Asymptotic Behavior Research Articles

Contaminated Gibbs-Type Priors

Gibbs-type priors are combinatorial processes widely used as key components in several Bayesian nonparametric models. By virtue of their flexibility and mathematical tractability, they turn out to be predominant priors in species sampling problems and mixture modeling. We introduce a new family of processes which extends the Gibbs-type one, by including a contaminant component in the model to account for an excess of observations with frequency one. We first investigate the induced random partition, the associated predictive distribution, the asymptotic behavior of the total number of blocks and the number of blocks with a given frequency: all the results we obtain are in closed form and easily interpretable. A remarkable aspect of contaminated Gibbs-type priors relies on their predictive structure, compared to the one of the standard Gibbs-type family: it depends on the additional sampling information on the number of observations with frequency one out of the observed sample. As a noteworthy example we focus on the contaminated version of the Pitman-Yor process, which turns out to be analytically tractable and computationally feasible. Finally we pinpoint the advantage of our construction in different applications: we show how it helps to improve predictive inference in a species-related dataset exhibiting a high number of species with frequency one; we also discuss the use of the proposed construction in mixture models to perform density estimation and outlier detection.

Asymptotics of the sticky particles evolution

We study the long-time asymptotic behavior of the Sticky Particles dynamics on the real line. The time average of the Sticky Particles Lagrangian map has a limit which arises as a general property of projections onto closed convex cones in Hilbert spaces. More notably, we prove that the map itself has an asymptotic limit in the case where the sticky particles dynamics is confined to a compact set.

Discrete-time staged progression epidemic models

ABSTRACT In the Staged Progression (SP) epidemic models, infected individuals are classified into a suitable number of states. The goal of these models is to describe as closely as possible the effect of differences in infectiousness exhibited by individuals going through the different stages. The main objective of this work is to study, from the methodological point of view, the behaviour of solutions of the discrete time SP models without reinfection and with a general incidence function. Besides calculating R 0 , we find bounds for the epidemic final size, characterize the asymptotic behaviour of the infected classes, give results about the final monotonicity of the infected classes, and obtain results regarding the initial dynamics of the prevalence of the disease. Moreover, we incorporate into the model the probability distribution of the number of contacts in order to make the model amenable to study its effect in the dynamics of the disease.

Ginzburg–Landau equation with fractional Laplacian on a upper- right quarter plane

We consider the initial-boundary value problem for the Ginzburg–Landau equation with fractional Laplacian on a upper-right quarter plane 0\\\\ u\\left(0,\\mathbf{x}\\right) = u_{0}\\left(\\mathbf{x}\\right),\\mathbf{x\\in D},\\\\ u\\left\\vert _{x_{i} = 0}\\right. = h_{j}\\left(t,x_{j}\\right),\\ x_{j}>0,\\ j = 1,2,t>0, \\end{array} \\right. $?> {ut(t,x)−∇βu(t,x)=|u(t,x)|σu(t,x),x∈D,t>0u(0,x)=u0(x),x∈D,u|xi=0=hj(t,xj), xj>0, j=1,2,t>0, where D={x1>0,x2>0} , β∈(32,2), σ>0 and ∇β is a fractional Laplacian defined as ∇βu=∑j1Γ(2−β)∫0xjuyjyj(xj−yj)β−1dy. We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the L2− based Sobolev spaces to the case of a multidimensional domain. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.

(Almost) complete characterization of the stability of a discrete-time Hawkes process with inhibition and memory of length two

Abstract We consider a Poisson autoregressive process whose parameters depend on the past of the trajectory. We allow these parameters to take negative values, modelling inhibition. More precisely, the model is the stochastic process $(X_n)_{n\ge0}$ with parameters $a_1,\ldots,a_p \in \mathbb{R}$ , $p\in\mathbb{N}$ , and $\lambda \ge 0$ , such that, for all $n\ge p$ , conditioned on $X_0,\ldots,X_{n-1}$ , $X_n$ is Poisson distributed with parameter $(a_1 X_{n-1} + \cdots + a_p X_{n-p} + \lambda)_+$ . This process can be regarded as a discrete-time Hawkes process with inhibition and a memory of length p. In this paper we initiate the study of necessary and sufficient conditions of stability for these processes, which seems to be a hard problem in general. We consider specifically the case $p = 2$ , for which we are able to classify the asymptotic behavior of the process for the whole range of parameters, except for boundary cases. In particular, we show that the process remains stochastically bounded whenever the solution to the linear recurrence equation $x_n = a_1x_{n-1} + a_2x_{n-2} + \lambda$ remains bounded, but the converse is not true. Furthermore, the criterion for stochastic boundedness is not symmetric in $a_1$ and $a_2$ , in contrast to the case of non-negative parameters, illustrating the complex effects of inhibition.

