Stable Noises Research Articles

First-passage problem for nonlinear systems under Lévy white noise through path integral method

In this paper, the first-passage problem for nonlinear systems driven by \(\alpha \)-stable Levy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of \(\alpha \)-stable random variables and processes, PIS is extended to deal with Levy white noises with any value of the stability index \(\alpha \). Application to linear and nonlinear systems considering different values of \(\alpha \) is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Gradient estimates for SDEs driven by multiplicative Lévy noise

Gradient estimates are derived, for the first time, for the semigroup associated to a class of stochastic differential equations driven by multiplicative Lévy noise. In particular, the estimates are sharp for α -stable type noises. To derive these estimates, a new derivative formula of Bismut–Elworthy–Li's type is established for the semigroup by using the Malliavin calculus and a finite-jump approximation argument.

Time Regularity of Generalized Ornstein–Uhlenbeck Processes with Lévy Noises in Hilbert Spaces

In this paper we first obtain a necessary condition for \(H\)-cadlag modification and \(H\)-weakly cadlag modification of generalized Ornstein–Uhlenbeck processes with Levy noises in Hilbert spaces \(H\). Then we give a necessary and sufficient condition for the \(H\)-cadlag modification and \(H\)-weakly cadlag modification of Ornstein–Uhlenbeck processes driven by cylindrical \(\alpha \)-semistable processes. Secondly, we investigate the properties of cylindrical cadlag modification and \(V\)-cylindrical cadlag modification. Applying the obtained results to diagonal Ornstein–Uhlenbeck processes with \(\alpha \)-stable noises, we show a necessary and sufficient condition for cylindrical cadlag modification and \(V\)-cylindrical cadlag modification in the symmetric case for \(\alpha \in (0,1)\) and give a sufficient condition in the general case for \(\alpha \in (0,2)\). Some examples illustrate the relations among the concepts of various cadlag modifications.

Stationary states in two-dimensional systems driven by bivariate Lévy noises.

Systems driven by α-stable noises could be very different from their Gaussian counterparts. Stationary states in single-well potentials can be multimodal. Moreover, a potential well needs to be steep enough in order to produce stationary states. Here it is demonstrated that two-dimensional (2D) systems driven by bivariate α-stable noises are even more surprising than their 1D analogs. In 2D systems, intriguing properties of stationary states originate not only due to heavy tails of noise pulses, which are distributed according to α-stable densities, but also because of properties of spectral measures. Consequently, 2D systems are described by a whole family of Langevin and fractional diffusion equations. Solutions of these equations bear some common properties, but also can be very different. It is demonstrated that also for 2D systems potential wells need to be steep enough in order to produce bounded states. Moreover, stationary states can have local minima at the origin. The shape of stationary states reflects symmetries of the underlying noise, i.e., its spectral measure. Finally, marginal densities in power-law potentials also have power-law asymptotics.

Small noise fluctuations of the CIR model driven by [formula omitted]-stable noises

The central limit theorems, the deviation inequality (and large deviation), and the moderate deviations for least squares estimators of parameters in the CIR type model driven byα-stable noises are established when the dispersion parameter ε→0 and the discrete observation frequency k→∞ simultaneously.

Non-Gaussian, non-dynamical stochastic resonance

The archetypal system demonstrating stochastic resonance is nothing more than a threshold triggered device. It consists of a periodic modulated input and noise. Every time an output crosses the threshold the signal is recorded. Such a digitally filtered signal is sensitive to the noise intensity. There exist the optimal value of the noise intensity resulting in the "most" periodic output. Here, we explore properties of the non-dynamical stochastic resonance in non-equilibrium situations, i.e. when the Gaussian noise is replaced by an $\alpha$-stable noise. We demonstrate that non-equilibrium $\alpha$-stable noises, depending on noise parameters, can either weaken or enhance the non-dynamical stochastic resonance.

Ergodicity of the stochastic real Ginzburg–Landau equation driven by [formula omitted]-stable noises

We study the ergodicity of the stochastic real Ginzburg–Landau equation driven by additive α-stable noises, showing that as α∈(3/2,2), this stochastic system admits a unique invariant measure. After establishing the existence of invariant measures by the same method as in Dong et al. (2011) [12], we prove that the system is strong Feller and accessible to zero. These two properties imply the ergodicity by a simple but useful criterion by Hairer (2008) [14]. To establish the strong Feller property, we need to truncate the nonlinearity and apply a gradient estimate established by Priola and Zabczyk (2011) [22] (or see Priola et al. (2012) [20] for a general version for the finite dimension systems). Because the solution has discontinuous trajectories and the nonlinearity is not Lipschitz, we cannot solve a control problem to get irreducibility. Alternatively, we use a replacement, i.e., the fact that the system is accessible to zero. In Section 3, we establish a maximal inequality for stochastic α-stable convolution, which is crucial for studying the well-posedness, the strong Feller property and the accessibility of the mild solution. We hope this inequality will also be useful for studying other SPDEs forced by α-stable noises.

