Positive Half-line Research Articles

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line

We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line in the limit as the number of particles goes to infinity. We establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles, which is a weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We show that this density process has good L2-regularity properties in the interior of the domain but may exhibit singularities on the boundary at a dense set of times. We make connections to the corresponding absorbing and reflecting SPDEs as the elastic parameter varies.

Hamiltonian for the Hilbert–Pólya conjecture

We introduce a Hamiltonian to address the Hilbert–Pólya conjecture. The eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary conditions on the positive half-line, vanish at the origin by the nontrivial zeros of the Riemann zeta function. Consequently, the eigenvalues are determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is true, the eigenvalues become real and represent the imaginary parts of the nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more generally, admits only real eigenvalues, then the RH follows. In our attempt to demonstrate the latter, we establish the existence of a similarity transformation of the introduced Hamiltonian that is self-adjoint on the domain specified by an appropriate boundary condition, which is satisfied by the eigenfunctions through the vanishing of the Riemann zeta function. Our result can be extended to a broader class of functions whose zeros lie on the critical line.

Reflected stochastic differential equations driven by standard and fractional Brownian motion

The reflection problem on the positive half-line with reflection at zero for a time-dependent stochastic differential equations driven by standard and fractional Brownian motion with Hurst parameter [Formula: see text] is considered. We prove the existence of weak solutions by using Euler scheme. Moreover, we show that pathwise uniqueness holds and a strong solution exists in the case of additive fractional noise and also up to a stopping time [Formula: see text] for the multiplicative case, but remains an open question beyond [Formula: see text].

Occupation time of a system of Brownian particles on the line with steplike initial condition.

We consider a system of noninteracting Brownian particles on the line with steplike initial condition and study the statistics of the occupation time on the positive half-line. We demonstrate that even at large times, the behavior of the occupation time exhibits long-lasting memory effects of the initialization. Specifically, we calculate the mean and the variance of the occupation time, demonstrating that the memory effects in the variance are determined by a generalized compressibility (or Fano factor), associated with the initial condition. In the particular case of the uncorrelated uniform initial condition we conduct a detailed study of two probability distributions of the occupation time: annealed (averaged over all possible initial configurations) and quenched (for a typical configuration). We show that at large times both the annealed and the quenched distributions admit large deviation form and we compute analytically the associated rate functions. We verify our analytical predictions via numerical simulations using importance sampling Monte Carlo strategy.

The Jacobi-Sobolev, Laguerre-Sobolev, and Gegenbauer-Sobolev differential equations and their interrelations

Recently, the author determined the higher-order differential operator having the Jacobi-Sobolev polynomials as its eigenfunctions for certain eigenvalues. These polynomials form an orthogonal system with respect to an inner product equipped with the Jacobi measure on the interval [ − 1 , 1 ] with parameters α ∈ N 0 , β > − 1 and two point masses N, S>0 at the right end point of the interval involving functions and their first derivatives. The first purpose of the present paper is to reveal how the Jacobi-Sobolev equation reduces to the differential equation satisfied by the Laguerre-Sobolev polynomials on the positive half line via a confluent limiting process as β → ∞ . Secondly, we explicitly establish the differential equation for the symmetric Gegenbauer-Sobolev polynomials by employing a quadratic transformation of the argument. Each of the three differential operators involved is of order 4 α + 10 and symmetric with respect to the corresponding Sobolev inner product.

The f↔f˜ Correspondence and Its Applications in Quantum Information Geometry.

Due to the classifying theorems by Petz and Kubo-Ando, we know that there are bijective correspondences between Quantum Fisher Information(s), operator means, and the class of symmetric, normalized operator monotone functions on the positive half line; this last class is usually denoted as Fop. This class of operator monotone function has a significant structure, which is worthy of study; indeed, any step in understanding Fop, besides being interesting per se, immediately translates into a property of the classes of operator means and therefore of Quantum Fisher Information(s). In recent years, the f↔f correspondence has been introduced, which associates a non-regular element of Fop to any regular element of the same set. In terms of operator means, this amounts to associating a mean with multiplicative character to a mean that has an additive character. In this paper, we survey a number of different settings where this technique has proven useful in Quantum Information Geometry. In Sections 1-4, all the needed background is provided. In Sections 5-14, we describe the main applications of the f↔f˜ correspondence.

Maximum Entropy Criterion for Moment Indeterminacy of Probability Densities.