Dynamics of Hepatitis B Virus Transmission with a Lévy Process and Vaccination Effects

This work proposes a novel stochastic model describing the propagation dynamics of the hepatitis B virus. The model takes into account numerous disease characteristics, and environmental disturbances were collected using Lévy jumps and the conventional Brownian motions. Initially, the deterministic model is developed, and the asymptotic behavior of the model’s solution near the equilibria is examined. The deterministic model is transformed into a stochastic model while retaining the Lévy jumps and conventional Brownian motions. Under specific assumptions, the stochastic system is shown to have a unique solution. The study further investigates the conditions that ensure the extinction and persistence of the infection. The numerical solutions to both stochastic and deterministic systems were obtained using the well-known Milstein and RK4 techniques, and the analytical findings are theoretically confirmed. The simulation suggests that the noise intensities have a direct relationship with the amplitudes of the stochastic curves around the equilibria of the deterministic system. Smaller values of the intensities imply negligible fluctuations of trajectories around the equilibria and, hence, better describe the extinction and persistence of the infection. It has also been found that both Brownian motions and the Lévy jump had a significant influence on the oscillations of these curves. A discussion of the findings of the study reveals other important aspects as well as some future research guidelines. In short, this study proposes a novel stochastic model to describe the propagation dynamics of the hepatitis B virus.

Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results

Abstract In this study, we consider a viscoelastic Shear beam model with no rotary inertia. Specifically, we study ρ 1 φ t t − κ ( φ x + ψ ) x + ( g ∗ φ x x ) ( t ) = 0 , − b ψ x x + κ ( φ x + ψ ) = 0 , \begin{array}{rcl}{\rho }_{1}{\varphi }_{tt}-\kappa {\left({\varphi }_{x}+\psi )}_{x}+\left(g\ast {\varphi }_{xx})\left(t)& =& 0,\\ -b{\psi }_{xx}+\kappa \left({\varphi }_{x}+\psi )& =& 0,\end{array} where the convolution memory function g g belongs to a class of L 1 ( 0 , ∞ ) {L}^{1}\left(0,\infty ) functions that satisfies g ′ ( t ) ≤ − ξ ( t ) ϒ ( g ( t ) ) , ∀ t ≥ 0 , g^{\prime} \left(t)\le -\xi \left(t)\Upsilon \left(g\left(t)),\hspace{1.0em}\forall t\ge 0, where ξ \xi is a positive nonincreasing differentiable function and ϒ \Upsilon is an increasing and convex function near the origin. Using just this general assumptions on the behavior of g g at infinity, we provide optimal and explicit general energy decay rates from which we recover the exponential and polynomial rates when ϒ ( s ) = s p \Upsilon \left(s)={s}^{p} and p p covers the full admissible range [ 1 , 2 ) \left[1,2) . Given this degree of generality, our results improve some of earlier related results in the literature.

Prescribed Asymptotic Behavior of Nonlinear Dynamic Equations Under Impulsive Perturbations

Prescribed Asymptotic Behavior of Nonlinear Dynamic Equations Under Impulsive Perturbations

Existence, uniqueness, and asymptotic behaviors of ground state solutions of Kirchhoff‐type equation with fourth‐order dispersion

In this paper, we focus on the following Schrödinger–Kirchhoff‐type problem with fourth‐order dispersion: where are constants and . We make use of Nehari manifold technique together with concentration‐compactness principle to prove that the above equation has at least a ground state solution for if , 6, and 7, and for if . Moreover, we also investigate the asymptotic behaviors of ground state solutions when some coefficients tend to zero. Among them, a uniqueness result about ground state solutions is obtained by implicit function theorem, and a blow‐up result is established by Pohozaev identity if dimension .

