Positive Half-line Research Articles

Ergodicity of an SPDE associated with a many-server queue

We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a first-come-first-serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1-\beta N^{-1/2}+o(N^{-1/2})$ for some $\beta>0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Ito equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [Aghajani and Ramanan (2016)] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in [Oper. Res. 29 (1981) 567–588]. The methods introduced here are more generally applicable for the analysis of a broader class of networks.

Convergence of the Successive Approximation Method in the Cauchy Problem for an Integro-Differential Equation with Quadratic Nonlinearity

The equations considered in this article have the form in which the time derivative of the unknown function is expressed as a double integral over the space variables of a weighted quadratic expression of the sought function. The domain of integration is unbounded and does not depend on time but depends on the space variable. We study the Cauchy problem in the function classes accompanying the equation with initial data on the positive half-line. In application to this problem, the convergence of the successive approximation method is justified. An estimate is given of the quality of the approximation depending on the number of the iterated solution. It is proved that, on any finite time interval, the posed Cauchy problem has at most one solution in the accompanying function class. An existence theorem is proved in the same class.

On Bound Electron Pairs on the Half-Line

Based on the quantum two-body problem introduced in [arXiv:1604.06693] we consider bound pairs of electrons moving on the positive half-line. The analysis is motivated by the ground-breaking work of Cooper who identified the pairing of electrons as a possible explanation for superconducting behaviour in metals. In this paper we are interested in the connection between the topologies of the underlying one- and two-particle configuration spaces and spectral properties of the Hamiltonian linked to a condensation of pairs. We derive explicit estimates for the energy gap and prove condensation for a gas of non-interacting pairs. Finally, we add some disorder to the system and prove destruction of the condensate.

On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions

ABSTRACTIn the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to the Laguerre weight and an additional point mass N>0 at the origin, these polynomials satisfy, for any , a linear differential equation of order . In the present paper we establish a new elementary representation of the corresponding 'Laguerre-type' differential operator and show its symmetry with respect to the underlying weighted scalar product. Furthermore, we discuss various other representations of the operator, mainly given in factorized form, and show their equivalence. Finally, by applying a limiting process to the Laguerre-type equation, we deduce new elementary representations for the higher-order differential equation satisfied by the Bessel-type functions on the positive half-line.

A remark on the effect of random singular two-particle interactions

In this note we study a two-particle bound system (molecule) moving on the positive half-line under the influence of randomly distributed singular two-particle interactions generated by a Poisson process. We give a rigorous definition of the underlying Hamiltonian and study its spectral properties. As a main result we prove that, with finite probability, the random interactions destroy the discrete part of the spectrum which is present in the free system. Most interestingly, this phenomenon is somewhat contrary to the role attributed to random interactions in the context of Anderson localisation where disorder is generally associated with a suppression of transport.

A Note on the Quasi-stationary Distribution of the Shiryaev Martingale on the Positive Half-Line

We obtain a closed-form formula for the quasi-stationary distribution of the classical Shiryaev martingale diffusion considered on the positive half-line $[A,+\infty)$ with $A>0$ fixed; the state s...

Continuity and growth of free multiplicative convolution semigroups

Let $\mu $ be a compactly supported probability measure on the positive half-line and let $\mu ^{\boxtimes t}$ be the free multiplicative convolution semigroup. We show that the support of $\mu ^{\boxtimes t}$ varies continuously as $t$ changes. We also obtain the asymptotic length of the support of these measures.

Socioeconomic fractality: A Poisson-process approach

Combining together socioeconomic inequality and fractals, the notion of socioeconomic fractality is introduced and explored. Two different socioeconomic perspectives are considered, and a Poisson-process modeling approach is applied. It is established that socioeconomic fractality is characterized by a power-law form: a power-law Poisson intensity over the positive half-line. The results presented here are beyond the realm of a random-variables modeling approach, and their connections to Pareto distributions and to other types of fractality are discussed.

Persistence of small noise and random initial conditions

AbstractThe effect of small noise in a smooth dynamical system is negligible on any finite time interval; in this paper we study situations where the effect persists on intervals increasing to ∞. Such an asymptotic regime occurs when the system starts from an initial condition that is sufficiently close to an unstable fixed point. In this case, under appropriate scaling, the trajectory converges to a solution of the unperturbed system started from a certainrandominitial condition. In this paper we consider the case of one-dimensional diffusions on the positive half-line; this case often arises as a scaling limit in population dynamics.

The Form of Multi-additive Symmetric Functions

In this paper we show that every n-additive symmetric function, between quite general structures like abelian groups and semigroups, can be factorized into a composition of additive function with the product of additive functions. We also show that every two n-additive functions, defined on a product of groups, with equal counterimages of the positive half-line must be proportional.

On Lennard-Jones-type potentials on the half-line

In this paper we study a particle under the influence of a Lennard-Jones potential moving in a simple quantum wire modelled by the positive half-line. Despite its physical significance, this potential is only rarely studied in the literature and due to its singularity at the origin it cannot be considered as a standard perturbation of the one-dimensional Laplacian. It is therefore our aim to provide a thorough description of the Hamiltonian in one dimension via the construction of a suitable quadratic form. Our results include a discussion of spectral and scattering properties which finally allows us to generalise some results from [Robinson1974] as well as [RadinSimon1978].

