $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n),$ the image under $(1-Δ)^{-s/2}$ of $\mathbf{bmo}(\mathbb{R}^n).$ One should note here that $Λ_{1}(\mathbb{R}^n) = Λ_{\ast}(\mathbb{R}^n).$ Such results were known earlier only for $n = s = 1$ with a proof that does not extend to the general case. Our results are expressed in terms of second differences. As a byproduct of our wavelet based proof, we also obtain the distance from $f \in Λ_{s}(\mathbb{R}^n)$ to $\mathrm{J}_{s}(\mathbf{bmo})(\mathbb{R}^n)$ in terms of the wavelet coefficients of $f.$ We additionally establish a third way to express this distance in terms of the size of the hyperbolic gradient of the harmonic extension of $f$ on the upper half-space $\mathbb{R}^{n+1}_{+}.$

math.CA↗

arXiv2021

Stretching and Rotation of Planar Quasiconformal Mappings on a Line

In this article, we examine stretching and rotation of planar quasiconformal mappings on a line. We show that for almost every point on the line, the set of complex stretching exponents (describing stretching and rotation jointly) is contained in the disk $ \overline{B}(1/(1-k^4),k^2/(1-k^4))$. This yields a quadratic improvement over the known optimal estimate for general sets of Hausdorff dimension $1$. Our proof is based on holomorphic motions and estimates for dimensions of quasicircles. We also give a lower bound for the dimension of the image of a $1$-dimensional subset of a line under a quasiconformal mapping.

math.CV↗

arXiv2021

On mappings of finite distortion that are quasiconformal in the unit disk

We study quasiconformal mappings of the unit disk that have planar extension with controlled distortion. For these mappings we prove a bound for the modulus of continuity of the inverse map, which somewhat surprisingly is almost as good as for global quasiconformal maps. Furthermore, we give examples which improve the known bounds for the three point property of generalized quasidisks. Finally, we establish optimal regularity of such maps when the image of the unit disk has cusp type singularities.

math.CV↗

arXiv2021

Riesz projection and bounded mean oscillation for Dirichlet series

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.

math.FA↗

arXiv2021

Localized Regularity of Planar Maps of Finite Distortion

In this article we study fine regularity properties for mappings of finite distortion. Our main theorems yield strongly localized regularity results in the borderline case in the class of maps of exponentially integrable distortion. Analogues of such results were known earlier in the case of quasiconformal mappings. Moreover, we study regularity for maps whose distortion has higher exponential integrability.

math.CV↗

arXiv2021

Random tree Besov priors -- Towards fractal imaging

We propose alternatives to Bayesian a priori distributions that are frequently used in the study of inverse problems. Our aim is to construct priors that have similar good edge-preserving properties as total variation or Mumford-Shah priors but correspond to well defined infinite-dimensional random variables, and can be approximated by finite-dimensional random variables. We introduce a new wavelet-based model, where the non zero coefficient are chosen in a systematic way so that prior draws have certain fractal behaviour. We show that realisations of this new prior take values in some Besov spaces and have singularities only on a small set $τ$ that has a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator, arising from the the new prior, in denoising problem.

math.ST↗

arXiv2020

Homogenization of iterated singular integrals with applications to random quasiconformal maps

We study homogenization of iterated randomized singular integrals and homeomorphic solutions to the Beltrami differential equation with a random Beltrami coefficient. More precisely, let $(F_j)_{j \geq 1}$ be a sequence of normalized homeomorphic solutions to the planar Beltrami equation $\overline{\partial} F_j (z)=μ_j(z,ω) \partial F_j(z),$ where the random dilatation satisfies $|μ_j|\leq k<1$ and has locally periodic statistics, for example of the type $$μ_j (z,ω)=ϕ(z)\sum_{n\in \mathbf{Z}^2}g(2^j z-n,X_{n}(ω)), $$ where $g(z,ω)$ decays rapidly in $z$, the random variables $X_{n}$ are i.i.d., and $ϕ\in C^\infty_0$. We establish the almost sure and local uniform convergence as $j\to\infty$ of the maps $F_j$ to a deterministic quasiconformal limit $F_\infty$. This result is obtained as an application of our main theorem, which deals with homogenization of iterated randomized singular integrals. As a special case of our theorem, let $T_1,\ldots , T_{m}$ be translation and dilation invariant singular integrals on ${\bf R}^d, $ and consider a $d$-dimensional version of $μ_j$, e.g., as defined above or within a more general setting. We then prove that there is a deterministic function $f$ such that almost surely as $j\to\infty$, $$ μ_j T_{m}μ_j\ldots T_1μ_j\to f \quad \textrm{weakly in } L^p,\quad 1 < p < \infty\ . $$

math.CV↗