Theory Of Diffusion Processes Research Articles

Diffusion kinetics in a multicomponent thermodynamic system at small deviations from the equilibrium state

The theory of diffusion processes in solids has achieved significant results in recent decades, but the development of methods for calculating diffusion in a multicomponent thermodynamic system is still an urgent task. Problems of diffusion in solid and liquid solutions with small deviations from the equilibrium state, or fluctuations, are of significant interest. The work develops a general methodology for calculating diffusion flows in a multicomponent thermodynamic system for small deviations from the equilibrium state. A connection has been established between the mechanical approach to the analysis of generalized systems and the phenomenological equations of nonequilibrium thermodynamics. Examples are given of the use of the developed methodology for the analysis of carbide transformations in chromium steel.

A stochastic log-logistic diffusion process: Statistical computational aspects and application to real data

This article introduces a new stochastic diffusion process based on the theory of diffusion processes whose mean function is proportional to the log-logistic growth curve. The main characteristics of the process are analyzed, including the transition probability density function, the mean functions and in particular, the auto-correlation function between two times of the process. The parameters of the process are estimated by maximum likelihood method using discrete sampling. The simulated annealing algorithm is applied after bounding the parametric space by a strategy procedure to solve the likelihood equations. The behavior of the diffusion process here derived is finally applied to study an example for the growth data of a microorganism culture.

Эллиптические уравнения со сдвигами произвольных направлений в полупространстве

In this paper, we investigate the half-space Dirichlet problem for elliptic differential-difference equations with superpositions of differential operators and translation operators acting in arbitrary directions parallel to the boundary hyperplane. The summability assumption is imposed on the boundary-value function of the problem. The specified equations, substantially generalizing classical elliptic partial differential equations, arise in various models of mathematical physics with nonlocal and (or) heterogeneous properties or the process or medium: multi-layer plates and envelopes theory, theory of diffusion processes, biomathematical applications, models of nonlinear optics, etc. The theoretical interest to such equations is caused by their nonlocal nature: they connect values of the desired function (and its derivatives) at different points (instead of the same one), which makes many classical methods unapplicable. For the considered problem, we establish the solvability in the sense of generalized functions, construct Poisson-like integral representations of solutions, and prove the infinite smoothness of the solution outside the boundary hyperplane and its uniform convergence to zero (together with all its derivatives) as the timelike variable tends to infinity. We find a power estimate of the velocity of the specified extinction of the solution and each its derivative.

Response of Quasi-Integrable and Resonant Hamiltonian Systems to Fractional Gaussian Noise

Abstract The major difficulty in studying the response of multi-degrees-of-freedom (MDOF) nonlinear dynamical systems driven by fractional Gaussian noise (fGn) is that the system response is not Markov process diffusion and thus the diffusion process theory cannot be applied. Although the stochastic averaging method (SAM) for quasi Hamiltonian systems driven by fGn has been developed, the response of the averaged systems still needs to be predicted by using Monte Carlo simulation. Later, noticing that fGn has rather flat power spectral density (PSD) in certain frequency band, the SAM for MDOF quasi-integrable and nonresonant Hamiltonian system driven by wideband random process has been applied to investigate the response of quasi-integrable and nonresonant Hamiltonian systems driven by fGn. The analytical solution for the response of an example was obtained and well agrees with Monte Carlo simulation. In the present paper, the SAM for quasi-integrable and resonant Hamiltonian systems is applied to investigate the response of quasi-integrable and resonant Hamiltonian system driven by fGn. Due to the resonance, the theoretical procedure and calculation will be more complicated than the nonresonant case. For an example, some analytical solutions for stationary probability density function (PDF) and stationary statistics are obtained. The Monte Carlo simulation results of original system confirmed the effectiveness of the analytical solutions under certain condition.

Disruptive innovation from the perspective of innovation diffusion theory

ABSTRACT Since it was first proposed by Christensen, disruptive innovation theory has provoked considerable debate in academia and industry. Initially limited to low-end innovations, most of the controversy around the theory has focused on other types of innovations that the theory excludes. This article examines both high-end and low-end disruptive innovations. Instead of technological development, the focus of analysis is on market groups, known as adopter categories. Adopter groups have various characteristics and sizes that play a critical role in the diffusion process of disruptive innovation. In addition, high-end and low-end should be very different in terms of relative advantage, one of the innovation attributes influencing the rate of adoption. To successfully ‘disrupt’ an incumbent, an innovation should have the potential to cross a critical chasm between early niche markets and majority mainstream markets. We conducted case studies on the rigid disk drives and the computer industry to illustrate the analytical approach based on the innovation diffusion process theory and the characteristics of various adopter categories.

