Multiplicative Gaussian Noise Research Articles

Intermittency for the Hyperbolic Anderson Model with rough noise in space

In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index H∈(14,12). Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the p-th moment the solution, for any p≥2. Condition H>14 turns out to be necessary for the existence of solution. Secondly, we show that this solution coincides with the one obtained by the authors in a recent publication, in which the solution is interpreted in the Itô sense. Finally, we prove that the solution of the equation in the Skorohod sense is weakly intermittent.

SPDEs with rough noise in space: Hölder continuity of the solution

We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index H∈(14,12). The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than H, and Hölder continuous in time of order smaller than γ, where γ=H for the wave equation and γ=H/2 for the heat equation.

Stochastic differential equations with variable structure driven by multiplicative Gaussian noise and sliding mode dynamic

This work is concerned with existence of weak solutions to discontinuous stochastic differential equations driven by multiplicative Gaussian noise and sliding mode control dynamics generated by stochastic differential equations with variable structure, that is with jump nonlinearity. The treatment covers the finite dimensional stochastic systems and the stochastic diffusion equation with multiplicative noise.

Backward uniqueness of stochastic parabolic like equations driven by Gaussian multiplicative noise

One proves here the backward uniqueness of solutions to stochastic semilinear parabolic equations and also for the tamed Navier–Stokes equations driven by linearly multiplicative Gaussian noises. Applications to approximate controllability of nonlinear stochastic parabolic equations with initial controllers are given. The method of proof relies on the logarithmic convexity property known to hold for solutions to linear evolution equations in Hilbert spaces with self-adjoint principal part.

Trichotomous noise induced stochastic resonance in a fractional oscillator with random damping and random frequency

.In this study, we investigate the stochastic resonance (SR) of a fractional linear oscillator subjected to multiplicative trichotomous noise and additive fractional Gaussian noise and driven by a periodic signal. Using the Shapiro-Loginov formula and the Laplace transformation technique, we acquire the exact expression of the first-order moment of the system’s steady response. Meanwhile, we discuss the evolutions of the output amplitude with the frequency of the periodic signal and noise parameters. We determine that bona fide SR, SR and reverse-resonance exist in this system. Specifically, the evolution of the output amplitude with the frequency of the periodic signal presents one-peak oscillation, double-peak oscillation and triple-peak oscillation. Moreover, the interplay of the trichotomous noise and memory can induce and diversify the stochastic multi-resonance (SMR) phenomena and give this linear system richer dynamic behavior. A hypersensitive response of the output amplitude to the noise intensity is demonstrated, which is not observed in systems driven by dichotomous noise.

On the intermittency front of stochastic heat equation driven by colored noises

We study the propagation of high peaks (intermittency fronts) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R} ^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.

Well-posedness and dynamics of the stochastic fractional magneto-hydrodynamic equations

The current paper is devoted to the well-posedness and dynamics of the stochastic 2D incompressible fractional Magneto-Hydrodynamic(MHD) equations driven by Gaussian multiplicative noise. The nonlocal fractional diffusion leads to a new difficulty in the convergence since higher order estimates cannot be obtained. The commutator estimates are introduced to overcome these difficulties. Using the stopping time technique and monotonicity arguments, the global existence and uniqueness of the weak solution are obtained in a fixed probability space. Finally, the existence of a random attractor for the random dynamical systems generated by the solution of stochastic MHD equation is presented.

Memory-induced sign reversals of the spatial cross-correlation for particles in viscoelastic shear flows

The behavior of shear-induced cross-correlation functions between particle fluctuations along orthogonal directions in the shear plane for harmonically trapped Brownian particles in a viscoelastic shear flow is studied. A generalized Langevin equation with a power-law-type memory kernel is used to model the complex structure of the viscoelastic media. Interaction with fluctuations of environmental parameters is modeled by a multiplicative white Gaussian noise, by an internal fractional Gaussian noise, and by an additive external white noise. It is shown that the presence of a memory has a profound effect on the behavior of the cross-correlation functions. Particularly, memory-induced reentrant sign reversals of the spatial cross-moment between orthogonal random displacements of a particle are established, i.e., an increase of the memory exponent can cause the sign reversal from positive to negative, but by a further increase of the memory exponent a reentrant transition from negative to positive values appears. Similarities and differences between the behavior of the models with additive internal and external noises are considered. It is shown that additive external and internal noises cause qualitatively different dependencies of the cross-correlation functions on the time lag. The occurrence of energetic instability due to the influence of multiplicative noise is also discussed.

