Fractional Brownian Noise Research Articles

Stochastic dynamical behaviors of SOS/ERK signaling pathway perturbated by external noise

The SOS/ERK cascades are key signaling pathways that regulate cellular processes ranging from cellular proliferation, differentiation and apoptosis to tumor formation. However, the properties of these signaling pathways are not well understood. More importantly, how stochastic perturbations of internal and external cellular environment affect these pathways remains unanswered. To answer these questions, we, in this paper, propose a stochastic model according to the biochemical reaction processes of the SOS/ERK pathways, and, respectively, research the dynamical behaviors of this model under the four kinds of noises: Gaussian noise, colored noise, Lévy noise and fraction Brown noise. Some important results are found that Gaussian and colored noises have less effect on the stability of the system when the strength of the noise is small; Lévy and fractional Brownian noises significantly change the trajectories of the system. Power spectrum analysis shows that Lévy noise induces a system with quasi-periodic trajectories. Our results not only provide an understanding of the SOS/ERK pathway, but also show generalized rules for stochastic dynamical systems.

Exponential Euler method for stiff stochastic differential equations with additive fractional Brownian noise

We discuss a system of stochastic differential equations with a stiff linear term and additive noise driven by fractional Brownian motions (fBms) with Hurst parameter H > 1 2 , which arise e.g. from spatial approximations of stochastic partial differential equations. For their numerical approximation, we present an exponential Euler scheme and show that it converges in the strong sense with an exact rate close to the Hurst parameter H. Further, based on E. Buckwar et al. [The numerical stability of stochastic ordinary differential equations with additive noise, Stoch. Dyn. 11 (2011), pp. 265–281], we conclude the existence of a unique stationary solution of the exponential Euler scheme that is pathwise asymptotically stable.

An Efficient Numerical Algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

An Efficient Numerical Algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

A Numerical Algorithm for Solving Nonlocal Nonlinear Stochastic Delayed Systems with Variable-Order Fractional Brownian Noise

A numerical technique was developed for solving nonlocal nonlinear stochastic delayed differential equations driven by fractional variable-order Brownian noise. Error analysis of the proposed technique was performed and discussed. The method was applied to the nonlocal stochastic fluctuations of the human body and the Nicholson’s blowfly models, and its accuracy and computational time were assessed for different values of the nonlocal order parameters. A comparison with other techniques available in the literature revealed the effectiveness of the proposed scheme.

A Computational Technique for Nonlinear Nonlocal Stochastic Dynamical Systems with Variable Order Fractional Brownian Noise

A Computational Technique for Nonlinear Nonlocal Stochastic Dynamical Systems with Variable Order Fractional Brownian Noise

Parameter estimation in mixed fractional stochastic heat equation

The paper is devoted to a stochastic heat equation with a mixed fractional Brownian noise. We investigate the covariance structure, stationarity, upper bounds and asymptotic behavior of the solution. Based on its discrete-time observations, we construct a strongly consistent estimator for the Hurst index H and prove the asymptotic normality for $H. Then assuming the parameter H to be known, we deal with joint estimation of the coefficients at the Wiener process and at the fractional Brownian motion. The quality of estimators is illustrated by simulation experiments.

Fractality of tics as a quantitative assessment tool for Tourette syndrome.

Tics manifest as brief, purposeless and unintentional movements or noises that, for many individuals, can be suppressed temporarily with effort. Previous work has hypothesized that the chaotic temporal nature of tics could possess an inherent fractality, that is, have neighbour-to-neighbour correlation at all levels of timescale. However, demonstrating this phenomenon has eluded researchers for more than two decades, primarily because of the challenges associated with estimating the scale-invariant, power law exponent—called the fractal dimension Df—from fractional Brownian noise. Here, we confirm this hypothesis and establish the fractality of tics by examining two tic time series datasets collected 6–12 months apart in children with tics, using random walk models and directional statistics. We find that Df is correlated with tic severity as measured by the YGTTS total tic score, and that Df is a sensitive parameter in examining the effect of several tic suppression conditions on the tic time series. Our findings pave the way for using the fractal nature of tics as a robust quantitative tool for estimating tic severity and treatment effectiveness, as well as a possible marker for differentiating typical from functional tics.