Stochastic stability and Hopf bifurcation analysis of a singular bio-economic model with stochastic fluctuations

In this paper, we study a class of singular stochastic bio-economic models described by differential-algebraic equations due to the influence of economic factors. Simplifying the model through a stochastic averaging method, we obtained a two-dimensional diffusion process of averaged amplitude and phase. Stochastic stability and Hopf bifurcations can be analytically determined based on the singular boundary theory of diffusion process, the Maximal Lyapunov exponent and the invariant measure theory. The critical value of the stochastic Hopf bifurcation parameter is obtained and the position of Hopf bifurcation drifting with the parameter increase is presented as a result. Practical example is presented to verify the effectiveness of the results.

Stochastic Stability of a Planner Gyropendulum System with Synchronous Motor Driven by Gaussian White Noises

In this paper, the stochastic moment stability and almost-sure stability of a planner gyropendulum system with synchronous motor under the white noises are investigated. By applying the theory of diffusion process, an eigenvalue problem for the moment Lyapunov exponent is formulated. Then, through a perturbation method and a Fourier cosine series expansion, the second-order expansion of the moment Lyapunov exponent is solved, which is just the leading eigenvalue of an infinite matrix. Finally, the convergence and validity of the procedure are numerically verified, and the effects of system and noise parameters on the moment Lyapunov exponent are discussed. It was found that the increase in both the noise intensity and coefficient of the synchronous motor torque will weaken the stability of the gyropendulum system, and when they reach certain values, the system becomes unstable. In addition, according to the relationship between the moment Lyapunov exponent and maximal Lyapunov exponent, the stable thresholds are also given.

Reliability of electrostatically actuated MEMS resonators to random mass disturbance

Abstract Random variable-mass systems have received widespread attention and shown prominent significance with the explosive development of micro- and nanotechnologies, so there is a growing need to study the influences of mass disturbance on the systems. This paper is devoted to investigating the reliability (first passage failure) of electrostatically actuated MEMS resonators subject to the random mass disturbance. The potential failure mode considered is the adhesion failure. First, the original system is approximately replaced by a system without mass disturbance based on an equivalent transformation. Then the first passage failure indicators of the approximate system are derived by using the combination approach of the stochastic averaging and the diffusion process theory of the first passage time. The comparisons between the analytical results and those from numerical simulation show that the proposed method works very well. The effects of mass disturbance on the system reliability are explored adequately. It turns out that the presence of mass disturbance can lead to a deterioration of the resonators’ reliability severely. It is remarkable that the system has multiple resonance regions with different intensities, and the reliability of the system in the case of primary resonance is much lower than that in the non-resonant case.

Modeling and Dynamic Analysis in a Hybrid Stochastic Bioeconomic System with Double Time Delays and Lévy Jumps

A double delayed hybrid stochastic prey-predator bioeconomic system with Lévy jumps is established and analyzed, where commercial harvesting on prey and environmental stochasticity on population dynamics are considered. Two discrete time delays are utilized to represent the maturation delay of prey and gestation delay of predator, respectively. For a deterministic system, positivity of solutions and uniform persistence of system are discussed. Some sufficient conditions associated with double time delays are derived to discuss asymptotic stability of interior equilibrium. For a stochastic system, existence and uniqueness of a global positive solution are studied. By using the invariant measure theory and singular boundary theory of diffusion process, existence of stochastic Hopf bifurcation and stochastic stability are investigated. By constructing appropriate Lyapunov functions, asymptotic dynamic behavior of the proposed hybrid stochastic system with double time delays and Lévy jumps is discussed. Numerical simulations are provided to show consistency with theoretical analysis.

Stochastic response analysis for nonlinear vibration systems with adjustable stiffness property under random excitation.

Nonlinear vibration systems with adjustable stiffness property have attracted considerable attentions for their prominent broadband performances. In the present manuscript, we consider the stochastic dynamical systems with adjustable stiffness and proposed a numerical method for the random responses analysis of the Gaussian white noise excited systems. A multi-dimensional Fokker-Plank-Kolmogorov equation governing the joint probability density of the mechanical states is derived according to the theory of diffusion processes. We solve the multi-dimensional equation using a splitting method and obtained the stationary probability densities and the mean-square responses directly. Two classical nonlinear vibration systems with adjustable stiffness, including the energy harvesting system and the Duffing system with Dahl friction, are presented as examples. Their comparisons with the results from Monte-Carlo simulations illustrate the effectiveness of the proposed procedure for both monostable and bistable cases, even for cases with strong excitation. In addition, the splitting method is efficient for higher-dimensional problem and has advantages of simple implementation, less storage of intermediate values and so on. Hence, in terms of the application scope, the proposed procedure is superior to the current mainstream methods for the random response evaluation of nonlinear vibration systems with adjustable stiffness.

