Stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise

The current paper is devoted to the study of the stochastic stability of FitzHugh-Nagumo systems perturbed by Gaussian white noise. First, the dynamics of stochastic FitzHugh-Nagumo systems are studied. Then, the existence and uniqueness of their invariant measures, which mix exponentially are proved. Finally, the asymptotic behaviors of invariant measures when size of noise gets to zero are investigated.

Suppressive Influence of Time- Space White Noise on the Explosion of Solutions of Stochastic Fokker- Planck Delay Differential Equations

It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.

Auto-Bcklund transformation and exact solutions of Wick-type stochastic Burgers equation

Burgers equation in random environment is studied. In order to give the exact solutions of random Burgers equation, we only consider the Wick-type stochastic Burgers equation which is the perturbation of the Burgers equation with variable coefficients by white noise W（t）=Bt, where Bt is a Brown motion. The auto-Baecklund transformation and stochastic soliton solutions of the Wick-type stochastic Burgers equation are shown by the homogeneous balance and Hermite transform. The generalization of the Wick-type stochastic Burgers equation is also studied.

Stochastic Stability of Burgers Equation

The current paper is devoted to stochastic Burgers equation with driving forcing given by white noise type in time and periodic in space. Motivated by the numerical results of Halter and Voss, we prove that the Burgers equation is stochastic stable in the sense that statistically steady regimes of ：fluid flows of stochastic Burgers equation converge to that of determinstic Burgers equation as noise tends to zero.

Stochastic Stability of FitzHugh-Nagumo Systems in Infinite Lattice Perturbed by Gaussian White Noise

The current paper is devoted to the study of stochastic stability of FitzHugh-Nagumo systems in infinite lattice perturbed by Gaussian white noise. We first study the dynamics of stochastic FitzHugh-Nagumo systems, then prove the existence and uniqueness of their equilibriums, which mix exponentially. Finally, we investigate asymptotic behavior of equilibriums when the size of noise gets to zero.

On the Effects of Different Interpretations of Stochastic Differential Equations

This paper addresses the problem of the interpretation of the stochastic differential equations (SDE). Even if from a theoretical point of view, there are infinite ways of interpreting them, in practice only Stratonovich’s and Itô’s interpretations and the kinetic form are important. Restricting the attention to the first two, they give rise to two different Fokker-Planck-Kolmogorov equations for the transition probability density function (PDF) of the solution. According to Stratonovich’s interpretation, there is one more term in the drift, which is not present in the physical equation, the so-called spurious drift. This term is not present in Itô’s interpretation so that the transition PDF’s of the two interpretations are different. Several examples are shown in which the two solutions are strongly different. Thus, caution is needed when a physical phenomenon is modelled by a SDE. However, the meaning of the spurious drift remains unclear.

分式白噪声(H>1/2)驱动的随机Burgers方程(英文)

考虑了一类由参数H>1/2的分式噪声驱动的一维随机Burgers方程,利用压缩映射原理证明了其取值于L2([0,1])的局部解的存在唯一性.

A Family of the Random Attractors for a Class of Generalized Kirchhoff-Type Equations

In this paper, we studied the existence of a family of the random attractor for a class of generalized Kirchhoff-type equations with a strong dissipation term. Firstly, according to Ornstein-Uhlenbeck process, we transformed the equation into a stochastic equation with random variables and multiplicative white noise. Secondly, we proved the existence of a bounded random absorbing set. Finally, by using the isomorphic mapping method and the compact embedding theorem, we get the stochastic dynamical system with a family of random attractors.

Effect of Gaussian Noise Amplitude on Translation Error Estimated Using Double-Wayland Algorithm

The Wayland algorithm has been improved in order to evaluate the degree of visible determinism for dynamical systems that generate time series. The objective of this study is to show that the Double-Wayland algorithm can distinguish between time series generated by a deterministic process and those generated by a stochastic process. The authors conducted numerical analysis of the van der Pol equation and a stochastic differential equation as a deterministic process and a Ganssian stochastic process, respectively. In case of large S/N ratios, the noise term did not affect the translation error derived from time series data, but affected that from the temporal differences of time series. In case of larger noise amplitudes, the translation error from the differences was calculated to be approximately 1 using the Double-Wayland algorithm, and it did not vary in magnitude. Furthermore, the translation error derived from the differenced sequences was considered stable against noise. This novel algorithm was applied to the detection of anomalous signals in some fields of engineering, such as the analysis of railway systems and bio-signals.

INFLUENCE OF NOISE AND DELAY ON REACTION-DIFFUSION RECURRENT NEURAL NETWORKS

In this paper, the influence of the noise and delay upon the stability property of reaction-diffusion recurrent neural networks （RNNs） with the time-varying delay is discussed. The new and easily verifiable conditions to guarantee the mean value exponential stability of an equilibrium solution are derived. The rate of exponential convergence can be estimated by means of a simple computation based on these criteria.

