One-dimensional Diffusion Process Research Articles

Diffusion-Based Staffing for Multitasking Service Systems with Many Servers

We consider a many-server queue in which each server can serve multiple customers in parallel. Such multitasking phenomena occur in various applications areas (e.g., in hospitals and contact centers), although the impact of the number of customers who are simultaneously served on system efficiency may vary. We establish diffusion limits of the queueing process under the quality-and-efficiency-driven scaling and for different policies of assigning customers to servers depending on the number of customers they serve. We show that for a broad class of routing policies, including routing to the least busy server, the same one-dimensional diffusion process is obtained in the heavy-traffic limit. In case of assignment to the most busy server, there is no state-space collapse, and the diffusion limit involves a custom regulator mapping. Moreover, we also show that assigning customers to the least (most) busy server is optimal when the cumulative service rate per server is concave (convex), motivating the routing policies considered. Finally, we also derive diffusion limits in the nonheavy-traffic scaling regime and in the heavy-traffic scaling regime where customers can be reassigned during service. Funding: The research of J. Storm is partly funded by the Netherlands Organization for Scientific Research (NWO) Gravitation project Networks [Grant 024.002.003].

A candidate method for prediction of the non-stationary response of strongly nonlinear systems under wide-band noise excitation

This paper expands the application of the RBFNN method to the prediction of non-stationary responses in strongly nonlinear systems subjected to wide-band noises (WBNs) excitation. Specifically, the SAM is initially employed to deduce the averaged Fokker–Planck–Kolmogorov (FPK) equation. The non-stationary responses of the energy of the system under WBNs are modeled as a one-dimensional Markov diffusion process. This yields a one-dimensional FPK equation that governs the NSRs of the system subject to WBNs. Subsequently, solving the corresponding FPK equation and obtaining the system’s NSRs are accomplished with the RBFNN method. Furthermore, four indicative cases exposed to WBNs are studied to verify the proposed scheme. Comparisons with simulations demonstrate the accuracy and efficiency of the developed technique.

Stable limit theorems for additive functionals of one-dimensional diffusion processes

Stable limit theorems for additive functionals of one-dimensional diffusion processes

A first-passage-place problem for integrated diffusion processes

AbstractLet ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$ , where Y(t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived.

Dynamic behavior and sliding mode control on a stochastic epidemic model with alertness and distributed delay

In this paper, a stochastic SEIR epidemic model with alertness and distributed delay is studied. The dynamic behavior of the model is analyzed and an integral sliding mode controller is designed. By using Lyapunov function method, sufficient conditions for the existence of unique global positive solution and stationary distribution are obtained. To analyze stochastic stability and stochastic bifurcation through singular boundary theory, we apply stochastic center manifold and stochastic averaging method to transform the model into a one-dimensional Markov diffusion process. The integral sliding mode controller is designed to confine the state trajectories of system to a sufficiently small band around the sliding mode surface, and prevent large outbreak of disease. Numerical simulations verify the correctness of the theoretical analysis and the effectiveness of the controller.

Nonlinear dynamic analysis of a stochastic delay wheelset system

Considering not only the stochastic track irregularity and the possible effect of stochastic parameter excitation, but also the time delay of spring, that is, its response is in place, but the force generated is not in place, a stochastic delay wheelset system is established. The infinite dimensional system is reduced to the finite dimensional stochastic differential equation by using the center manifold, and further reduced to a one-dimensional diffusion process by using the stochastic averaging method. The stability of the wheelset system is obtained by analyzing the singular boundary theory and calculating the maximum Lyapunov exponent. The conditions and types of stochastic bifurcation in wheelset system are obtained by combining probability density function. The numerical simulation verifies the correctness of the theoretical analysis and shows that the time delay affects the critical hunting instability speed of wheelset. The stochastic term affects the lateral displacement of the stochastic delay wheelset system.

Forecast analysis and sliding mode control on a stochastic epidemic model with alertness and vaccination

In this paper, a stochastic SEIR epidemic model is studied with alertness and vaccination. The goal is to stabilize the infectious disease system quickly. The dynamic behavior of the model is analyzed and an integral sliding mode controller with distributed compensation is designed. By using Lyapunov function method, the sufficient conditions for the existence and uniqueness of global positive solutions and the existence of ergodic stationary distributions are obtained. The stochastic center manifold and stochastic average method are used to simplify the system into a one-dimensional Markov diffusion process. The stochastic stability and Hopf bifurcation are analyzed using singular boundary theory. An integral sliding mode controller with non-parallel distributed compensation is designed by linear matrix inequality (LMI) method, which realizes the stability of system and prevents the outbreak of epidemic disease. The correction of theoretical analysis and the effectiveness of controller are validated using numerical simulation performed in MATLAB/Simulink.

General criteria for the study of quasi-stationarity

For Markov processes with absorption, we provide general criteria ensuring the existence and the exponential non-uniform convergence in weighted total variation norm to a quasi-stationary distribution. We also characterize a subset of its domain of attraction by an integrability condition, prove the existence of a right eigenvector for the semigroup of the process and the existence and exponential ergodicity of the Q-process. These results are applied to one-dimensional and multi-dimensional diffusion processes, to pure jump continuous time processes, to reducible processes with several communication classes, to perturbed dynamical systems and discrete time processes evolving in discrete state spaces.