The Hubbert diffusion process: Estimation via simulated annealing and variable neighborhood search procedures—application to forecasting peak oil production

Accurately charting the progress of oil production is a problem of great current interest. Oil production is widely known to be cyclical: in any given system, after it reaches its peak, a decline will begin. With this in mind, Marion King Hubbert developed his peak theory in 1956 based on the bell‐shaped curve that bears his name. In the present work, we consider a stochastic model based on the theory of diffusion processes and associated with the Hubbert curve. The problem of the maximum likelihood estimation of the parameters for this process is also considered. Since a complex system of equations appears, with a solution that cannot be guaranteed by classical numerical procedures, we suggest the use of metaheuristic optimization algorithms such as simulated annealing and variable neighborhood search. Some strategies are suggested for bounding the space of solutions, and a description is provided for the application of the algorithms selected. In the case of the variable neighborhood search algorithm, a hybrid method is proposed in which it is combined with simulated annealing. In order to validate the theory developed here, we also carry out some studies based on simulated data and consider 2 real crude oil production scenarios from Norway and Kazakhstan.

The spectral expansion approach to index transforms and connections with the theory of diffusion processes

Many important index transforms can be constructed via the spectral theory of Sturm-Liouville differential operators. Using the spectral expansion method, we investigate the general connection between the index transforms and the associated parabolic partial differential equations. We show that the notion of Yor integral, recently introduced by the second author, can be extended to the class of Sturm-Liouville integral transforms. We furthermore show that, by means of the Feynman-Kac theorem, index transforms can be used for studying Markovian diffusion processes. This gives rise to new applications of index transforms to problems in mathematical finance.

Optimal Dividend and Issuance of Equity Policies in the Presence of Interest

We investigate the dividend and equity issuance problem in the presence of interest rate. The evolution of the financial reserves of an insurance company, where management payout dividends and issue new equity, is described by a stochastic differential equation. The work of Lokka and Zervos [1] is extended by including the interest rate component into the model in order to make the model more realistic. The aim is to maximise the expected discounted dividends pay-out until the time of bankruptcy. In order to investigate this problem, the stochastic control theory for diffusion processes will be used. In order to handle the problem, the Hamilton-Jacobi-Bellman equation (HJB) is derived and solved. The second order ordinary differential equation associated with the problem turns out to belong to the class of Kummer’s confluent hyper-geometric differential equations. This category of equations is not easy to solve. The equation is non-dimensionalised and change of variables is effected in two different stages. The results show that interest rate affected the rate at which the value function and threshold level change.

Dynamic analysis of a hybrid bioeconomic plankton system with double time delays and stochastic fluctuations

In this paper, we establish a double delayed hybrid bioeconomic plankton system with stochastic fluctuations and commercial harvesting on zooplankton, where maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Stochastic fluctuations are incorporated into the proposed system in form of Gaussian white noises to depict stochastic environmental factors in plankton system. For deterministic system without double time delays, existence of singularity induced bifurcation is studied due to variation of economic interest of commercial harvesting, and state feedback controllers are designed to eliminate singularity induced bifurcation. For deterministic system with double time delays, positivity and uniform persistence of solutions are studied, and some sufficient conditions associated with asymptotic stability of interior equilibrium are investigated. For stochastic system without double time delays, stochastic stability and existence of stochastic Hopf bifurcation are discussed based on singular boundary theory of diffusion process and invariant measure theory. For stochastic system with double time delays, existence and uniqueness of global positive solution are investigated, and asymptotic behaviors of the interior equilibrium are studied by constructing appropriate Lyapunov functions. Numerical simulations are carried out to validate theoretical analysis.

Dynamical analysis in a bioeconomic phytoplankton zooplankton system with double time delays and environmental stochasticity

This paper presents a double delayed bioeconomic phytoplankton zooplankton system with commercial harvesting on zooplankton and environmental stochasticity. Maturation delay for toxin producing phytoplankton and gestation delay for zooplankton are considered. Environmental stochasticity is incorporated into the proposed system in form of Gaussian white noises. Some sufficient conditions are derived to show that the proposed system has a unique global positive solution. In absence of double time delays, stochastic stability and existence of stochastic Hopf bifurcation are studied based on invariant measure theory and singular boundary theory of diffusion process for the proposed system. In presence of double time delays, asymptotic behaviors of the interior equilibrium are discussed by constructing some appropriate Lyapunov functions.