Asymptotic behavior of the overlap gap between infinite words

Asymptotic behavior of the overlap gap between infinite words

An analysis of stability on the L2‐critical inhomogeneous nonlinear Schrödinger equation

We study the inhomogeneous nonlinear Schrödinger equation in the ‐critical case ( ): We establish a detailed analysis of solutions as the initial data approaches the unstable ground state , which satisfies the stationary equation . The results provide information about singularity formation of the blow‐up solutions and about large‐time asymptotic behavior of the global solutions.

Asymptotic behavior of thermoelastic systems of laminated Timoshenko beams with Kelvin-Voigt damping

Asymptotic behavior of thermoelastic systems of laminated Timoshenko beams with Kelvin-Voigt damping

Word Measures on Unitary Groups: Improved Bounds for Small Representations

Abstract Let $F$ be a free group of rank $r$ and fix some $w\in F$. For any compact group $G$ we can define a measure $\mu _{w,G}$ on $G$ by (Haar-)uniformly sampling $g_{1},...,g_{r}\in G$ and evaluating $w(g_{1},...,g_{r})$. In [23], Magee and Puder study the behavior of the moments of $\mu _{w,U(n)}$ as a function of $n$, establishing a connection between their asymptotic behavior and certain algebraic invariants of $w$, such as its commutator length. We employ geometric insights to refine their analysis, and show that the asymptotic behavior of the moments is also governed by the primitivity rank of $w$. Additionally, we also apply our methods to prove a special case of a conjecture of Hanany and Puder [13, Conjecture 1.13] regarding the asymptotic behavior of expected values of irreducible characters of $U(n)$ under $\mu _{w,U(n)}$.

High Perturbations of a Fractional Kirchhoff Equation with Critical Nonlinearities

This paper concerns a fractional Kirchhoff equation with critical nonlinearities and a negative nonlocal term. In the case of high perturbations (large values of α, i.e., the parameter of a subcritical nonlinearity), existence results are obtained by the concentration compactness principle together with the mountain pass theorem and cut-off technique. The multiplicity of solutions are further considered with the help of the symmetric mountain pass theorem. Moreover, the nonexistence and asymptotic behavior of positive solutions are also investigated.

Global Dynamics of a Social Hierarchy-Stratified Malaria Model: Insight from Fractional Calculus

In this study, a mathematical model for the transmission dynamics of malaria among different socioeconomic groups in the human population interacting with a susceptible-infectious vector population is presented and analysed using a fractional-order derivative of the Caputo type. The total human population is stratified into two distinguished classes of lower and higher income individuals, with each class further subdivided into susceptible, infectious, and recovered populations. The socio hierachy-structured fractional-order malaria model is analyzed through the application of different dynamical system tools. The theory of positivity and boundedness based on the generalized mean value theorem is employed to investigate the basic properties of solutions of the model, while the Banach fixed point theory approach is used to prove the existence and uniqueness of the solution. Furthermore, unlike the existing related studies, comprehensive global asymptotic dynamics of the fractional-order malaria model around both disease-free and endemic equilibria are explored by generalizing the usual classical methods for establishing global asymptotic stability of the steady states. The asymptotic behavior of the trajectories of the system are graphically illustrated at different values of the fractional (noninteger) order.

A Stochastic Semi-Parametric SEIR Model with Infectivity in an Incubation Period

This paper introduces stochastic disturbances into a semi-parametric SEIR model with infectivity in an incubation period. The model combines the randomness of disease transmission and the nonlinearity of transmission rate, providing a flexible framework for more accurate description of the process of infectious disease transmission. On the basis of the discussion of the deterministic model, the stochastic semi-parametric SEIR model is studied. Firstly, we use Lyapunov analysis to prove the existence and uniqueness of global positive solutions for the model. Secondly, the conditions for disease extinction are established, and appropriate stochastic Lyapunov functions are constructed to discuss the asymptotic behavior of the model’s solution at the disease-free equilibrium point of the deterministic model. Finally, the specific transmission functions are enumerated, and the accuracy of the results are demonstrated through numerical simulations.