Universal Poisson-process limits for general random walks

This paper considers ensembles of general, independent and identically distributed, random walks. Taking the ensemble-size to grow infinitely large, and also taking the running-time of the random walks to grow infinitely large, universal Poisson-process limits are obtained. Specifically, it is established that the positions of general linear random walks converge universally to Poisson processes, over the real line, with uniform and exponential intensities. And, it is established that the positions of general geometric random walks converge universally to Poisson processes, over the positive half-line, with harmonic and power intensities. Corollaries to these universal convergence results yield the extreme-value statistics of Gumbel, Weibull, and Frechet.

On the set of L-space surgeries for links

We study the qualitative structure of the set LS of integral L-space surgery slopes for links with two components. It is known that the set of L-space surgery slopes for a nontrivial L-space knot is always a positive half-line. However, already for two-component torus links the set LS has a very complicated structure, e.g. in some cases it can be unbounded from below. In order to probe the geometry of this set, we ask if it is bounded from below for more general L-space links with two components.For algebraic two-component links we provide three complete characterizations for the boundedness from below: one in terms of the Alexander polynomial, one in terms of the embedded resolution graph, and one in terms of the so-called h-function introduced by the authors in [9]. It turns out that LS is bounded from below for most algebraic links. If it is unbounded from below, it must contain a negative half-line parallel to one of the axes. We also give a sufficient condition for boundedness for arbitrary L-space links.

Quantum return probability of a system of N non-interacting lattice fermions

We consider N non-interacting fermions performing continuous-time quantum walks on a one-dimensional lattice. The system is launched from a most compact configuration where the fermions occupy neighboring sites. We calculate exactly the quantum return probability (sometimes referred to as the Loschmidt echo) of observing the very same compact state at a later time t. Remarkably, this probability depends on the parity of the fermion number—it decays as a power of time for even N, while for odd N it exhibits periodic oscillations modulated by a decaying power law. The exponent also slightly depends on the parity of N, and is roughly twice smaller than what it would be in the continuum limit. We also consider the same problem, and obtain similar results, in the presence of an impenetrable wall at the origin constraining the particles to remain on the positive half-line. We derive closed-form expressions for the amplitudes of the power-law decay of the return probability in all cases. The key point in the derivation is the use of Mehta integrals, which are limiting cases of the Selberg integral.

Spectral analysis of stable processes on the positive half-line

We study the spectral expansion of the semigroup of a general stable process killed on the first exit from the positive half-line. Starting with the Wiener-Hopf factorization we obtain the q-resolvent density for the killed process, from which we derive the spectral expansion of the semigroup via the inverse Laplace transform. The eigenfunctions and co-eigenfunctions are given rather explicitly in terms of the double sine function and they give rise to a pair of integral transforms which generalize the classical Fourier sine transform. Our results provide the first explicit example of a spectral expansion of the semigroup of a non-symmetric Levy process killed on the first exit from the positive half-line.

A note on the quasi-stationary distribution of the Shiryaev martingale on the positive half-line

В работе получена явная формула для квазистационарного распределения классического мартингального диффузионного процесса Ширяева, рассматриваемого на луче $[A,+\infty)$, где $A>0$ фиксировано; предполагается, что левая граница фазового пространства процесса является поглощающей. Вывод формулы осуществляется аналитически, посредством решения соответствующей вырожденной задачи Штурма-Лиувилля; последняя была впервые рассмотрена в [7, п. 7.8.2], однако оставалась нерешенной до настоящего момента.

A Study of Wavelets on Positive Half Line

A Study of Wavelets on Positive Half Line

Continual Version of the Perron Effect of Change in Values of Characteristic Exponents

A nonlinear perturbed differential system with a linear approximation is considered. An open question has been the validity of a (continual) version of the Perron effect when the set of Lyapunov exponents of all nontrivial solutions (necessarily infinitely extendable on the right) of the corresponding nonlinear perturbed system with a perturbation of arbitrary higher order of smallness in a neighborhood of the origin is measurable, lies entirely on the positive half-line, and has the cardinality of the continuum and even a positive Lebesgue measure. The positive answer to this question is given by the presented theorem, which generally determines an explicit representation of the Lyapunov exponents of all nontrivial solutions to the nonlinear system in terms of their initial values.

Yaglom limit for stable processes in cones

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha$-stable L\'evy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.

Selberg integral theory and Muttalib–Borodin ensembles

We study Muttalib–Borodin ensembles — particular eigenvalue PDFs on the half-line — with classical weights, i.e. Laguerre, Jacobi or Jacobi prime. We show how the theory of the Selberg integral, involving also Jack and Schur polynomials, naturally leads to a multi-parameter generalisation of these particular Muttalib–Borodin ensembles, and also to the explicit form of underlying biorthogonal polynomials of a single variable. A suitable generalisation of the original definition of the Muttalib–Borodin ensemble allows for negative eigenvalues. In the cases of generalised Gaussian, symmetric Jacobi and Cauchy weights, we show that the problem of computing the normalisations and the biorthogonal polynomials can be reduced down to Muttalib–Borodin ensembles with classical weights on the positive half-line.