Stochastic Integro-differential Equations Research Articles

Anomalous Diffusion and the Generalized Langevin Equation

The generalized Langevin equation (GLE) is a stochastic integro-differential equation that is commonly used to describe the velocity of microparticles that move randomly in viscoelastic fluids. Suc...

Exponential stability for neutral stochastic partial integro-differential equations of second order with poisson jumps

This paper studies the existence, uniqueness and the exponential stability in p-th moment of the mild solution of neutral second order stochastic partial differential equations with infinite delay and Poisson jumps. The existence and uniqueness of the mild solution of neutral second order stochastic differential equation is first established by means of Banach fixed point principle and stochastic analysis. The exponential stability in the p-th moment for the mild solution to impulsive neutral stochastic integrodifferential equations with Poisson jump is obtained by establishing an integral inequality.

Reconstructing Complex Fluid Properties from the Behavior of Fluctuating Immersed Particles

Complex fluids have long been characterized by two functions that summarize the fluid's elastic and viscous properties, respectively called the storage ($G'(\omega)$) and loss ($G''(\omega)$) moduli. A fundamental observation in this field, which is called passive microrheology, is that information about these bulk fluid properties can be inferred from the path statistics of immersed, fluctuating microparticles. In this work, we perform a systematic study of the multistep protocol that forms the foundation of this field. Particle velocities are assumed to be well described by the Generalized Langevin Equation, a stochastic integro-differential equation uniquely characterized by a memory kernel $G_r(t)$, which is hypothesized to be inherited from the surrounding fluid. We investigate the covariation between a particle's velocity process and the non-Markovian fluctuations that force it, and we establish rigorous justification for a key relationship between a particle's Mean Squared Displacement and its memo...

Symbolic and numeric scheme for solution of linear integro-differential equations with random parameter uncertainties and Gaussian stochastic process input

• A system of linear stochastic integro-differential equations is under consideration. • Gaussian nonstationary noises and random model uncertainties excite the system. • The problem is to find the mean and covariance functions of the state vector process. • Green matrix-valued functions are used to solve the problem. • As example, a response of a thermoviscoelastic system is estimated by our scheme. The paper describes a theoretical apparatus and an algorithmic part of application of the Green matrix-valued functions for time-domain analysis of systems of linear stochastic integro-differential equations. It is suggested that these systems are subjected to Gaussian nonstationary stochastic noises in the presence of model parameter uncertainties that are described in the framework of the probability theory. If the uncertain model parameter is fixed to a given value, then a time-history of the system will be fully represented by a second-order Gaussian vector stochastic process whose properties are completely defined by its conditional vector-valued mean function and matrix-valued covariance function. The scheme that is proposed is constituted of a combination of two subschemes. The first one explicitly defines closed relations for symbolic and numeric computations of the conditional mean and covariance functions, and the second one calculates unconditional characteristics by the Monte Carlo method. A full scheme realized on the base of Wolfram Mathematica and Intel Fortran software programs, is demonstrated by an example devoted to an estimation of a nonstationary stochastic response of a mechanical system with a thermoviscoelastic component. Results obtained by using the proposed scheme are compared with a reference solution constructed by using a direct Monte Carlo simulation.

Solution of the finite Milne problem in stochastic media with RVT Technique

This paper presents the solution to the Milne problem in the steady state with isotropic scattering phase function. The properties of the medium are considered as stochastic ones with Gaussian or exponential distributions and hence the problem treated as a stochastic integro-differential equation. To get an explicit form for the radiant energy density, the linear extrapolation distance, reflectivity and transmissivity in the deterministic case the problem is solved using the Pomraning-Eddington method. The obtained solution is found to be dependent on the optical space variable and thickness of the medium which are considered as random variables. The random variable transformation (RVT) technique is used to find the first probability density function (1-PDF) of the solution process. Then the stochastic linear extrapolation distance, reflectivity and transmissivity are calculated. For illustration, numerical results with conclusions are provided.

Attracting and quasi-invariant sets of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion

The paper is devoted to investigating a class of neutral stochastic integro-differential equations with impulses driven by fractional Brownian motion. By establishing two new impulsive integral inequalities which improve the inequalities established by Li (Neurocomputing 177:620-627, 2016) and Long et al. (Stat. Probab. Lett. 82(9):1699-1709, 2012), attracting and quasi-invariant sets of the system are obtained. Moreover, exponential stability of the mild solution is established with sufficient conditions.

