arXiv subjects

Mateusz Kwaśnicki

Publications and source records attributed to Mateusz Kwaśnicki.

At least 37 records · Page 2Linked to original sources

arXiv2024

The $\ell^p$ norm of the Riesz--Titchmarsh transform for even integer $p$

The long-standing conjecture that for $p \in (1, \infty)$ the $\ell^p(\mathbb Z)$ norm of the Riesz--Titchmarsh discrete Hilbert transform is the same as the $L^p(\mathbb R)$ norm of the classical Hilbert transform, is verified when $p = 2 n$ or $\frac{p}{p - 1} = 2 n$, for $n \in \mathbb N$. The proof, which is algebraic in nature, depends in a crucial way on the sharp estimate for the $\ell^p(\mathbb Z)$ norm of a different variant of this operator for the full range of $p$. The latter result was recently proved by the authors in [Bañuelos, Kwaśnicki, On the $\ell^p$-norm of the discrete Hilbert transform, Duke Math. J. 168(3) (2019): 471-504].

math.CA↗

arXiv2023

Fleming-Viot couples live forever

We prove a non-extinction result for Fleming-Viot-type systems of two particles with dynamics described by an arbitrary symmetric Hunt process under the assumption that the reference measure is finite. Additionally, we describe an invariant measure for the system, we discuss its ergodicity, and we prove that the reference measure is a stationary measure for the embedded Markov chain of positions of the surviving particle at successive branching times.

math.PR↗

arXiv2023

Axioms for Constant Function Market Makers

We study axiomatic foundations for different classes of constant-function automated market makers (CFMMs). We focus particularly on separability and on different invariance properties under scaling. Our main results are an axiomatic characterization of a natural generalization of constant product market makers (CPMMs), popular in decentralized finance, on the one hand, and a characterization of the Logarithmic Scoring Rule Market Makers (LMSR), popular in prediction markets, on the other hand. The first class is characterized by the combination of independence and scale invariance, whereas the second is characterized by the combination of independence and translation invariance. The two classes are therefore distinguished by a different invariance property that is motivated by different interpretations of the numéraire in the two applications. However, both are pinned down by the same separability property. Moreover, we characterize the CPMM as an extremal point within the class of scale invariant, independent, symmetric AMMs with non-concentrated liquidity provision. Our results add to a formal analysis of mechanisms that are currently used for decentralized exchanges and connect the most popular class of DeFi AMMs to the most popular class of prediction market AMMs.

cs.GT↗

arXiv2023

Bell-shaped sequences

A nonnegative real function $f$ is said to be bell-shaped if it converges to zero at $\pm\infty$ and the $n$th derivative of $f$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$ In a similar way, we may say that a nonnegative sequence $a_k$ is bell-shaped if it converges to zero and the $n$th iterated difference of $a_k$ changes sign $n$ times for every $n = 0, 1, 2, \ldots$ Bell-shaped functions were recently characterised by Thomas Simon and the first author. In the present paper we provide an analogous description of bell-shaped sequences. More precisely, we identify bell-shaped sequences with convolutions of Pólya frequency sequences and completely monotone sequences, and we characterise the corresponding generating functions as exponentials of appropriate Pick functions.

math.CA↗

arXiv2019

Boundary traces of shift-invariant diffusions in half-plane

We study boundary traces of shift-invariant diffusions: two-dimensional diffusions in the upper half-plane $\mathbb{R} \times [0, \infty)$ (or in $\mathbb{R} \times [0, R)$) invariant under horizontal translations. We prove that the corresponding trace processes are Lévy processes with completely monotone jumps, and, conversely, every Lévy process with completely monotone jumps is a boundary trace of some shift-invariant diffusion. Up to some natural transformations of space and time, this correspondence is bijective. We also reformulate this result in the language of additive functionals of the Brownian motion in $[0, \infty)$ (or in $[0, R)$), and Brownian excursions. Our main tool is the recent extension of Krein's spectral theory of strings, due to Eckhardt and Kostenko.