We deal with absolutely continuous probability distributions with finite all-positive integer-order moments. It is well known that any such distribution is either uniquely determined by its moments (M-determinate), or it is non-unique (M-indeterminate). In this paper, we follow the maximum entropy approach and establish a new criterion for the M-indeterminacy of distributions on the positive half-line (Stieltjes case). Useful corollaries are derived for M-indeterminate distributions on the whole real line (Hamburger case). We show how the maximum entropy is related to the symmetry property and the M-indeterminacy.

Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line

Cutpoints of (1,2) and (2,1) Random Walks on the Lattice of Positive Half Line

Tamed stability of finite difference schemes for the transport equation on the half-line

In this paper, we prove that, under precise spectral assumptions, some finite difference approximations of scalar leftgoing transport equations on the positive half-line with numerical boundary conditions are ℓ 1 \ell ^1 -stable but ℓ q \ell ^q -unstable for any q > 1 q>1 . The proof relies on the accurate description of the Green’s function for a particular family of finite rank perturbations of Toeplitz operators whose essential spectrum belongs to the closed unit disk and with a simple eigenvalue of modulus 1 1 embedded into the essential spectrum.

Asymptotics of noncolliding q-exchangeable random walks

We consider a process of noncolliding q-exchangeable random walks on making steps 0 (‘straight’) and −1 (‘down’). A single random walk is called q-exchangeable if under an elementary transposition of the neighboring steps the probability of the trajectory is multiplied by a parameter . Our process of m noncolliding q-exchangeable random walks is obtained from the independent q-exchangeable walks via the Doob’s h-transform for a nonnegative eigenfunction h (expressed via the q-Vandermonde product) with the eigenvalue less than 1. The system of m walks evolves in the presence of an absorbing wall at 0. The repulsion mechanism is the q-analogue of the Coulomb repulsion of random matrix eigenvalues undergoing Dyson Brownian motion. However, in our model, the particles are confined to the positive half-line and do not spread as Brownian motions or simple random walks. We show that the trajectory of the noncolliding q-exchangeable walks started from an arbitrary initial configuration forms a determinantal point process, and express its kernel in a double contour integral form. This kernel is obtained as a limit from the correlation kernel of q-distributed random lozenge tilings of sawtooth polygons. In the limit as , with γ > 0 fixed, and under a suitable scaling of the initial data, we obtain a limit shape of our noncolliding walks and also show that their local statistics are governed by the incomplete beta kernel. The latter is a distinguished translation invariant ergodic extension of the two-dimensional discrete sine kernel.

The damped vibrating string equation on the positive half-line

In this paper, the existence of a solution to the problem describing the small vertical vibration of an elastic string on the positive half-line is investigated in the case when both viscous and material damping coefficients are present. The result is obtained by transforming the original partial differential equation into an appropriate abstract second-order ordinary differential equation in a suitable infinite dimensional space. The abstract problem is then studied using the combination of the Kakutani fixed point theorem together with the approximation solvability method and the weak topology. The applied procedure enables obtaining the existence result also for problems depending on the first derivative, without any strict compactness assumptions put on the right-hand side and on the fundamental system generated by the linear term. The paper ends by applying the obtained result to the studied mathematical model describing the small vertical vibration of an elastic string with a nonlinear Balakrishnan–Taylor-type damping term.

Exponentially decaying solutions for models with delayed and advanced arguments: Nonlinear effects in linear differential equations

We consider a scalar linear mixed differential equation with several terms, both delayed and advanced arguments and a bounded right-hand side. Assuming that the deviations of the argument are bounded, we present sufficient conditions when there exists a unique bounded solution on the positive half-line. Explicit tests for a bounded solution of a homogeneous equation to decay exponentially are obtained. Existence of exponentially decaying solutions for this class of differential equations is studied for the first time, and we illustrate sharpness of the results with examples. We show that the standard approach when convergence of all solutions is stated does not work for mixed equations; in addition to an exponentially decaying, there may be a growing solution. All the coefficients and the mixed arguments are assumed to be Lebesgue measurable functions, not necessarily continuous. Though the equation is linear, some properties, as well as the methods applied, are more typical for nonlinear models, for example, fixed-point theorems are used in the proofs.

Periodic Gabor frames on positive half line

In this paper, we introduce the concept of periodic Gabor frames on positive half line. Firstly, we establish a necessary and sufficient condition for a periodic Gabor system to be a Gabor frame. Then, we present some equivalent characterizations of Parseval Gabor frames on positive half line by means of some fundamental equations in time domain.