Financial Interpretation of Feller’s Factorization

The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.

On density functions related to discrete time maximum of some one-dimensional diffusion processes

This article focuses on the probability density functions related to the discrete time maximum of some one-dimensional diffusion processes. Firstly, we shall consider solutions of one-dimensional stochastic differential equations and prove an integration by parts formula on the discrete time maximum of the solutions. The smoothness, expressions and upper bounds of the density function will be obtained by the formula. Secondly, Gaussian processes will be dealt with. For some Gaussian processes, we shall obtain asymptotic behaviors of the density functions related to the discrete time maximum of the processes. The Malliavin calculus and Laplace’s method play important roles for the proofs.

On one-dimensional diffusion processes with moving membranes

Journal of Applied Mathematics and Computational Mechanics, Prace Naukowe Instytutu Matematyki i Informatyki, Politechnika Częstochowska, Scientific Research of the Institute of Mathematics and Computer Science, Czestochowa University of Technology

On an Ergodic Two-Sided Singular Control Problem

Motivated by applications in natural resource management, risk management, and finance, this paper is focused on an ergodic two-sided singular control problem for a general one-dimensional diffusion process. The control is given by a bounded variation process. Under some mild conditions, the optimal reward value as well as an optimal control policy are derived by the vanishing discount method. Moreover, the Abelian and Cesàro limits are established. Then a direct solution approach is provided at the end of the paper.

HOW LARGE IS THE JUMP DISCONTINUITY IN THE DIFFUSION COEFFICIENT OF A TIME-HOMOGENEOUS DIFFUSION?

We consider high-frequency observations from a one-dimensional time-homogeneous diffusion process Y. We assume that the diffusion coefficient $\sigma $ is continuously differentiable in y, but with a jump discontinuity at some level y, say $y=0$ . We first study sign-constrained kernel estimators of functions of the left and right limits of $\sigma $ at $0$ . These functions intricately depend on both limits. We propose a method to extricate these functions by searching for bandwidths where the kernel estimators are stable by iteration. We finally provide an estimator of the discontinuity jump size. We prove its convergence in probability and discuss its rate of convergence. A Monte Carlo study shows the finite sample properties of this estimator.

Thermal properties and pyrolysis kinetics of phosphate-rock acid-insoluble residues

In the phosphorous–sulphur two-step process for the clean production of phosphoric acid, a phosphate-rock acid-insoluble residue (PAIR) is a solid filter residue obtained via the phosphoric acid acidolysis of phosphate rock (PR). PAIR combined with other raw materials can be used to prepare cement, ceramics and glasses, opening a potential avenue for large-scale PAIR utilisation. However, the preparation of such materials requires high-temperatures calcination. Understanding the high-temperature thermal properties of PAIR can enable its more targeted comprehensive utilisation or disposal. In this study, the thermal properties and pyrolysis kinetics of PAIR were systematically studied using a multiple heating rate method based on thermogravimetric analysis and a kinetic model. Results showed that from room temperature to 1200 °C, the main changes in the PAIR were the complete removal of fluorine and sulphur, partial removal of phosphorus and conversion of quartz to cristobalite. Moreover, during these processes, H2O(g), NH3, N2, CO2, SO2, P2O5(g), CO, CF3+ and organic gases were volatilised. Herein, the pyrolysis kinetics of PAIR is divided into five stages. Stage 1 (conversion rate ɑ: 0.05–0.2) conforms to the random nucleation and growth as well as the Avrami–Erofeev (n = 2/3) mechanism; the corresponding mechanism function is F(ɑ) = [−Ln(1 − ɑ)]2/3. Stage 2 (ɑ: 0.2–0.4) conforms to the first-order chemical reaction mechanism; the corresponding mechanism function is F(ɑ) = −Ln(1 − ɑ). Stage 3 (ɑ: 0.4–0.6) conforms to the phase boundary–controlled reaction and one-dimensional movement mechanism; the corresponding mechanism function is F(ɑ) = ɑ. Stage 4 (ɑ: 0.6–0.8) conforms to the three-dimensional diffusion process (Jander model); the corresponding mechanism function is F(ɑ) = [1 − (1 − ɑ)1/3]2. Stage 5 (ɑ: 0.6–0.95) conforms to the one-dimensional diffusion process; the corresponding mechanism function is F(ɑ) = ɑ2.

