Stable Levy Noise Research Articles

Statistical solution to SDEs with $$\alpha $$-stable Lévy noise via deep neural network

The probability density function of the stochastic differential equations with $$\alpha $$ -stable Levy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.

Strong solutions for the stochastic Navier-Stokes equations on the 2D rotating sphere with stable Lévy noise

The Navier-Stokes equation with rough data arises in many problems of fluid dynamics but mathematical analysis of such problems is notoriously difficult. In this paper we consider a two-dimensional fluid moving on the surface of a rotating sphere under the influence of an impulsive force that is very irregular in time. More precisely, we assume that the impulsive force is associated to a Brownian Motion subordinated by a stable subordinator. Then we prove the existence and uniqueness of a strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by a stable Lévy noise. This strong solution turns out to exist globally in time.

Modulating bifurcations in a self-sustained birhythmic system by $$\varvec{\alpha }$$-stable Lévy noise and time delay

Birhythmical nature has been widely encountered and has aroused considerable interest in controlling dynamic behaviors of a self-sustained birhythmic system. However, researchers focus a lot on bifurcations induced by idealized Gaussian noises, the Gaussianity of which can be violated in natural phenomena. The birhythmic oscillator is well suited for modeling biological systems, especially for simulating enzymatic reactions. There are many relevant articles, but few examine the effects of non-Gaussian noises on dynamics of birhythmic systems. $$\alpha $$-stable Levy noise is more appropriate to characterize complicated biological surroundings. Besides, a few investigations simultaneously introduce the Levy noise and the inevitable time delay. Control of bifurcations in a birhythmic system driven by $$\alpha $$-stable Levy noise and time delay is numerically studied in this work, with three noise and two delay parameters treated as control parameters. Modulating the stability index and noise intensity can govern dynamics of the system, but regulating the skewness parameter cannot. More abundant bifurcations are available from adjusting the strength of delayed feedback and time delay. These results may be conducive to further exploring bifurcations in the real-world applications.

Synchronisation of networked Kuramoto oscillators under stable Lévy noise

Abstract We study the Kuramoto model on several classes of network topologies examining the dynamics under the influence of Levy noise. Such noise exhibits heavier tails than Gaussian and allows us to understand how ’shocks’ influence the individual oscillator and collective system behaviour. Skewed α -stable Levy noise, equivalent to fractional diffusion perturbations, are considered. We perform numerical simulations for Erdős–Renyi (ER) and Barabasi–Albert (BA) scale free networks of size N = 1000 while varying the Levy index α for the noise. We find that synchrony now assumes a surprising variety of forms, not seen for Gaussian-type noise, and changing with α : a noise-generated drift, a smooth α dependence of the point of cross-over of ER and BA networks in the degree of synchronisation, and a severe loss of synchronisation at low values of α . We also show that this robustness of the BA network across most values of α can also be understood as a consequence of the Laplacian of the graph working within the fractional Fokker–Planck equation of the linearised system, close to synchrony, with both eigenvalues and eigenvectors alternately contributing in different regimes of α .

Stochastic Resonance in Continuous and Spiking Neuron Models With Levy Noise

Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jump-noise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the ItO calculus show that white Levy noise will benefit subthreshold neuronal signal detection if the noise process's scaled drift velocity falls inside an interval that depends on the threshold values. These results generalize earlier "forbidden interval" theorems of neuronal "stochastic resonance" (SR) or noise-injection benefits. Global and local Lipschitz conditions imply that additive white Levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation (SDE). Simulation results show that the same noise benefits still occur for some infinite-variance stable Levy noise processes even though the theorems themselves apply only to finite-variance Levy noise. The proves the two ItO-theoretic lemmas that underlie the new Levy noise-benefit theorems.

On regular points in Burgers turbulence with stable noise initial data

Abstract We study the set of regular points (i.e. the points which have not been involved into shocks up to time t ) for the inviscid Burgers equation in dimension 1 when initial velocity is a stable Levy noise. We prove first that when the noise is not completely asymmetric and has index α ∈(1/2,1), the set of regular points is discrete a.s. and regenerative. Then, we show that in the case of the Cauchy noise, the set of regular points is uncountable, with Minkowsky dimension 0.

Structure of Shocks in Burgers Turbulence¶with Stable Noise Initial Data

Burgers equation can be used as a simplified model for hydrodynamic turbulence. The purpose of this paper is to study the structure of the shocks for the inviscid equation in dimension 1 when the initial velocity is given by a stable Levy noise with index α∈ (1/2,2]. We prove that Lagrangian regular points exist (i.e. there are fluid particles that have not participated in shocks at any time between 0 and t) if and only if α≤ 1 and the noise is not completely asymmetric, and that otherwise the shock structure is discrete. Moreover, in the Cauchy case α= 1, we show that there are no rarefaction intervals, i.e. at time t >0$, there are fluid particles in any non-empty open interval.

Large-Deviations Estimates in Burgers Turbulence with Stable Noise Initial Data

We consider the inviscid Burgers equation where the initial datum is given by a stable (Levy) noise. The asymptotic behavior of the tail distribution of the solution is described; the decay is much faster in the case when the stable noise is completely skewed to the left.