Nadaraya-Watson estimator for stochastic processes driven by stable Lévy motions

We discuss the nonparametric Nadaraya-Watson (N-W) estimator of the drift function for ergodic stochastic processes driven by $\alpha$-stable noises and observed at discrete instants. Under geometrical mixing condition, we derive consistency and rate of convergence of the N-W estimator of the drift function. Furthermore, we obtain a central limit theorem for stable stochastic integrals. The central limit theorem has its own interest and is the crucial tool for the proofs. A simulation study illustrates the finite sample properties of the N-W estimator.

Stochastic resonance in an ensemble of bistable systems under stable distribution noises and nonhomogeneous coupling

In this paper, stochastic resonance of an ensemble of coupled bistable systems driven by noise having an α-stable distribution and nonhomogeneous coupling is investigated. The α-stable distribution considered here is characterized by four intrinsic parameters: α∈(0,2] is called the stability parameter for describing the asymptotic behavior of stable densities; β∈[-1,1] is a skewness parameter for measuring asymmetry; γ∈(0,∞) is a scale parameter for measuring the width of the distribution; and δ∈(-∞,∞) is a location parameter for representing the mean value. It is demonstrated that the resonant behavior is optimized by an intermediate value of the diversity in coupling strengths. We show that the stability parameter α and the scale parameter γ can be well selected to generate resonant effects in response to external signals. In addition, the interplay between the skewness parameter β and the location parameter δ on the resonance effects is also studied. We further show that the asymmetry of a Lévy α-stable distribution resulting from the skewness parameter β and the location parameter δ can enhance the resonance effects. Both theoretical analysis and simulation are presented to verify the results of this paper.

Statistically instable processes: Connection with flicker, nonequilibrium, fractal and colored noise

Analytical expressions that link statistical instability parameters with process’s spectrum are obtained. It is shown that statistical stability is determined solely by the character of spectral power density dependence on frequency. It is revealed that statistically stable noises are those with rising intensity when frequency is increased, white noise, and equilibrium flicker noise described by the dependence 1/fβ where the spectrum shape parameter is 0 < β < 1 as well as fractal Gaussian noise. Statistically instable noises are nonequilibrium flicker noises with spectral power density 1/fβ when β≥1. It is determined that not only random nonstationary and deterministic processes are statistically instable as it was considered earlier, but stationary processes as well.

Parameter-induced stochastic resonance in overdamped system with stable noise

Parameter-induced stochastic resonance is an important method of detecting weak signal from noise, but under stable noise background, this method has not been reported. In this paper, we study the parameter-induced stochastic resonance in an overdamped system with stable noise. Our investigation discloses that the stochastic resonance can be realized by tuning the system parameter under stable noise background; when the nonlinear term parameter is turned, the resonant effect becomes weakened as the stability index decreases. But when the linear term parameter is turned, the resonant effect becomes strengthened as the stability index decreases. Our observation is significant for understanding the positive role of stable noise in weak signal detection, which is helpful for understanding the effects of different stable noises on stochastic resonance systems.

Exponential ergodicity and regularity for equations with Lévy noise

We prove exponential convergence to the invariant measure, in the total variation norm, for solutions of SDEs driven by α -stable noises in finite and in infinite dimensions. Two approaches are used. The first one is based on Liapunov’s function approach by Harris, and the second on Doeblin’s coupling argument in [8]. Irreducibility and uniform strong Feller property play an essential role in both approaches. We concentrate on two classes of Markov processes: solutions of finite dimensional equations, introduced in [27], with Hölder continuous drift and a general, non-degenerate, symmetric α -stable noise, and infinite dimensional parabolic systems, introduced in [29], with Lipschitz drift and cylindrical α -stable noise. We show that if the nonlinearity is bounded, then the processes are exponential mixing. This improves, in particular, an earlier result established in [28], with a different method.

Synthesis of multifractional Gaussian noises based on variable-order fractional operators

In this paper, a synthesis method, which is based on variable-order fractional operators, for multifractional Gaussian noises (mGn) is proposed by studying the relationship of white Gaussian noise (wGn), mGn, and multifractional Brownian motion (mBm). Furthermore, a synthesis method for multifractional α ‐ stable processes, the generalization of mGn, is proposed in order to more accurately characterize the processes with local scaling characteristics and heavy tailed distributions. Synthetic examples of mGn and multifractional α ‐ stable noises are provided for illustration.