On the nonparametric inference of coefficients of self-exciting jump-diffusion

In this paper, we consider a one-dimensional diffusion process with jumps driven by a Hawkes process. We are interested in the estimations of the volatility function and of the jump function from discrete high-frequency observations in a long time horizon which remained an open question until now. First, we propose to estimate the volatility coefficient. For that, we introduce a truncation function in our estimation procedure that allows us to take into account the jumps of the process and estimate the volatility function on a linear subspace of L2(A) where A is a compact interval of R. We obtain a bound for the empirical risk of the volatility estimator, ensuring its consistency, and then we study an adaptive estimator w.r.t. the regularity. Then, we define an estimator of a sum between the volatility and the jump coefficient modified with the conditional expectation of the intensity of the jumps. We also establish a bound for the empirical risk for the non-adaptive estimators of this sum, the convergence rate up to the regularity of the true function, and an oracle inequality for the final adaptive estimator. Finally, we give a methodology to recover the jump function in some applications. We conduct a simulation study to measure our estimators’ accuracy in practice and discuss the possibility of recovering the jump function from our estimation procedure.

One-dimensional Wiener process with the properties of partial reflection and delay

In this paper, we construct the two-parameter semigroup of operators associated with a certain one-dimensional inhomogeneous diffusion process and study its properties. We are interested in the process on the real line which can be described as follows. At the interior points of the half-lines separated by a point, the position of which depends on the time variable, this process coincides with the Wiener process given there and its behavior on the common boundary of these half-lines is determined by a kind of the conjugation condition of Feller-Wentzell's type. The conjugation condition we consider is local and contains only the first-order derivatives of the unknown function with respect to each of its variables.
 The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding conjugation problem for a second order linear parabolic equation to which the above problem is reduced. Its classical solvability is obtained by the boundary integral equations method under the assumption that the initial function is bounded and continuous on the whole real line, the parameters characterizing the Feller-Wentzell conjugation condition are continuous functions of the time variable, and the curve defining the common boundary of the domains is determined by the function which is continuously differentiable and its derivative satisfies the Hölder condition with exponent less than $1/2$.

Effect of Carbonized 2-Methylnaphthalene on the Hydrogen Storage Performance of MgH2

Mg-based hydride materials (MgH2) are in the spotlight of hydrogen storage due to their high gravimetric density. Yet, its large-scale utilization is limited by the poor thermodynamic stability and slow kinetics. Herein, we report a novel and straightforward way to prepare MgH2 with amorphous carbon by cosintering 2-methylnaphthalene (CMN) organics with pure Mg and a hydriding combustion synthesis method, where the amorphous carbon formed from the CMN not only improves the dehydrogenation/hydrogenation capacity but also enhances the kinetics of the Mg/MgH2 system. The dehydrogenation capacity of the CMN-MgH2 composite reaches 4.88 wt % of H2 at 623 K, nearly 2 times of pure MgH2, and its onset dehydrogenation temperature decreases to 560 K, 90 K lower than that of pure MgH2; in addition, at a lower temperature of 473 K, the composite remarkably absorbs 4.54 wt % of H2 within 42 s while the absorption is only 0.71 wt % H2 for the pure MgH2. Moreover, the activation energy greatly decreases from 165.35 to 101.52 kJ/mol. Further research reveals that the evolution of hydrogenation changes from a three-dimensional diffusion process to a one-dimensional diffusion process, attributed to the formation of the amorphous carbon. This work is expected to provide inspiration to design and prepare effective additives for the improvement of hydrogen storage performance of Mg-based hydrides.

Brownian Motion of Magnetic Skyrmions in One- and Two-Dimensional Systems

Magnetic skyrmions, which are topologically protected particle-like spin textures, exhibit Brownian motion at ambient temperatures in a ferromagnetic thin film. The potential applications of a skyrmions’ diffusive motion to the ultra-low energy stochastic and Brownian computation make research on its thermal dynamics necessary. Herein we experimentally demonstrate the mobility anomaly of the diffusion coefficients in one-dimensional diffusion processes. In addition, we observed a chiral property in the velocity–position correlation functions. The results are discussed in the context of Brownian motion in the shallow potential fluctuations. The data indicate the importance of the chiral property of skyrmions in Brownian motion.

Dynamic analysis of a stochastic toxin-mediated predator-prey model in aquatic environments

This paper considers a stochastic aquatic toxin-mediated predator-prey model. Our main aim is to investigate the effects of environmental toxins and white noise on the population dynamics of this model. First, we prove that there is a global positive solution for any given initial value. Then we show the existence of a unique stationary distribution applying the Lyapunov function method and establish extinction conditions using the invariant density of a one-dimensional diffusion process. In addition, by solving the corresponding Fokker-Plauck equation, we obtain the explicit probability density function around the quasi-stable equilibrium of the transformed stochastic system without pollution. Finally, we use numerical tests to illustrate the main results and reveal some interesting biological phenomena, such as small white noise can reduce the risk of extinction, and the same toxin concentration has different effects on predators and their prey, etc.

Markov chain approximation and measure change for time-inhomogeneous stochastic processes

In this paper, we propose a general time-inhomogeneous continuous-time Markov chain (CTMC) framework for the approximation of the general one-dimensional and two-dimensional time-inhomogeneous diffusion processes. For the approximating CTMC, we can perform a change of measure, choose the minimal relative entropy measure to determine the measure uniquely, and finally establish the convergence. Therefore, the proposed methodology covers the stochastic processes that are hard to perform a change of measure, and is applicable to valuation problems driven by models not only under the risk-neutral probability measure but also under the physical probability measure.