Two-time-scale stochastic differential delay equations driven by multiplicative fractional Brownian noise: Averaging principle

The main goal of this article is to study an averaging principle for a class of two-time-scale stochastic differential delay equations in which the slow-varying process includes a multiplicative fractional Brownian noise with Hurst parameter H∈(12,1) and the fast-varying process is a rapidly-changing diffusion. We would like to emphasize that the approach proposed in this paper is based on the fact that a stochastic integral with respect to fractional Brownian motion with Hurst parameter in (12,1) can be defined as a generalized Stieltjes integral. In particular, to prove a limit theorem for the averaging principle, we will introduce a sequence of stopping times to control the size of multiplicative fractional Brownian noise. Then, inspired by the Khasminskii's approach, an averaging principle is developed in the sense of convergence in the p-th moment uniformly in time.

Asymptotic Behavior of Large Gaussian Correlated Wishart Matrices

We consider high-dimensional Wishart matrices \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\), associated with a rectangular random matrix \({\mathcal {X}}_{n,d}\) of size \(n\times d\) whose entries are jointly Gaussian and correlated. Even if we will consider the case of overall correlation among the entries of \({\mathcal {X}}_{n,d}\), our main focus is on the case where the rows of \({\mathcal {X}}_{n,d}\) are independent copies of an n-dimensional stationary centered Gaussian vector of correlation function s. When s belongs to \(\ell ^{4/3}({\mathbb {Z}})\), we show that a proper normalization of \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\) is close in Wasserstein distance to the corresponding Gaussian ensemble as long as d is much larger than \(n^3\), thus recovering the main finding of Bubeck et al. (Random Struct Algorithms 49(3):503–532, 2016) and Jiang and Li (J Theor Probab 28(3):804–847, 2015) and extending it to a larger class of matrices. We also investigate the case where s is the correlation function associated with the fractional Brownian noise of parameter H. This example is very rich, as it gives rise to a great variety of phenomena with very different natures, depending on how H is located with respect to 1/2, 5/8 and 3/4. Notably, when \(H>3/4\), our study highlights a new probabilistic object, which we have decided to call the Rosenblatt–Wishart matrix. Our approach crucially relies on the fact that the entries of the Wishart matrices we are dealing with are double Wiener–Itô integrals, allowing us to make use of multivariate bounds arising from the Malliavin–Stein method and related ideas. To conclude the paper, we analyze the situation where the row-independence assumption is relaxed and we also look at the setting of random p-tensors (\(p\ge 3\)), a natural extension of Wishart matrices.

$${\mathbb {L}}^p$$-solutions of deterministic and stochastic convective Brinkman–Forchheimer equations

In the first part of this work, we establish the existence and uniqueness of a local mild solution to deterministic convective Brinkman–Forchheimer (CBF) equations defined on the whole space, by using properties of the heat semigroup and fixed point arguments based on an iterative technique. Moreover, we prove that the solution exists globally. The second part is devoted for establishing the existence and uniqueness of a pathwise mild solution upto a random time to the stochastic CBF equations perturbed by Lévy noise by exploiting the contraction mapping principle. Then by using stopping time arguments, we show that the pathwise mild solution exists globally. We also discuss the local and global solvability of the stochastic CBF equations forced by fractional Brownian noise.

Well-posedness for Hardy–Hénon parabolic equations with fractional Brownian noise

We study the Hardy-Hénon parabolic equations on \( \mathbb {R}^{N}\) (N=2 or 3) under the effect of an additive fractional Brownian noise with Hurst parameter \(H>\max \left( 1/2, N/4\right) .\) We show local existence and uniqueness of a mid \(L^{q}\)-solution under suitable assumptions on q.

Numerical Approximation of Optimal Convergence for Fractional Elliptic Equations with Additive Fractional Gaussian Noise

We study numerical approximation for one-dimensional stochastic elliptic equations with integral fractional Laplacian and the additive Gaussian noise of power-law: $1/f^\beta$ noise and fractional Brownian noise. We present an optimal convergence of our method using spectral expansions of noises. We first establish the well-posedness of a corresponding deterministic problem and show the stability of solutions for the rough data via negative norms in weighted Sobolev spaces. We also analyze the regularity of the noise and approximation properties of their finite truncations. Next, we show the optimal error estimates of our method for a wide range of parameters in the order of fractional operator and the fractional Gaussian noise. Finally, we present several numerical examples to illustrate the mean-square convergence orders and verify our optimal convergence rates.