An asymptotic method for quasi-integrable Hamiltonian system with multi-time-delayed feedback controls under combined Gaussian and Poisson white noises

In the present paper, we consider an approximate approach for predicting the responses of the quasi-integrable Hamiltonian system with multi-time-delayed feedback control under combined Gaussian and Poisson white noise excitations. Two-step approximation is taken here to obtain the responses of such system. First, based on the property of the system solution, the time-delayed system state variables are approximated by using the system state variables without time delay. After this approximation, the system is converted to the one without time delay but with delay time as parameters. Then, stochastic averaging method for quasi-integrable Hamiltonian system under combined Gaussian and Poisson white noises is applied to simplify the converted system to obtain the averaged stochastic integro-differential equations and generalized Fokker–Planck–Kolmogorov equations for both non-resonant and resonant cases. Finally, two examples are worked out to show the detailed procedure of proposed method for the illustrative purpose. And the influences of the time delay on the responses of the systems are also discussed. In addition, the validity of the results obtained by present method is verified by Monte Carlo simulation.

On Large Deviations of Stochastic Integrodifferential Equations with Brownian Motion

In this paper, a Freidlin-Wentzell type large deviation principle is established for the stochastic integrodifferential equation driven by finite dimensional Brownian motion. Both the additive and multiplicative noise cases are considered here. Large deviation principle for additive noise case is established via contraction principle whilst weak convergence approach is employed to obtain the same for the multiplicative noise case.

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.

Ga-Polar (In,Ga)N/GaN Quantum Wells Versus N-Polar (In,Ga)N Quantum Disks in GaN Nanowires: A Comparative Analysis of Carrier Recombination, Diffusion, and Radiative Efficiency

We investigate the radiative and nonradiative recombination processes in planar (In,Ga)N/GaN(0001) quantum wells and (In,Ga)N quantum disks embedded in GaN$(000\bar{1})$ nanowires using photoluminescence spectroscopy under both continuous-wave and pulsed excitation. The photoluminescence intensities of these two samples quench only slightly between 10 and 300 K, which is commonly taken as evidence for high internal quantum efficiencies. However, a side-by-side comparison shows that the absolute intensity of the Ga-polar quantum wells is two orders of magnitude higher than that of the N-polar quantum disks. A similar difference is observed for the initial decay time of photoluminescence transients obtained by time-resolved measurements, indicating the presence of a highly efficient nonradiative decay channel for the quantum disks. In apparent contradiction to this conjecture, the decay of both samples is observed to slow down dramatically after the initial rapid decay. Independent of temperature, the transients approach a power law for longer decay times, reflecting that recombination occurs between individual electrons and holes with varying spatial separation. Employing a coupled system of stochastic integro-differential equations taking into account both radiative and nonradiative Shockley-Read-Hall recombination of spatially separate electrons and holes as well as their diffusion, we obtain simulated transients matching the experimentally obtained ones. The results reveal that even dominant nonradiative recombination conserves the power law decay for (In,Ga)N/GaN{0001} quantum wells and disks.

Asymptotic Stability of Neutral Impulsive Stochastic Partial Differential Equation of Sobolev Type with Poisson Jumps

This paper is concerned with the p-th asymptotical stability of the mild solution of an impulsive neutral stochastic integro-differential equation of Sobolev type involving Poisson jumps and non-instantaneous impulses. By utilizing Leray–Schauder fixed point theorem and compact semigroup, a set of sufficient conditions establishing the asymptotical stability of the mild solution to stochastic system in the p-moment is derived. An example is also considered for illustrating abstract results.

Well-posedness and dynamics of a fractional stochastic integro-differential equation

In this paper we investigate the well-posedness and dynamics of a fractional stochastic integro-differential equation describing a reaction process depending on the temperature itself. Existence and uniqueness of solutions of the integro-differential equation is proved by the Lumer–Phillips theorem. Besides, under appropriate assumptions on the memory kernel and on the magnitude of the nonlinearity, the existence of random attractor is achieved by obtaining first some a priori estimates. Moreover, the random attractor is shown to have finite Hausdorff dimension.

Optimal feedback control law for some stochastic integrodifferential equations on Hilbert spaces

In this work, we consider a class of partially observed stochastic integrodifferential equations on Hilbert spaces subject to measurement uncertainty. We prove the existence of optimal feedback control law from a class of operator valued functions furnished with the product topology. This work is an extension of [2] for uncertain systems governed by stochastic differential equations on Hilbert spaces ; whereby the measurement uncertainty is a square integrable stochastic process. We illustrate the abstract results proved by analyzing an example.