math.PR↗

arXiv2019

Characterisation of the class of bell-shaped functions

A non-negative function $f$ is said to be 'bell-shaped' if $f$ tends to zero at $\pm \infty$ and the $n$-th derivative of $f$ changes its sign $n$ times for every $n = 0, 1, 2, \ldots$ We provide a complete characterisation of the class of bell-shaped functions: we prove that every bell-shaped function is a convolution of a 'Pólya frequency function' and an *absolutely monotone-then-completely monotone* function. An equivalent condition in terms of the holomorphic extension of the Fourier transform is also given. As a corollary, various properties of bell-shaped functions follow. In particular, we prove that bell-shaped probability distributions are infinitely divisible, and that the zeroes of the $n$-th derivative of a bell-shaped function grow at a linear rate as $n \to \infty$.

math.PR↗

arXiv2019

Harmonic extension technique for non-symmetric operators with completely monotone kernels

We identify a class of non-local integro-differential operators $K$ in $\mathbb{R}$ with Dirichlet-to-Neumann maps in the half-plane $\mathbb{R} \times (0, \infty)$ for appropriate elliptic operators $L$. More precisely, we prove a bijective correspondence between Lévy operators $K$ with non-local kernels of the form $ν(y - x)$, where $ν(x)$ and $ν(-x)$ are completely monotone functions on $(0, \infty)$, and elliptic operators $L = a(y) \partial_{xx} + 2 b(y) \partial_{x y} + \partial_{yy}$. This extends a number of previous results in the area, where symmetric operators have been studied: the classical identification of the Dirichlet-to-Neumann operator for the Laplace operator in $\mathbb{R} \times (0, \infty)$ with $-\sqrt{-\partial_{xx}}$, the square root of one-dimensional Laplace operator; the Caffarelli--Silvestre identification of the Dirichlet-to-Neumann operator for $\nabla \cdot (y^{1 - α} \nabla)$ with $(-\partial_{xx})^{α/2}$ for $α\in (0, 2)$; and the identification of Dirichlet-to-Neumann maps for operators $a(y) \partial_{xx} + \partial_{yy}$ with complete Bernstein functions of $-\partial_{xx}$ due to Mucha and the author. Our results rely on recent extension of Krein's spectral theory of strings by Eckhardt and Kostenko.

math.AP↗

arXiv2019

Random walks are determined by their trace on the positive half-line

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $\max(X_n, 0)$ for $n = 1, 2, \ldots$, as conjectured recently by Loïc Chaumont and Ron Doney. Equivalently, the law of $X_n$ is determined by its upward space-time Wiener-Hopf factor. Our methods are complex-analytic.

math.PR↗

arXiv2018

A new class of bell-shaped functions

We provide a large class of functions $f$ that are bell-shaped: the $n$-th derivative of $f$ changes its sign exactly $n$ times. This class is described by means of Stieltjes-type representation of the logarithm of the Fourier transform of $f$, and it contains all previously known examples of bell-shaped functions, as well as extended generalised gamma convolutions, including all density functions of stable distributions. The proof involves representation of $f$ as the convolution of a Pólya frequency function and a function which is absolutely monotone on $(-\infty, 0)$ and completely monotone on $(0, \infty)$. In the final part we disprove three plausible generalisations of our result.

math.CA↗

arXiv2018

Fluctuation theory for Lévy processes with completely monotone jumps

We study the Wiener-Hopf factorization for Lévy processes $X_t$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener-Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_t$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_t$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(ξ)$ of $X_t$, including a peculiar structure of the curve along which $f(ξ)$ takes real values.

math.PR↗

arXiv2017

On the $l^p$-norm of the discrete Hilbert transform

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

math.CA↗

arXiv2017

Extension technique for complete Bernstein functions of the Laplace operator

We discuss representation of certain functions of the Laplace operator $Δ$ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies $(-Δ)^{1/2}$, the square root of the $d$-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the $(d + 1)$-dimensional Laplace operator $Δ_{t,x}$ in $(0, \infty) \times \mathbf{R}^d$. Caffarelli and Silvestre extended this to fractional powers $(-Δ)^{α/2}$, which correspond to operators $\nabla_{t,x} (t^{1 - α} \nabla_{t,x})$. We provide an analogous result for all complete Bernstein functions of $-Δ$ using Krein's spectral theory of strings. Two sample applications are provided: a Courant--Hilbert nodal line theorem for harmonic extensions of the eigenfunctions of non-local Schrödinger operators $ψ(-Δ) + V(x)$, as well as an upper bound for the eigenvalues of these operators. Here $ψ$ is a complete Bernstein function and $V$ is a confining potential.