Parameter estimation for a class of stochastic differential equations driven by small stable noises from discrete observations

We study the least squares estimation of drift parameters for a class of stochastic differential equations driven by small α-stable noises, observed at n regularly spaced time points t i = i/n, i = 1, …, n on [0,1]. Under some regularity conditions, we obtain the consistency and the rate of convergence of the least squares estimator (LSE) when a small dispersion parameter ɛ → 0 and n → ∞ simultaneously. The asymptotic distribution of the LSE in our setting is shown to be stable, which is completely different from the classical cases where asymptotic distributions are normal.

Synchronization of systems of Marcus canonical equations driven by [formula omitted]-stable noises

Dynamical systems driven by Gaussian noises have been considered extensively in modeling, simulation and theory. However, complex systems in engineering and science are often subject to non-Gaussian fluctuations or uncertainties. A coupled dynamical system under non-Gaussian Lévy noise is considered. After discussing cocycle property, stationary orbits and random attractors, a synchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfy certain dissipativity conditions. The synchronization result implies that coupled dynamical systems share a dynamical feature in certain asymptotic sense.

Least squares estimator for Ornstein–Uhlenbeck processes driven by [formula omitted]-stable motions

We study the problem of parameter estimation for generalized Ornstein–Uhlenbeck processes driven by α -stable noises, observed at discrete time instants. Least squares method is used to obtain an asymptotically consistent estimator. The strong consistency and the rate of convergence of the estimator have been studied. The estimator has a higher order of convergence in the general stable, non-Gaussian case than in the classical Gaussian case.

Mobile communication array processing based on radial‐basis function neural network and fractional lower order statistics

AbstractAlpha stable distribution is better for modelling impulsive noises than Gaussian distribution in signal processing. This class of process has no close form of probability density function and finite second order moments. This paper addresses the direction of arrival estimation and beamforming problems of mobile communication system with linear antenna arrays in high impulsive noise environment. In order to reduce the computational complexity, the problems of DOA and beamforming are approached as a nonlinear mapping that can be modelled using a suitable radial‐basis function neural network (RBFNN). This paper also proposes the application of a three–layer RBFNN to perform the DOA estimation and beamforming in presence of alpha stable distribution noises. The performance of the network is compared to that of the fractional lower order statistics based algorithms. Simulations show that the RBFNN is appropriate to approach the DOA estimation and beamforming. At the same time, the RBFNN substantially reduces the computation complexity. Copyright © 2008 John Wiley &amp; Sons, Ltd.

Stationary states in Langevin dynamics under asymmetric Lévy noises

Properties of systems driven by white non-Gaussian noises can be very different from these of systems driven by the white Gaussian noise. We investigate stationary probability densities for systems driven by alpha-stable Lévy-type noises, which provide natural extension to the Gaussian noise having, however, a new property, namely a possibility of being asymmetric. Stationary probability densities are examined for a particle moving in parabolic, quartic, and in generic double well potential models subjected to the action of alpha-stable noises. Relevant solutions are constructed by methods of stochastic dynamics. In situations where analytical results are known they are compared with numerical results. Furthermore, the problem of estimation of the parameters of stationary densities is investigated.

Linearly constrained minimum-"geometric power" adaptive beamforming using logarithmic moments of data containing heavy-tailed noise of unknown statistics

This letter presents a new adaptive beamforming approach, against arbitrary algebraically tailed impulse noise of otherwise unknown statistics. (This includes all symmetric α -stable noises with infinite variance or even infinite mean.) This new beamformer iteratively minimizes the "geometric power" of the beamformer's output Y, subject to a prespecified set of linear constraints. This geometric power is defined in terms of the logarithmic moment E\{log|Y|}, as an alternative to the customary "fractional lower order moments" (FLOM). This logarithmic-moment beamformer offers these advantages over the FLOM beamformer: 1) simpler computationally in general, 2) needing no prior information nor estimation of the numerical value of the impulse noise's effective characteristic exponent, and 3) applicable to a wider class of heavy-tailed impulse noises.

Lévy-Brownian motion on finite intervals: Mean first passage time analysis

We present the analysis of the first passage time problem on a finite interval for the generalized Wiener process that is driven by Lévy stable noises. The complexity of the first passage time statistics (mean first passage time, cumulative first passage time distribution) is elucidated together with a discussion of the proper setup of corresponding boundary conditions that correctly yield the statistics of first passages for these non-Gaussian noises. The validity of the method is tested numerically and compared against analytical formulas when the stability index alpha approaches 2, recovering in this limit the standard results for the Fokker-Planck dynamics driven by Gaussian white noise.