Parameter Estimation From Quantized Observations in Multiplicative Noise Environments

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

Variational Solution of Stochastic Schrödinger Equations With Power-Type Nonlinearity

We investigate a Schrödinger problem with multiplicative Gaussian noise term and power-type nonlinearity on a bounded one-dimensional domain. In order to prove the existence and uniqueness of the variational solution, a further process will be introduced which allows to transform the stochastic nonlinear Schrödinger problem into a pathwise one. Galerkin approximations and compact embedding results are used.

Moment equations for a piecewise deterministic PDE

We analyze a piecewise deterministic PDE consisting of the diffusion equation on a finite interval Ω with randomly switching boundary conditions and diffusion coefficient. We proceed by spatially discretizing the diffusion equation using finite differences and constructing the Chapman–Kolmogorov (CK) equation for the resulting finite-dimensional stochastic hybrid system. We show how the CK equation can be used to generate a hierarchy of equations for the r-th moments of the stochastic field, which take the form of r-dimensional parabolic PDEs on that couple to lower order moments at the boundaries. We explicitly solve the first and second order moment equations (r = 2). We then describe how the r-th moment of the stochastic PDE can be interpreted in terms of the splitting probability that r non-interacting Brownian particles all exit at the same boundary; although the particles are non-interacting, statistical correlations arise due to the fact that they all move in the same randomly switching environment. Hence the stochastic diffusion equation describes two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment. Finally, in the limit of fast switching, we use a quasi-steady state approximation to reduce the piecewise deterministic PDE to an SPDE with multiplicative Gaussian noise in the bulk and a stochastically-driven boundary.

C205 ヒトのバランス操作の確率動的モデル構築に向けて(OS12 ロボットおよび人間のダイナミクスと制御2)

In this study, we conduct experiments of human balancing task and present a stochastic dynamic model of the human balancing task. The parameters of the stochastic dynamic model are identified using particle swarm optimizers. Tn order to examine stochastic behavior of the human balancing, we experimentally estimate probability density function of the state quantities. The experimental result implies that the behavior of the human balance seems to be fat-tailed distribution. This experimental observation was reproduced by the model that has additive and multiplicative white Gaussian noise. This work has implications for designing human-like motions of artificial agents such as human-like partner robots.

Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

Optimising threshold levels for information transmission in binary threshold networks: Independent multiplicative noise on each threshold

The problem of optimising the threshold levels in multilevel threshold system subject to multiplicative Gaussian and uniform noise is considered. Similar to previous results for additive noise, we find a bifurcation phenomenon in the optimal threshold values, as the noise intensity changes. This occurs when the number of threshold units is greater than one. We also study the optimal thresholds for combined additive and multiplicative Gaussian noise, and find that all threshold levels need to be identical to optimise the system when the additive noise intensity is a constant. However, this identical value is not equal to the signal mean, unlike the case of additive noise. When the multiplicative noise intensity is instead held constant, the optimal threshold levels are not all identical for small additive noise intensity but are all equal to zero for large additive noise intensity. The model and our results are potentially relevant for sensor network design and understanding neurobiological sensory neurons such as in the peripheral auditory system.

Existence, uniqueness and regularity for a class of semilinear stochastic Volterra equations with multiplicative noise

We consider a class of semilinear Volterra type stochastic evolution equation driven by multiplicative Gaussian noise. The memory kernel, not necessarily analytic, is such that the deterministic linear equation exhibits a parabolic character. Under appropriate Lipschitz-type and linear growth assumptions on the nonlinear terms we show that the unique mild solution is mean-p Hölder continuous with values in an appropriate Sobolev space depending on the kernel and the data. In particular, we obtain pathwise space–time (Sobolev–Hölder) regularity of the solution together with a maximal type bound on the spatial Sobolev norm. As one of the main technical tools we establish a smoothing property of the derivative of the deterministic evolution operator family.