math.AP↗

arXiv2017

Minimal Hermite-type eigenbasis of the discrete Fourier transform

There exist many ways to build an orthonormal basis of $\mathbb{R}^N$, consisting of the eigenvectors of the discrete Fourier transform (DFT). In this paper we show that there is only one such orthonormal eigenbasis of the DFT that is optimal in the sense of an appropriate uncertainty principle. Moreover, we show that these optimal eigenvectors of the DFT are direct analogues of the Hermite functions, that they also satisfy a three-term recurrence relation and that they converge to Hermite functions as $N$ increases to infinity.

math.CA↗

arXiv2017

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process. In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space. Our results are expressed in terms of Pruitt's functions $K(r)$ and $L(r)$, measuring local activity and the amount of large jumps of the Lévy process, respectively.

math.PR↗

arXiv2016

Pólya's conjecture fails for the fractional Laplacian

The analogue of Pólya's conjecture is shown to fail for the fractional Laplacian (-Delta)^{alpha/2} on an interval in 1-dimension, whenever 0 < alpha < 2. The failure is total: every eigenvalue lies below the corresponding term of the Weyl asymptotic. In 2-dimensions, the fractional Pólya conjecture fails already for the first eigenvalue, when 0 < alpha < 0.984.

math.SP↗

arXiv2016

Eigenvalues of the fractional Laplace operator in the unit ball

We describe a highly efficient numerical scheme for finding two-sided bounds for the eigenvalues of the fractional Laplace operator (-Delta)^{alpha/2} in the unit ball D in R^d, with a Dirichlet condition in the complement of D. The standard Rayleigh-Ritz variational method is used for the upper bounds, while the lower bounds involve the less-known Aronszajn method of intermediate problems. Both require explicit expressions for the fractional Laplace operator applied to a linearly dense set of functions in L^2(D). We use appropriate Jacobi-type orthogonal polynomials, which were studied in a companion paper "Fractional Laplace operator and Meijer G-function". Our numerical scheme can be applied analytically when polynomials of degree two are involved. This is used to partially resolve the conjecture of T. Kulczycki, which claims that the second smallest eigenvalue corresponds to an antisymmetric function: we prove that this is the case when either d <= 2 and 0 < alpha <= 2, or d <= 9 and alpha = 1, and we provide strong numerical evidence for d <= 9 and general alpha in (0,2].

math.AP↗

arXiv2015

Contractivity and ground state domination properties for non-local Schrödinger operators

We study supercontractivity and hypercontractivity of Markov semigroups obtained via ground state transformation of non-local Schrödinger operators based on generators of symmetric jump-paring Lévy processes with Kato-class confining potentials. This class of processes has the property that the intensity of single large jumps dominates the intensity of all multiple large jumps, and the related operators include pseudo-differential operators of interest in mathematical physics. We refine these contractivity properties by the concept of $L^p$-ground state domination and its asymptotic version, and derive sharp necessary and sufficient conditions for their validity in terms of the behaviour of the Lévy density and the potential at infinity. As a consequence, we obtain for a large subclass of confining potentials that, on the one hand, supercontractivity and ultracontractivity, on the other hand, hypercontractivity and asymptotic ultracontractivity of the transformed semigroup are equivalent properties. This is in stark contrast to classical Schrödinger operators, for which all these properties are known to be different.

math.PR↗

arXiv2015

Fractional Laplace operator and Meijer G-function

We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|^2, or generalized hypergeometric functions of -|x|^2, multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1-|x|^2)_+^{alpha/2} (-Delta)^{alpha/2} with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper "Eigenvalues of the fractional Laplace operator in the unit ball".

math.AP↗