Effusion of stochastic processes on a line

We consider the problem of leakage or effusion of an ensemble of independent stochastic processes from a region where they are initially randomly distributed. The case of Brownian motion, initially confined to the left half line with uniform density and leaking into the positive half line is an example which has been extensively studied in the literature. Here we derive new results for the average number and variance of the number of leaked particles for arbitrary Gaussian processes initially confined to the negative half line and also derive its joint two-time probability distribution, both for the annealed and the quenched initial conditions. For the annealed case, we show that the two-time joint distribution is a bivariate Poisson distribution. We also discuss the role of correlations in the initial particle positions on the statistics of the number of particles on the positive half line. We show that the strong memory effects in the variance of the particle number on the positive real axis for Brownian particles, seen in recent studies, persist for arbitrary Gaussian processes and also at the level of two-time correlation functions.

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

Stochastic kinetics under combined action of two noise sources.

We are exploring two archetypal noise-induced escape scenarios: Escape from a finite interval and from the positive half-line under the action of the mixture of Lévy and Gaussian white noises in the overdamped regime, for the random acceleration process and higher-order processes. In the case of escape from finite intervals, the mixture of noises can result in the change of value of the mean first passage time in comparison to the action of each noise separately. At the same time, for the random acceleration process on the (positive) half-line, over the wide range of parameters, the exponent characterizing the power-law decay of the survival probability is equal to the one characterizing the decay of the survival probability under action of the (pure) Lévy noise. There is a transient region, the width of which increases with stability index α, when the exponent decreases from the one for Lévy noise to the one corresponding to the Gaussian white noise driving.

Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference Kernel and with Convex Nonlinearity on the Positive Half-Line

Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference Kernel and with Convex Nonlinearity on the Positive Half-Line

Conditioned limit theorems for hyperbolic dynamical systems

AbstractLet $({\mathbb X}, T)$ be a subshift of finite type equipped with the Gibbs measure $\nu $ and let f be a real-valued Hölder continuous function on ${\mathbb X}$ such that $\nu (f) = 0$ . Consider the Birkhoff sums $S_n f = \sum _{k=0}^{n-1} f \circ T^{k}$ , $n\geqslant 1$ . For any $t \in {\mathbb R}$ , denote by $\tau _t^f$ the first time when the sum $t+ S_n f$ leaves the positive half-line for some $n\geqslant 1$ . By analogy with the case of random walks with independent and identically distributed increments, we study the asymptotic as $ n\to \infty $ of the probabilities $ \nu (x\in {\mathbb X}: \tau _t^f(x)>n) $ and $ {\nu (x\in {\mathbb X}: \tau _t^f(x)=n) }$ . We also establish integral and local-type limit theorems for the sum $t+ S_n f(x)$ conditioned on the set $\{ x \in {\mathbb X}: \tau _t^f(x)>n \}.$

Uniqueness of the Solution of a Class of Integral Equations with Sum-Difference. Kernel and with Convex Nonlinearity on the Positive Half-Line

Работа посвящена изучению вопроса единственности и исследованию некоторых качественных свойств решения одного класса интегральных уравнений с суммарно-разностным ядром на положительной полупрямой и с выпуклой вниз нелинейностью. Данный класс уравнений в частном случае возникает в динамической теории $p$-адических открыто-замкнутых струн для скалярного поля тахионов. Такие уравнения играют весьма важную роль также при исследовании вопросов существования и единственности решения нелинейных интегральных уравнений в математической теории географического распространения эпидемии в рамках модели Дикмана-Капера. В настоящей работе доказана теорема единственности решения рассматриваемого уравнения в классе неотрицательных (ненулевых) и ограниченных на $\mathbb{R}^+$ функций, тем самым окончательно решена открытая проблема В. С. Владимирова о единственности роллинговых решений нелинейных $p$-адических уравнений. При одном дополнительном ограничении на ядро уравнения доказано также, что решение представляет собой выпуклую вверх функцию на множестве $[0,+\infty)$, производная которой принадлежит пространству $L_1(0,+\infty)$. В конце работы приведены конкретные модельные уравнения из указанных выше приложений, к которым применены полученные результаты. Библиография: 21 название.

The Rate of Accumulation of Negative Eigenvalues to Zero and the Absolutely Continuous Spectrum

For a bounded real-valued function V on ℝd, we consider two Schrödinger operators H+ = −Δ+V and H− = −Δ − V . We prove that if the negative spectra H+ and H−are discrete and the negative eigenvalues of H+ and H− tend to zero sufficiently fast, then the absolutely continuous spectra cover the positive half-line [0,∞).