A Dyson Brownian Motion Model for Weak Measurements in Chaotic Quantum Systems

We consider a toy model for the study of monitored dynamics in many-body quantum systems. We study the stochastic Schrödinger equation resulting from continuous monitoring with a rate Γ of a random Hermitian operator, drawn from the Gaussian unitary ensemble (GUE) at every time t. Due to invariance by unitary transformations, the dynamics of the eigenvalues {λα}α=1n of the density matrix decouples from that of the eigenvectors, and is exactly described by stochastic equations that we derive. We consider two regimes: in the presence of an extra dephasing term, which can be generated by imperfect quantum measurements, the density matrix has a stationary distribution, and we show that in the limit of large size n→∞ it matches with the inverse-Marchenko–Pastur distribution. In the case of perfect measurements, instead, purification eventually occurs and we focus on finite-time dynamics. In this case, remarkably, we find an exact solution for the joint probability distribution of λ’s at each time t and for each size n. Two relevant regimes emerge: at short times tΓ=O(1), the spectrum is in a Coulomb gas regime, with a well-defined continuous spectral distribution in the n→∞ limit. In that case, all moments of the density matrix become self-averaging and it is possible to exactly characterize the entanglement spectrum. In the limit of large times tΓ=O(n), one enters instead a regime in which the eigenvalues are exponentially separated log(λα/λβ)=O(Γt/n), but fluctuations ∼O(Γt/n) play an essential role. We are still able to characterize the asymptotic behaviors of the entanglement entropy in this regime.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

Abstract In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra-Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schl"{a}fli integral of the Volterra-Bessel function. In this respect, we study the asymptotic analysis of the Volterra-Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

Theoretico-experimental analysis of bistability in the oscillatory response of a TaOx ReRAM to pulse train stimuli

Fading memory is the capability of a physical system to approach a unique asymptotic behaviour, irrespective of the initial conditions, when stimulated by an input from a certain class. Standard stimuli from the AC periodic class typically induce fading memory effects in non-volatile memristors, as uncovered for the first time back in 2016. Very recently, a deep investigation of resistance switching phenomena in a TaOx resistive random access memory cell revealed the capability of the nano-device to exhibit one of two possible oscillatory behaviours, depending upon the initial condition, when subject to a particular periodic excitation. This interesting finding was, however, left unexplained. Bistability is the simplest form of local fading memory. In a system, endowed with local fading memory under a given stimulus, the initial condition does not affect the long-term behaviour of the state as long as it is drawn from the basin of attraction of either of the distinct coexisting state-space attractors (two limit cycles for the periodically forced memristor acting as a bistable oscillator). Here, the history of the system, encoded in the initial condition, is, thus, erasable only locally through ad hoc stimulation. Motivated by the discovery of local history erase effects in our resistive random access memory cell, this study applies a powerful system-theoretic tool, enabling the analysis of the response of first-order systems to square pulse train-based periodic stimuli, known as the time-average state dynamic route, to an accurate physics-based mathematical model, earlier fitted to the nano-device, to determine a strategy for specifying the parameters of an excitation signal, consisting of the sequence of two square pulses of opposite polarity per period so as to induce various forms of monostability or multistability in the non-volatile memristor. In particular, as an absolute novelty in the literature, experimental measurements validate the theoretical prediction on the capability of the device to operate as one of two distinct oscillators, depending upon the initial condition, under a specific pulse train excitation signal. The coexistence of multiple oscillatory operating modes in the periodically forced resistive random access memory cell, an example par excellence of their unique non-linear dynamics, may inspire the development and circuit implementation of novel sensing and mem-computing paradigms.

Asymptotic behavior of null controllability cost for parabolic equations with vanishing diffusivity under Robin and Neumann boundary conditions

In this paper we study the null-controllability cost of a transport-diffusion system under Robin conditions with distributed control and in which the transport coefficient is a gradient field. First, we provide some conditions on transport coefficient and boundary potential to show that the control cost decays exponentially when the viscosity vanishes and the control time is sufficiently large. On the other hand, if the range of the control region by the transport flow does not cover that of Ω, we prove that the control cost explodes exponentially for the Neumann conditions case with vanishing viscosity and all control time.