arXiv subjects

Palle Jorgensen

Publications and source records attributed to Palle Jorgensen.

At least 19 recordsLinked to original sources

arXiv2026

Subdiagrams and invariant measures for generalized Bratteli diagrams

The results of this paper contribute to the study of invariant measures of Borel dynamical systems that can be modeled using generalized Bratteli diagrams. In this context, we study tail invariant measures on the path spaces of generalized Bratteli diagrams, allowing countably infinite vertex sets at each level. Our main focus is on subdiagrams of generalized Bratteli diagrams and the problem of extending tail invariant probability measures from vertex and edge subdiagrams to the ambient diagram. We establish necessary and sufficient conditions for the finiteness of such extensions, formulated in terms of incidence matrices and associated stochastic matrices. Several classes of generalized Bratteli diagrams and their subdiagrams are analyzed in detail, including simple, stationary, and bounded size diagrams. We develop constructive, step-by-step procedures for measure extension and for approximating invariant measures by measures supported on suitable subdiagrams. In addition, we provide explicit examples of generalized Bratteli diagrams that admit no probability tail invariant measures, a phenomenon absent for standard Bratteli diagrams with finite vertex sets. Finally, we address convergence questions for sequences of invariant measures arising from approximations by subdiagrams, clarifying the relationship between combinatorial structure and measure-theoretic behavior.

math.DS↗

arXiv2025

An explicit construction of heat kernels and Green's functions in measure spaces

We explicitly construct a heat kernel as a Neumann series for certain function spaces, such as $L^{1}$, $L^{2}$, and Hilbert spaces, associated to a locally compact Hausdorff space $\mathfrak{X}$ with Borel $σ$-algebra $\mathcal{B}$, and endowed with additional measure-theoretic data. Our approach is an adaptation of classical work due to Minakshishundaram and Pleijel, and it requires as input a parametrix or small time approximation to the heat kernel. The methodology developed in this article applies to yield new instances of heat kernel constructions, including normalized Laplacians on finite and infinite graphs as well as Hilbert spaces with reproducing kernels.

math.CA↗

arXiv2025

Analytic continuation of time in Brownian motion. Stochastic distributions approach

With the use of Hida's white noise space theory space theory and spaces of stochastic distributions, we present a detailed analytic continuation theory for classes of Gaussian processes, with focus here on Brownian motion. For the latter, we prove and make use a priori bounds, in the complex plane, for the Hermite functions; as well as a new approach to stochastic distributions. This in turn allows us to present an explicit formula for an analytically continued white noise process, realized this way in complex domain. With the use of the Wick product, we then apply our complex white noise analysis in a derivation of a new realization of Hilbert space-valued stochastic integrals

math.PR↗

arXiv2024

Fourier Series for Two-Dimensional Singular-Fibered Measures

In this paper we study 2D Fourier expansions for a general class of planar measures $μ$, generally singular, but assumed compactly supported in $\mathbb{R}^2$. We focus on the following question: When does $L^2(μ)$ admit a 2D system of Fourier expansions? We offer concrete conditions allowing an affirmative answer to the question for a large class of Borel probability measures, and we present an explicit Fourier duality for these cases. Our 2D Fourier analysis relies on a detailed conditioning-analysis. For a given $μ$, it is based on the corresponding systems of 1D measures consisting of a marginal measure and associated family of conditional measures computed from $μ$ by the Rokhlin Disintegration Theorem. Our identified $L^2(μ)$-Fourier expansions are special in two ways: For our measures $μ$, the Fourier expansions are generally non-orthogonal, but nonetheless, they lend themselves to algorithmic computations. Second, we further stress that our class of 2D measures $μ$ considered here go beyond what exists in the literature. In particular, our measures do not require affine iterated function system (IFS) properties, but we do study grid IFS measures in detail and provide some technical criteria guaranteeing their admission of Fourier expansions. Our analyses make use of estimates for the Hausdorff dimensions of the measure supports. An important class of examples addressed in this paper is fractal Bedford-McMullen carpets.