Deception, blindness and disorientation in particle swarm optimization applied to noisy problems

Particle swarm optimization (PSO) is a population-based algorithm designed to find good solutions to optimization problems. However, if the problems are subject to noise, the quality of its results significantly deteriorates. Previous works have addressed such a deterioration by developing noise mitigation mechanisms to target specific issues such as handling the inaccurate memories of the particles and aiding the particles to correctly select their neighborhood best solutions. However, in spite of the improvements achieved, it still remains uncertain the extent to which these issues affect the particles, and the underlying reasons for the deterioration of the quality of the results. In this article, we formally define deception, blindness and disorientation as the conditions responsible for such a deterioration, and we develop a set of population statistics to measure the extent to which these conditions affect the particles throughout the search process. The population statistics are computed for the regular PSO algorithm and for PSO with equal resampling (PSO-ER) on 20 large-scale benchmark functions subject to different levels of multiplicative Gaussian noise. The key findings that we reveal with the population statistics on optimization problems subject to noise are the following: (a) the quality of the results significantly deteriorates as particles suffer from large proportions of deception and blindness; (b) the presence of deception, blindness and disorientation, and their effects on the quality of results, makes the performance of the swarms sensitive to optimization problems subject to noise; (c) the incorporation of resampling methods into PSO significantly improves the quality of the results; and (d) it is better to first address the conditions of blindness and disorientation before addressing deception.

Fully phase image encryption using double random-structured phase masks in gyrator domain.

We propose a method for fully phase image encryption based on double random-structured phase mask encoding in the gyrator transform (GT) domain. The security of the system is strengthened by parameters used in the construction of a structured phase mask (SPM) based on a devil's vortex Fresnel lens (DVFL). The input image is recovered using the correct parameters of the SPMs, transform orders of the GT, and conjugate of the random phase masks. The use of a DVFL-based SPM enhances security by increasing the key space for encryption, and also overcomes the problem of axis alignment associated with an optical setup. The proposed scheme can also be implemented optically. The computed values of mean squared error between the retrieved and the original image show the efficacy of the proposed scheme. We have also investigated the scheme's sensitivity to the encryption parameters, and robustness against occlusion and multiplicative Gaussian noise attacks.

A well-balanced and adaptive variational model for the removal of mixed noise

Image denoising is one of the fundamental problems concerning image processing. Over the last decade mathematical models based on partial differential equations and variational techniques have led to superior results related to denoising problems. The additive noise models have been studied extensively, however, the reconstruction of images corrupted by nonadditive noise has not yet been thoroughly studied. In this paper, a novel variational method for the reconstruction of images corrupted by non-uniformly distributed noise is presented. The proposed model includes a balance between the data term and the regularization term in the energy functional, which takes into account the statistical control of the parameters and the position of the noisy points related to the edges presented in the image. The parameters are determined by the given initial noisy image. The obtained results have shown the effectiveness and robustness of the proposed model and in restoring images with multiplicative noise or mixed Gaussian noise, while preserving edges and small structures belonging to the image.

Ghost stochastic resonance induced by a power-law distributed noise in the FitzHugh–Nagumo neuron model

We numerically investigate the ghost stochastic resonance phenomenon induced by a power-law distributed noise in the neuron FitzHugh–Nagumo model. The input noise considered is produced by a Langevin process including both multiplicative and additive Gaussian noise sources. In this process, the power-law decay exponent of the resulting noise distribution is governed by the off-set of the multiplicative noise, thus allowing us to probe both regimes of Gaussian and strongly non-Gaussian noises. Ghost stochastic resonance, i.e., stochastic resonance in a missing fundamental harmonic, occurs in this model. Deviations from the Gaussianity of the input noise are shown to reduce both the additive noise intensity corresponding to the optimal identification of the missing fundamental as well as the number of firing events at the ghost stochastic resonance condition.

Order-by-disorder in classical oscillator systems

We consider classical nonlinear oscillators on hexagonal lattices. When the coupling between the elements is repulsive, we observe coexisting states, each one with its own basin of attraction. These states differ by their degree of synchronization and by patterns of phase-locked motion. When disorder is introduced into the system by additive or multiplicative Gaussian noise, we observe a non-monotonic dependence of the degree of order in the system as a function of the noise intensity: intervals of noise intensity with low synchronization between the oscillators alternate with intervals where more oscillators are synchronized. In the latter case, noise induces a higher degree of order in the sense of a larger number of nearly coinciding phases. This order-by-disorder effect is reminiscent to the analogous phenomenon known from spin systems. Surprisingly, this non-monotonic evolution of the degree of order is found not only for a single interval of intermediate noise strength, but repeatedly as a function of increasing noise intensity. We observe noise-driven migration of oscillator phases in a rough potential landscape.