Improved error estimates for a modified exponential Euler method for the semilinear stochastic heat equation with rough initial data

A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.

平稳噪声驱动下稳定流形的逼近

文章研究一类受平稳噪声驱动的随机微分方程的动力学行为的逼近.首先证明该系统的解逐点收敛到由加性白噪声驱动的随机微分方程的解,然后证明它们的稳定流形的存在性及其逼近关系.

p-moment stability of stochastic impulsive differential equations and its application in impulsive control

The exponential p-moment stability of stochastic impulsive differential equations is addressed. A new theorem to ensure the p-moment stability is established for the trivial solution of the stochastic impul- sive differential system. As an application of the theorem proposed, the problem of controlling chaos of Lorenz system which is excited by parameter white-noise excitation is considered using impulsive control method. Finally, numerical simulation results are given to verify the feasibility of our approach.

Auxiliary Equations Approach for the Stochastic Unsteady Navier-Stokes Equations with Additive Random Noise

This paper presents a Martingale regularization method for the stochas-tic Navier–Stokes equations with additive noise.The original system is split into two equivalent parts,the linear stochastic Stokes equations with Martingale solution and the stochastic modified Navier–Stokes equations with relatively-higher regular-ities.Meanwhile,a fractional Laplace operator is introduced to regularize the noise term.The stability and convergence of numerical scheme for the pathwise modified Navier–Stokes equations are proved.The comparisons of non-regularized and reg-ularized noises for the Navier–Stokes system are numerically presented to further demonstrate the efficiency of our numerical scheme.

An LDG Method for Stochastic Cahn-Hilliard Type Equation Driven by General Multiplicative Noise Involving Second-Order Derivative

In this paper,we propose a local discontinuous Galerkin(LDG)method for themulti-dimensional stochastic Cahn-Hilliard type equation in a general form,which involves second-order derivative Du in the multiplicative noise.The stability of our scheme is proved for arbitrary polygonal domain with triangular meshes.We get the sub-optimal error estimate O(h^(k))if the Cartesian meshes with Q^(k) elements are used.Numerical examples are given to display the performance of the LDG method.

Poisson Stable Solutions for Stochastic Differential Equations with Lévy Noise

In this paper,we use a unified framework to study Poisson stable(including stationary,periodic,quasi-periodic,almost periodic,almost automorphic,Birkhoff recurrent,almost recurrent in the sense of Bebutov,Levitan almost periodic,pseudo-periodic,pseudo-recurrent and Poisson stable)solutions for semilinear stochastic differential equations driven by infinite dimensional L′evy noise with large jumps.Under suitable conditions on drift,diffusion and jump coefficients,we prove that there exist solutions which inherit the Poisson stability of coefficients.Further we show that these solutions are globally asymptotically stable in square-mean sense.Finally,we illustrate our theoretical results by several examples.

基于随机微分方程的流域汇流模型

针对水文过程中存在的许多随机不确定性因素,本文以Nash模型为基础,利用随机微分方程理论,在确定性汇流模型中引入随机输入项,建立了输入具有白噪声特性的随机Nash汇流模型,求解得到出流过程均值和方差的解析解和数值解,并利用出流过程的均值和方差确定各时刻出流过程的概率分布。该方法借助于各时刻的方差得到流量过程的分布概率,从而考虑预报的不确定性,为防洪决策提供预报值的不确定度,有利于降低水库调洪风险率。

具有HollingⅡ型功能性作用函数的随机恒化器模型的渐近性态

考虑了一类营养的转化率受到随机噪声干扰,具有HollingⅡ型功能性反应函数的随机恒化器模型.通过构造Liapunov函数,利用停时、伊藤公式证明了模型正解的全局存在唯一性.研究了模型解的长期渐近性态,主要揭示在不同条件下模型的解围绕其相应确定性模型的各类平衡点的振荡行为.

随机镇定与反镇定概述

本文概述了随机镇定与反镇定理论的研究现状.主要回顾了一个微分方程的随机镇定与反镇定普遍理论及其发展,并围绕该理论的应用和扩展从四个方面阐述连续时间系统噪声镇定理论的当前发展概况.此外,本文还概述了离散时间系统随机镇定方面的最新进展.

具有可加噪声的耗散KdV型方程的随机吸引子

考虑具有可加噪声的耗散KdV型方程在一维有界区域上的渐进行为,主要目的是建立整体吸引子的存在性.首先证明解的存在性和唯一性并得到解的先验估计,然后讨论该系统的随机吸收集的存在性,最后证明整体吸引子在有界确定性集合的范围内存在.