Typical dynamics and fluctuation analysis of slow–fast systems driven by fractional Brownian motion

This paper studies typical dynamics and fluctuations for a slow–fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we characterize the asymptotic dynamics of the slow component to two orders (i.e. the typical dynamics and the fluctuations). The limiting distribution of the fluctuations turns out to depend upon the manner in which the small-noise parameter is taken to zero relative to the scale-separation parameter. We study also an extension of the original model in which the relationship between the two small parameters leads to a qualitative difference in limiting behavior. The results of this paper provide an approximation, to two orders, to dynamical systems perturbed by small fractional Brownian noise and incorporating multiscale effects.

Generalized Self-Similar First Order Autoregressive Generator (GSFO-ARG) for Internet Traffic

Internet traffic data such as the number of transmitted packets and time spent on the transmission of Internet protocols (IPs) have been shown to exhibit self-similar property which can contain the long memory property, particularly in a heavy Internet traffic. Simulating this type of dataset is an important aspect of delay avoidance planning, especially when trying to mimic real-life processing of packets on the Internet. Most of the existing procedures often assumed the process follows a Gaussian distribution, and thus long memory processes such as Fractional Brownian Motion (FBM) and Fractional Gaussian Noise (FGN) among others are used. These approaches often result in estimation errors arising from the use of inappropriate distribution. However, it has been established that the distribution of Internet processes are heavy-tailed. Therefore, in this paper, a new method that is capable of generating heavy-tailed self-similar traffic is proposed based on the first-order autoregressive AR (1) process. The proposed method is compared with some of the existing methods at varying values of the self-similar index and sample sizes. The imposed self-similarity indices were estimated using the Range/Standard deviation statistic (R/S). Performance analysis was achieved using the absolute percentage errors. The results showed that the proposed method has a lower average error when compared with other competing methods.

Tail distribution of the integrated Jacobi diffusion process

In this paper, we study the distribution of the integrated Jacobi diffusion processes with Brownian noise and fractional Brownian noise. Based on techniques of Malliavin calculus, we develop a unified method to obtain explicit estimates for the tail distribution of these integrated diffusions.

Active particles with fractional rotational Brownian motion

We study the two-dimensional overdamped motion of an active particle whose orientational dynamics is subject to fractional Brownian noise, whereas its position is affected by self-propulsion and Brownian fluctuations. From a Langevin-like model of active motion with constant swimming speed, we derive the corresponding Fokker–Planck equation, from which we find the angular probability density of the particle orientation for arbitrary values of the Hurst exponent that characterizes the fractional rotational noise. We provide analytical expressions for the velocity autocorrelation function and the translational mean-squared displacement, which show that active diffusion effectively emerges in the long-time limit for all values of the Hurst exponent. The corresponding expressions for the active diffusion coefficient and the effective rotational diffusion time are also derived. Our results are compared with numerical simulations of active particles with rotational motion driven by fractional Brownian noise, with which we find an excellent agreement.

Regularity properties of the stochastic flow of a skew fractional Brownian motion

In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

Generalized Ornstein-Uhlenbeck model for active motion.

We investigate a one-dimensional model of active motion, which takes into account the effects of persistent self-propulsion through a memory function in a dissipative-like term of the generalized Langevin equation for particle swimming velocity. The proposed model is a generalization of the active Ornstein-Uhlenbeck model introduced by G. Szamel [Phys. Rev. E 90, 012111 (2014)10.1103/PhysRevE.90.012111]. We focus on two different kinds of memory which arise in many natural systems: an exponential decay and a power law, supplemented with additive colored noise. We provide analytical expressions for the velocity autocorrelation function and the mean-squared displacement, which are in excellent agreement with numerical simulations. For both models, damped oscillatory solutions emerge due to the competition between the memory of the system and the persistence of velocity fluctuations. In particular, for a power-law model with fractional Brownian noise, we show that long-time active subdiffusion occurs with increasing long-term memory.

Regularity properties of the solution to a stochastic heat equation driven by a fractional Gaussian noise on [formula omitted

We study the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S2. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.

2D Navier–Stokes equation with cylindrical fractional Brownian noise

We consider the Navier–Stokes equation on the 2D torus, with a stochastic forcing term which is a cylindrical fractional Wiener noise of Hurst parameter H. Following Albeverio and Ferrario (Ann Probab 32(2):1632–1649, 2004) and Da Prato and Debussche (J Funct Anal 196(1):180–210, 2002) which dealt with the case $$H=\frac{1}{2}$$ , we prove a local existence and uniqueness result when $$\frac{7}{16}< H<\frac{1}{2}$$ and a global existence and uniqueness result when $$\frac{1}{2}<H<1$$ .