math.FA↗

arXiv2022

Representation theory and multilevel filters

We present a general setting where wavelet filters and multiresolution decompositions can be defined, beyond the classical $\mathbf L^2(\mathbb R,dx)$ setting. This is done in a framework of {\em iterated function system} (IFS) measures; these include all cases studied so far, and in particular the Julia set/measure cases. Every IFS has a fixed order, say $N$, and we show that the wavelet filters are indexed by the infinite dimensional group $G$ of functions from $X$ into the unitary group $U_N$. We call $G$ the loop group because of the special case of the unit circle.

math.FA↗

arXiv2022

Finitely additive functions in measure theory and applications

We consider, and make precise, a certain extension of the Radon-Nikodym derivative operator, to functions which are additive, but not necessarily sigma-additive, on a subset of a given sigma-algebra. We give applications to probability theory; in particular, to the study of $μ$-Brownian motion, to stochastic calculus via generalized Itô-integrals, and their adjoints (in the form of generalized stochastic derivatives), to systems of transition probability operators indexed by families of measures $μ$, and to adjoints of composition operators.

math.PR↗

arXiv2021

White Noise Space Analysis and Multiplicative Change of Measures

In this paper we display a family of Gaussian processes, with explicit formulas and transforms. This is presented with the use of duality tools in such a way that the corresponding path-space measures are mutually singular. We make use of a corresponding family of representations of the canonical commutation relations (CCR) in an infinite number of degrees of freedom. A key feature of our construction is explicit formulas for associated transforms; these are infinite-dimensional analogues of Fourier transforms. Our framework is that of Gaussian Hilbert spaces, reproducing kernel Hilbert spaces, and Fock spaces. The latter forms the setting for our CCR representations. We further show, with the use of representation theory, and infinite-dimensional analysis, that our pairwise inequivalent probability spaces (for the Gaussian processes) correspond in an explicit manner to pairwise disjoint CCR representations.

math-ph↗

arXiv2020

New characterizations of reproducing kernel Hilbert spaces and applications to metric geometry

We give two new global and algorithmic constructions of the reproducing kernel Hilbert space associated to a positive definite kernel. We further present ageneral positive definite kernel setting using bilinear forms, and we provide new examples. Our results cover the case of measurable positive definite kernels, and we give applications to both stochastic analysisand metric geometry and provide a number of examples.

math.FA↗

arXiv2019

New boundaries for positive definite functions

With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $K$ we analyze associated Gaussian processes $V$. Properties of the Gaussian processes will be derived from certain factorizations of $K$, arising as a covariance kernel of $V$. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $K$. Our results entail an analysis of a partial order on families of p.d. kernels, a duality for operators and frames, optimization, Karhunen--Loève expansions, and factorizations. Applications include a new boundary analysis for the Drury-Arveson kernel, and for certain fractals arising as iterated function systems; and an identification of optimal feature spaces in machine learning models.

math.FA↗

arXiv2019

A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spaces

We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral theory of operators in Hilbert space, to optimization, to large sparse systems, to iterated function systems (IFS), and to fractal harmonic analysis. We present a new recursive iteration scheme involving as input a prescribed sequence of selfadjoint projections. Applications include random Kaczmarz recursions, their limits, and their error-estimates.

math.FA↗

arXiv2019

Superposition, reduction of multivariable problems, and approximation

We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes for multivariable problems, covering both a finite, and an infinite, number of variables. Starting with functions in "many" variables, we offer constructive reductions into superposition, with component terms, that make use of only functions in one variable, and specified choices of coordinate directions. Our proofs are transform based, using explicit transforms, Fourier and Radon; as well as multivariable Shannon interpolation.

math.FA↗

arXiv2019

On reproducing kernels, and analysis of measures

Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive definite kernel is also the covariance kernel of a Gaussian process. Given a fixed sigma-finite measure $μ$, we consider positive definite kernels defined on the subset of the sigma algebra having finite $μ$ measure. We show that then the corresponding Hilbert factorizations consist of signed measures, finitely additive, but not automatically sigma-additive. We give a necessary and sufficient condition for when the measures in the RKHS, and the Hilbert factorizations, are sigma-additive. Our emphasis is the case when $μ$ is assumed non-atomic. By contrast, when $μ$ is known to be atomic, our setting is shown to generalize that of Shannon-interpolation. Our RKHS-approach further leads to new insight into the associated Gaussian processes, their Itô calculus and diffusion. Examples include fractional Brownian motion, and time-change processes.

math.FA↗

arXiv2018

Decomposition of Gaussian processes, and factorization of positive definite kernels

