White Noise Type Research Articles

Stochastic nonlinear beam equations

An extensible beam equation with a stochastic force of a white noise type is studied, Lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution.

Averaging Principle for Stochastic Perturbations of Multifrequency Systems

We consider the averaging principle for deterministic and stochastic perturbations of multidimensional dynamical systems for which coordinates can be introduced in such a way that the "fast" coordinates change in a torus (for Hamiltonian systems, "action-angle coordinates"). Stochastic perturbations of the white-noise type are considered. Our main assumption is that the set of action values for which the frequencies of the motion on corresponding tori are rationally dependent (and so the motion reduces to a torus of smaller dimension) has Lebesgue measure zero. Our results about stochastic perturbations imply some new results for averaging of purely deterministic perturbations.

Acoustic source localization using phase difference spectrum images

The localization of a single acoustic source on the horizontal plane using phase difference spectrum images is discussed. The azimuth for source is identified from the general linear relationship, which is extracted from the measured phase difference spectrum after filtering. The phase difference spectrum is introduced as the quasi-stationary cross-spectral phase between the sound signals detected simultaneously by two sensors. Acoustic source localization in an anechoic chamber having a metal base plate using two types of sound signals, white noise and voice, indicated that the phase difference spectrum was not affected with respect to the sound pressure level but was affected with respect to the azimuth for source. Although the phase difference spectrum measured in a reverberant room had less continuity as a function of frequency, a linear distribution of the images obtained from the data (dots) was observed on the frequency - phase difference plane. Using the phase difference spectrum images, the azimuths for various sources, which radiated any kind of broadband sound on separated time schedule, were precisely identified even in the reverberant room.

The Hammerstein–Wiener Model for Identification of Stochastic Systems

Identification of nonlinear stochastic systems in the class of Hammerstein–Wiener models is studied. The problem is specific due to the nonlinearities of the investigated object. Hammerstein–Wiener models are constructed with regard for the disturbances of the type of white noise and martingale sequence at the output of the object. A two-stage recursive identification algorithm is designed. Necessary and sufficient conditions for the strong consistency of parameter estimates found by this algorithm are formulated. The results are applied to adaptive tracking of the output of an object.

STABILITY FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH DIRICHLET WHITE-NOISE BOUNDARY CONDITIONS

In this paper we study the long-time behaviour a stochastic parabolic equation perturbed through the boundary; the perturbation is represented by a nonhomogeneous Dirichlet boundary condition of white-noise type. The existence of the solution for an equation of this kind was the object of a previous paper of the same authors. The estimates proved therein allow to study the asymptotic properties of the solution; we show that there exists a unique invariant measure that is exponentially mean square stable.

On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation

We investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear Schrodinger equation. We prove that, contrary to the deterministic case, any initial data gives birth to a solution which develops singularities. Moreover, the singularities appear immediately. We use a stochastic generalization of the variance identity and a control argument.

Numerical simulation of focusing stochastic nonlinear Schrödinger equations

In this paper, we numerically investigate nonlinear Schrödinger equations with a stochastic contribution which is of white noise type and acts either as a potential (multiplicative noise) or as a forcing term (additive noise). In the subcritical case, we recover similar results as in the case of the Korteweg–de Vries equation. In the critical or supercritical case, we observe that depending on its smoothness, the noise may have different effects. Spatially smooth noises amplify blow-up phenomena, whereas delta correlated multiplicative noises prevent blow-up formation. Note that in this latter case, very few results are known, both from a theoretical and a numerical point of view.

Stochastic partial differential equations with Dirichlet white-noise boundary conditions

Abstract The paper is devoted to one-dimensional nonlinear stochastic partial differential equations of parabolic type with non homogeneous Dirichlet boundary conditions of white-noise type. We formulate a set of conditions that a random field must satisfy to solve the equation. We show that a unique solution exists and that we can write it in terms of the stochastic kernel related to the problem. This formulation allows us to study the basic properties of the solution, as the continuity and the boundary-layer behavior, by means of Malliavin calculus.

Transient structures of wave patterns arising in the wave regeneration of subalpine coniferous forests.

In wave-regeneration phenomena observed in the subalpine coniferous forests, mainly consisting of Abies species, the blighted forests present various shapes in the course of development, spots at the initial stage turning into arches and finally into long whitish stripes. Because the wave-regeneration could not be followed in the field without long term studies, a simple model has been elaborated to simulate the various different dieback structures observed in the real forests. This model, based on cellular automata, is employed to analyze the power spectral density of canopy tree height fluctuations in the wave-regenerated forests. The results demonstrate that almost all the dieback structures observed in the field can be generated by this simple model, by varying the wind direction and its strength by some stochasticity. The power spectrum density presents various shapes in the course of development, white noise type at the initial stage turning into Lorentz type and finally into 1/f type power spectrum (spatial Fourier frequency).

On random perturbations of Hamiltonian systems with many degrees of freedom

We consider a class of random perturbations of Hamiltonian systems with many degrees of freedom. We assume that the perturbations consist of two components: a larger one which preserves the energy and destroys all other first integrals, and a smaller one which is a non-degenerate white noise type process. Under these assumptions, we show that the long time behavior of such a perturbed system is described by a diffusion process on a graph corresponding to the Hamiltonian of the system. The graph is homeomorphic to the set of all connected components of the level sets of the Hamiltonian. We calculate the differential operators which govern the process inside the edges of the graph and the gluing conditions at the vertices.