Stochastic averaging of quasi partially integrable and resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for quasi partially integrable and resonant Hamiltonian systems excited by the combined Gaussian and Poisson white noises is proposed. The averaged stochastic integro-differential equations (SIDEs) and generalized Fokker-Planck-Kolmogorov (GFPK) equation for the quasi partially integrable Hamiltonian system with r(1<r<n) independent first integrals in involution and β(1≤β≤r−2) resonant relations are derived. The dimension of averaged SIDEs and the GFPK equation is equal to the number of the independent first integrals in involution plus the number of resonant relations of the associated Hamiltonian system. A 4-degree-of-freedom (DOF) quasi partially integrable and resonant Hamiltonian system under combined Gaussian and Poisson white noise excitations are calculated as an example to illustrate the application of the proposed method. Different resonant cases and non-resonant case are worked out for the purpose of comparison. The Monte Carlo simulations are also carried out to verify the effectiveness and accuracy of the proposed stochastic averaging method.

Large Deviations for Nonlinear Ito Type Stochastic Integrodifferential Equations

Large Deviations for Nonlinear Ito Type Stochastic Integrodifferential Equations

Asymptotic behavior of second-order impulsive partial stochastic functional neutral integrodifferential equations with infinite delay

In this paper, the existence and asymptotic stability in p-th moment of mild solutions to a class of second-order impulsive partial stochastic functional neutral integrodifferential equations with infinite delay in Hilbert spaces is considered. By using H?lder?s inequality, stochastic analysis, fixed point strategy and the theory of strongly continuous cosine families with the Hausdorff measure of noncompactness, a new set of sufficient conditions is formulated which guarantees the asymptotic behavior of the nonlinear second-order stochastic system. These conditions do not require the nonlinear terms are assumed to be Lipschitz continuous. An example is also discussed to illustrate the efficiency of the obtained results.

Some results in stochastic functional integro-differential equations with infinite delays

In this paper, we study the results on averaging principle and stability of mild solutions for stochastic functional integro-differential equation with non-Lipschitz condition. We establish the result by the method of successive approximation and Bihari's inequality under the theory of resolvent operators. Finally, an example is provided for demonstration.

Application of orthonormal Bernstein polynomials to construct a efficient scheme for solving fractional stochastic integro-differential equation

Fractional stochastic integro-differential equations arise in many problems in mechanics, finance, biology, medical, social sciences. In many situation, we can not solve these equations analytically, therefore we try to get an approximate solution for these equations. In this article, we apply operational matrices method based on orthonormal Bernstein polynomials (OBPs) to solve fractional stochastic integro-differential equations. For this aim, we introduce Bernstein polynomials and OBPs and explain that how an arbitrary function can be expanded by using these polynomials. We get operational matrix and stochastic operational matrix based on OBPs. By applying these matrices fractional stochastic integro-differential equations convert to a linear system of algebraic equations which can be solved by a appropriate numerical method. Also, convergence and error analysis of the proposed method is discussed and an upper error bound of the numerical method are given. Finally, Accuracy and efficiency of the present method are shown with two examples.

Asymptotic stability of fractional impulsive neutral stochastic partial integro-differential equations with infinite delay

ABSTRACTIn this article, we investigate the existence and asymptotic stability in p-th moment of a mild solution to a class of neutral stochastic integro-differential equation of fractional order involving non-instantaneous impulses with infinite delay in a Hilbert space. A new set of sufficient conditions proving existence and asymptotic stability of mild solution is derived by utilizing solution operator, functional analysis, stochastic analysis and fixed point technique. Finally, an example is provided to illustrate the obtained abstract result.

Statistical steady state in turbulent droplet condensation

Motivated by systems in which droplets grow and shrink in a turbulence-driven supersaturation field, we investigate the problem of turbulent condensation in a general manner. Using direct numerical simulations, we show that the turbulent fluctuations of the supersaturation field offer different conditions for the growth of droplets which evolve in time due to turbulent transport and mixing. Based on this, we propose a Lagrangian stochastic model for condensation and evaporation of small droplets in turbulent flows. It consists of a set of stochastic integro-differential equations for the joint evolution of the squared radius and the supersaturation along the droplet trajectories. The model has two parameters fixed by the total amount of water and the thermodynamic properties, as well as the Lagrangian integral time scale of the turbulent supersaturation. The model reproduces very well the droplet size distributions obtained from direct numerical simulations and their time evolution. A noticeable result is that, after a stage where the squared radius simply diffuses, the system converges exponentially fast to a statistical steady state independent of the initial conditions. The main mechanism involved in this convergence is a loss of memory induced by a significant number of droplets undergoing a complete evaporation before growing again. The statistical steady state is characterized by an exponential tail in the droplet mass distribution. These results reconcile those of earlier numerical studies, once these various regimes are considered.