We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization for p.d. kernels is intuitively motivated by matrix factorizations, but in infinite dimensions, subtle measure theoretic issues must be addressed. Consider a given p.d. kernel $K$, presented as a covariance kernel for a Gaussian process $V$. We then give an explicit duality for these two seemingly different notions of factorization, for p.d. kernel $K$, vs for Gaussian process $V$. Our result is in the form of an explicit correspondence. It states that the analytic data which determine the variety of factorizations for $K$ is the exact same as that which yield factorizations for $V$. Examples and applications are included: point-processes, sampling schemes, constructive discretization, graph-Laplacians, and boundary-value problems.

math.FA↗

arXiv2018

Realizations and Factorizations of Positive Definite Kernels

Given a fixed sigma-finite measure space $\left(X,\mathscr{B},ν\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic processes. In the interesting cases, the given measure $ν$ is infinite, but sigma-finite. We introduce such positive definite kernels $K\left(\cdot,\cdot\right)$ with the two variables from the subset of the sigma-algebra $\mathscr{B}$, sets having finite $ν$ measure. Our setting and results are motivated by applications. The latter are covered in the second half of the paper. We first make precise the notions of realizations and factorizations for $K$; and we give necessary and sufficient conditions for $K$ to have realizations and factorizations in $L^{2}\left(ν\right)$. Tools in the proofs rely on probability theory and on spectral theory for unbounded operators in Hilbert space. Applications discussed here include the study of reversible Markov processes, and realizations of Gaussian fields, and their Ito-integrals.

math.SP↗

arXiv2018

Purely atomic representations of higher-rank graph C*-algebras

We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.

math.OA↗

arXiv2018

Martin boundaries of representations of the Cuntz algebra

In a number of recent papers, the idea of generalized boundaries has found use in fractal and in multiresolution analysis; many of the papers having a focus on specific examples. Parallel with this new insight, and motivated by quantum probability, there has also been much research which seeks to study fractal and multiresolution structures with the use of certain systems of non-commutative operators; non-commutative harmonic/stochastic analysis. This in turn entails combinatorial, graph operations, and branching laws. The most versatile, of these non-commutative algebras are the Cuntz algebras; denoted $\mathcal{O}_{N}$, $N$ for the number of isometry generators. $N$ is at least 2. Our focus is on the representations of $\mathcal{O}_{N}$. We aim to develop new non-commutative tools, involving both representation theory and stochastic processes. They serve to connect these parallel developments. In outline, boundaries, Poisson, or Martin, are certain measure spaces (often associated to random walk models), designed to encode the asymptotic behavior, e.g., how trajectories diverge when the number of steps goes to infinity. We stress that our present boundaries (commutative or non-commutative) are purely measure-theoretical objects. Although, as we show, in some cases our boundaries may be compared with more familiar topological boundaries.

math.FA↗

arXiv2018

Monic representations of finite higher-rank graphs

In this paper we define the notion of monic representation for the $C^*$-algebras of finite higher-rank graphs with no sources, and undertake a comprehensive study of them. Monic representations are the representations that, when restricted to the commutative $C^*$-algebra of the continuous functions on the infinite path space, admit a cyclic vector. We link monic representations to the $Λ$-semibranching representations previously studied by Farsi, Gillaspy, Kang, and Packer, and also provide a universal representation model for nonnegative monic representations.

math.OA↗

arXiv2017

Positive definite (p.d.) functions vs p.d. distributions

We give explicit transforms for Hilbert spaces associated with positive definite functions on $\mathbb{R}$, and positive definite tempered distributions, incl., generalizations to non-abelian locally compact groups. Applications to the theory of extensions of p.d. functions/distributions are included. We obtain explicit representation formulas for positive definite tempered distributions in the sense of L. Schwartz, and we give applications to Dirac combs and to diffraction. As further applications, we give parallels between Bochner's theorem (for continuous p.d. functions) on the one hand, and the generalization to Bochner/Schwartz representations for positive definite tempered distributions on the other; in the latter case, via tempered positive measures. Via our transforms, we make precise the respective reproducing kernel Hilbert spaces (RKHSs), that of N. Aronszajn and that of L. Schwartz. Further applications are given to stationary-increment Gaussian processes.

math.FA↗