An innovation approach to non-Gaussian time series analysis

The paper shows that the use of both types of random noise, white noise and Poisson noise, can be justified when using an innovations approach. The historical background for this is sketched, and then several methods of whitening dependent time series are outlined, including a mixture of Gaussian white noise and a compound Poisson process: this appears as a natural extension of the Gaussian white noise model for the prediction errors of a non-Gaussian time series. A statistical method for the identification of non-linear time series models with noise made up of a mixture of Gaussian white noise and a compound Poisson noise is presented. The method is applied to financial time series data (dollar-yen exchange rate data), and illustrated via six models.

An innovation approach to non-Gaussian time series analysis

The paper shows that the use of both types of random noise, white noise and Poisson noise, can be justified when using an innovations approach. The historical background for this is sketched, and then several methods of whitening dependent time series are outlined, including a mixture of Gaussian white noise and a compound Poisson process: this appears as a natural extension of the Gaussian white noise model for the prediction errors of a non-Gaussian time series. A statistical method for the identification of non-linear time series models with noise made up of a mixture of Gaussian white noise and a compound Poisson noise is presented. The method is applied to financial time series data (dollar-yen exchange rate data), and illustrated via six models.

Stochastic Analysis of Shape Imperfection in RC Cooling Tower Shells

In this paper, the response variability of a reinforced concrete cooling tower shell due to shape imperfections is investigated, based on statistical stochastic analysis. In general, shape imperfections are modeled as ideal shapes, such as axisymmetric cosine shape and/or cyclic shape imperfections. However, these kinds of idealized imperfections cannot well enough represent the real shape imperfections in actual cooling tower shells. In this study, the shape imperfection of the cooling tower is assumed to be globally distributed stochastic fields rather than localized ones. The two geometrical parameters, i.e., thickness and radius, of a cooling tower shell are chosen as the spatially varying stochastic fields with predetermined statistical terms. The amount of additional responses revealed that the coefficients of variation (COVs) of stresses are larger than those of displacements, and that the additional response due to randomness in the cooling tower radius is larger than that in the shell thickness. The COVs of these two parameters show quite opposite trends. As the type of stochastic field approaches white noise, the response variability is decreased in the case of randomness in thickness, but increased in the case of radius.

On the Phase Portrait of a Harmonic Oscillator with Friction Perturbed by a Random Process of White Noise Type

We study the phase portrait of a harmonic oscillator with friction for the case in which random perturbations of white noise type expressed in the Ito form act at a certain angle to the phase velocity vector.

Some biological effects of microwave irradiation

Morphological analysis and morphometry using immunoperoxidase methods showed that microwave (MW) radiation with a white noise type spectrum and radiation power density close to that emitted by the object stimulated proliferation of human fibroblastsin vitro. This effect is realized via stimulation of DNA synthesis. MV irradiation with the same parameters increased adhesive capacity of human blood neutrophils.

The Stochastic Korteweg–de Vries Equation inL2([formula omitted

We present here two results of global existence and uniqueness of the Cauchy problem concerning the Korteweg–de Vries equation forced by a random term of white noise type in the functional framework of weighted spaces onL2(R). This work is motivated by both physical experiments where a forced generation of weakly nonlinear waves by localized disturbances was observed [1,15] and a result of uniqueness in weighted space in the homogeneous case [8,14].

Parametric Synthesis of One Class of Dynamic Systems under Random External Disturbances

The problem of suboptimal parametric optimization of nonlinear nonstationary controllable systems under random external disturbances of white noise type is considered in the present paper. The method of solution incorporates V.F. Krotov's idea of global estimates.

Korteweg–de Vries solitons under additive stochastic perturbations

The evolution of solitonic solutions of the Korteweg--de Vries equation subject to additive noise is investigated using numerical techniques. Various types of additive white Gaussian noise are considered. The averaged solution amplitudes exhibit in all cases algebraic decay, verifying Wadati's universality conjecture. If the noise is time dependent, or position and time dependent, algebraic decay is obtained for intermediate times too. These intermediate-time results agree well with the outcome of an experiment on ion-acoustic soliton propagation in a noisy plasma. The distribution of soliton first passage times in a noisy medium is also discussed.

A note on the statistics of ordered peaks in stationary stochastic processes

Numerical simulation experiments have been carried out to evaluate the relative performance of three approximate formulations owing to Amini and Trifunac, [1] Gupta and Trifunac [2] and Basu et al. [3] for probability distributions of ordered peaks in stationary stochastic processes. The first two formulations are based on the assumption that the unordered peaks are statistically independent; whereas the formulation of Basu et al. [3] considers the dependence via Markov transition probability. In the formulation of Gupta and Trifunac, [2] the probability distribution of nth order peak is inherently conditioned by the fact that at most ( n−1) peaks can occur with higher amplitudes and, thus, the ordered peaks are necessarily dependent. In this paper, ensembles of stationary time-histories are generated for three PSDFs representing a narrow-band, a broad-band and a band-limited white noise type of processes. Comparison of the results for the expected value and standard deviation for various orders of peaks, obtained by averaging over these ensembles, with the corresponding results obtained from the above mentioned three approximations, defined in terms of the moments of the PSDFs, has shown that the formulation of Gupta and Trifunac [2] describes the data well.

Self-similar crack-generation effects in the fracture process in brittle materials

Using acoustic-emission data banks we have computed time and space correlation functions for the purpose of investigation of crack-propagation self-similarity during the fracture process in brittle materials. It is shown that the whole fracture process may be represented as a two-stage process. In the first stage, the crack propagation is uniform and uncorrelated in space, having a time spectral density of the white-noise type and a correlation fractal dimension approximately equal to that of 3D Euclidean space. In the second stage, this fractal dimension decreases significantly, reaching the value of 2.2-2.4, characteristic for the fracture surfaces, while the time spectral density exhibits a significant low-frequency increase becoming of -noise type. The resulting fractal shows no multifractal behaviour, appearing to be a single fractal.