Плотности вероятностей процессов процентных ставок доходности

The processes of short-term interest rates rise to many changes in market indices, as well as form the basis of determining the value of market assets and commercial contracts. A special role they play in calculating the term structure of the yield. Therefore, the development of mathematical models of these processes is extremely interesting for financial analysts and market research issues. There are many versions of change of short-term risk-free interest rates in the framework of the theory of diffusion processes. However, there is still no such a model, which would be the basis for building a term structure of yields close to existing on a real financial market. It is interesting to analyze the existing models in order to clarify their features in a probabilistic sense, in more detail than has been done by their creators and users. To this end, the paper examines the marginal probability density of the diffusion processes generated by sixteen models of short-term interest rates, that allow obtain densities in an analytical form. Here will be made such analysis for the family of models used by the authors of papers that are widely known in their fit to the actual time series of yield. All considered models belong to the class of diffusion that generate processes, where a specific setting of drift and volatility defines one or another particular model. Some models, such as the Vasicek model, Cox – Ingersoll – Ross, geometric Brownian motion, Ahn – Gao, are well documented in the literature, but nevertheless their properties are listed here for convenience of comparison with other, less well-known or unstudied models. Other densities are described for the first time. The proposed analysis will be useful to the reader to determine the most appropriate models of short-term rates in the determination of the term structure of zero-coupon yield approximating actually observed, as far as possible, by the best way. Analysis scheme reduces to solution of the forward Kolmogorov equation for the stationary probability density and, if necessary, discuss its features and the first four moments are calculated, usually of interest in practice. It is shown that for the models the coefficients of skewness and kurtosis, defined by moments third and fourth order, depend on a single parameter, that called as the form parameter density, which, in turn, is determined only by the ratio of the variance to the square of the expectation (this corresponds to the square of the so-called coefficient of variation).

The diffusion dynamics of choice: From durable goods markets to fashion first names

Goods, styles, ideologies are adopted by society through various mechanisms. In particular, adoption driven by innovation is extensively studied by marketing economics. Mathematical models are currently used to forecast sales of innovative goods. Inspired by the theory of diffusion processes developed for marketing economics, we propose a modelling framework for the mechanism of fashion, which we apply to first names. Analyses of French, Dutch, and US national databases validate our modelling approach for thousands of first names, covering, on average, more than 50% of the yearly incidence in each database. In these cases, it is thus possible to study how first names become popular and when they run out of fashion. Furthermore, we uncover a clear distinction between popularity and fashion: less popular names, typically not included in studies of fashion, may be driven by fashion, as well. © 2016 The Authors Complexity Published byWiley Periodicals, Inc. Complexity 21: 362–369, 2016

Blow-up and nonexistence of solutions of some semilinear degenerate parabolic equations

In this paper we study a class of semilinear degenerate parabolic equations arising in mathematical finance and in the theory of diffusion processes. We show that blow-up of spatial derivatives of smooth solutions in finite time occurs to initial boundary value problems for a class of degenerate parabolic equations. Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function. Finally, the phenomenon of blow-up is verified by a numerical experiment.

BEST statistics of Markovian fluxes: a tale of Eulerian tours and Fermionic ghosts

We provide an exact expression for the statistics of the fluxes of Markov jump processes at all times, improving on asymptotic results from large deviation theory. The main ingredient is a generalization of the BEST theorem in enumeratoric graph theory to Eulerian tours with open ends. In the long-time limit we reobtain Sanov’s theorem for Markov processes, which expresses the exponential suppression of fluctuations in terms of relative entropy. The finite-time power-law term, increasingly important with the system size, is a spanning-tree determinant that, by introducing Grassmann variables, can be absorbed into the effective Lagrangian of a Fermionic ghost field on a metric space, coupled to a gauge potential. With reference to concepts in nonequilibrium stochastic thermodynamics, the metric is related to the dynamical activity that measures net communication between states, and the connection is made to a previous gauge theory for diffusion processes.

On a Theme by Heckscher-Ohlin: A Diffusion Model for Spatial Dynamics in Factor Prices

An analytical study is presented for the cross-sectional distribution of factor prices over time and across space. A drift-diffusion model is proposed to describe the dynamic process governing the fluctuations around the equilibrium distribution. The model is mechanical and descriptive in nature, and illustrates that the growth distribution of factor prices can be generated by a single stochastic process that builds upon the theory of diffusion processes. An empirical application of the proposed model, to the evolution of the distribution of incomes for 186 countries, recorded from 1993 up to 2007, illustrates the applicability of the proposed method and suggests that diffusion may be a preferable technique for the analysis of the spatial dynamics in factor prices.