Universality in stochastic enzymatic futile cycle

The enzymatic futile cycle model, which is believe to regulate functional mechanism of bimolecular networks and associated signal processing, is solved analytically within the stochastic framework. The obtained probability distributions of substrate and product at stochastic to deterministic transition exhibit Poisson distribution, and with large ⟨x⟩ limit Normal distribution which are independent of thermodynamic variables indicating universal behavior of molecular distribution. The dynamics of the substrate and product, by simulating the reaction network using stochastic simulation algorithm, exhibit switching mechanism driven by noise in the system. We also observe various distinct noise driven patterns which may correspond to various cellular states. We propose that this noise induce switching mechanism and patterns could be a key element to regulate and control signal processing in molecular networks of cellular system, and may exhibit in the phenotype.     

Stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses

The dependence of stochastic resonance in small-world neuronal networks with hybrid electrical–chemical synapses on the probability of chemical synapse and the rewiring probability is investigated. A subthreshold periodic signal is imposed on one single neuron within the neuronal network as a pacemaker. It is shown that, irrespective of the probability of chemical synapse, there exists a moderate intensity of external noise optimizing the response of neuronal networks to the pacemaker. Moreover, the effect of pacemaker driven stochastic resonance of the system depends largely on the probability of chemical synapse. A high probability of chemical synapse will need lower noise intensity to evoke the phenomenon of stochastic resonance in the networked neuronal systems. In addition, for fixed noise intensity, there is an optimal chemical synapse probability, which can promote the propagation of the localized subthreshold pacemaker across neural networks. And the optimal chemical synapses probability turns even larger as the coupling strength decreases. Furthermore, the small-world topology has a significant impact on the stochastic resonance in hybrid neuronal networks. It is found that increasing the rewiring probability can always enhance the stochastic resonance until it approaches the random network limit.     

A study on stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point with the moment method

This paper analyzes the stochastic resonance induced by a novel transition of one-dimensional bistable system in the neighborhood of bifurcation point with the method of moment, which refer to the transition of system motion among a potential well of stable fixed point before bifurcation of original system and double-well potential of two coexisting stable fixed points after original system bifurcation at the presence of internal noise. The results show: the semi-analytical result of stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point may be obtained, and the semi-analytical result is in accord with the one of Monte Carlo simulation qualitatively, the occurrence of stochastic resonance is related to the bifurcation of noisy nonlinear dynamical system moment equations, which induce the transfer of energy of ensemble average (Ex) of system response in each frequency component and make the energy of ensemble average of system response concentrate on the frequency of input signal, stochastic resonance occurs.     

Noise-induced vegetation transitions in the Grazing Ecosystem

The Grazing Ecosystem is a special case of predator-prey systems, which has attracted widespread attention since a long time ago. Because of the ubiquity of noise, there is a growing need to research the influences of noise on the Grazing Ecosystem. This paper is devoted to investigating the transition behaviors of the high vegetation biomass in the Grazing Ecosystem subjected to Gaussian noise and Lévy noise, respectively. Firstly, the original system is translated into the Itô stochastic differential equation, which is utilized to derive the analytical expression of the escape probability through the Dirichlet boundary value problems. Then the transitions between the two vegetation potential wells are explored by calculating the size of the stochastic basin of attraction based on the escape probability. The comparison between the analytical results and the ones through Monte Carlo simulations shows that the proposed method works very well. It turns out that the Gaussian white noise intensity, Lévy noise stability parameter and herbivore density have different impact mechanisms on the basin stability of high density vegetation in the stochastic Grazing Ecosystem.     

Multifractal detrended fluctuation analysis of analog random multiplicative processes

We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function.     

Natural gradient algorithm for stochastic distribution systems with output feedback

In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm.     

Sufficient conditions for the eventual strong Feller property for degenerate stochastic evolutions

The strong Feller property is an important quality of Markov semigroups which helps for example in establishing uniqueness of invariant measure. Unfortunately degenerate stochastic evolutions, such as stochastic delay equations, do not possess this property. However the eventual strong Feller property is sufficient in establishing uniqueness of invariant probability measure. In this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong Feller property. The results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay, with an application in neural networks.     

Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise

This paper is devoted to study a class of stochastic differential equations with Lévy noise. In comparison to the standard Gaussian noise, Lévy noise is more versatile and interesting with a wider range of applications. However, Lévy noise makes the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. We propose several sufficient conditions under which we investigate the long-time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability. Also, we discuss two types of continuity of the solution: continuous in probability and continuous in the pth moment. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.     

Stochastic delay Lotka-Volterra model

We reveal in this paper that the environmental noise will not only suppress a potential population explosion in the stochastic delay Lotka-Volterra model but will also make the solutions to be stochastically ultimately bounded. To reveal these interesting facts, we stochastically perturb the delay Lotka-Volterra model into the Itô form , and show that although the solution to the original delay equation may explode to infinity in a finite time, with probability one that of the associated stochastic delay equation does not. We also show that the solution of the stochastic equation will be stochastically ultimately bounded without any additional condition on the matrix A.     

由时变Lévy噪声驱动的随机微分方程的平均值原理

该文讨论了一类由时变Lévy噪声驱动的随机微分方程(LSDE)的平均值原理,提出了其均值化方程,在均方和以概率意义下得到了均值化方程的解收敛到原LSDE的解,给出了一个具体例子.     

Positive Harris recurrence for degenerate diffusions with internal variables and randomly perturbed time-periodic input

We consider a multidimensional time-homogeneous dynamical system and add a randomly perturbed time-dependent deterministic signal to some of its components, giving rise to a high-dimensional system of stochastic differential equations, which is driven by possibly very low-dimensional noise. Equations of this type commonly occur in biology when modeling neurons or in statistical mechanics for certain Hamiltonian systems. We provide verifiable conditions on the original deterministic dynamical system under which the solution to the respective stochastic system features a point in the interior of its state space, which can be proved to be attainable by deterministic control arguments, and at which a local Hörmander condition holds. Together with a Lyapunov condition, it follows that the corresponding process is positive Harris recurrent.     

The exponential integrator scheme for stochastic partial differential equations: Pathwise error bounds

We present an error analysis for the pathwise approximation of a general semilinear stochastic evolution equation in d dimensions. We discretise in space by a Galerkin method and in time by using a stochastic exponential integrator. We show that for spatially regular (smooth) noise the number of nodes needed for the noise can be reduced and that the rate of convergence degrades as the regularity of the noise reduces (and the noise becomes rougher).     

Impact of two types of time delays and cross-correlated multiplicative and additive noises on stability and stochastic resonance for a metapopulation system

In this paper, we investigate the stability and the shift between the extinction state and the stable one of a large density and the stochastic resonance (SR) for a metapopulation system subjected to two types of time delay terms, cross-correlation noises and multiplicative signal. By using the fast descent method and the method of small delay approximation, the expressions of the effective potential function and the signal-to-noise ratio (SNR) are obtained. We denote by Q the intensity of the multiplicative noise, and M the intensity of the additive noise, θ and τ the two time delay terms introduced into the metapopulation system. Our main results show some facts that time delay θ and the strength of correlation noise λ can restrain the development of the metapopulation, while the other term of time delay τ can accelerate the expansion of the population from the extinction state to the large stable one. We discover that it is possible to enhance the signal-to-noise ratio by adjusting the intensities of the multiplicative, additive noises and the time delays of the stochastic metapopulation system     

Contribution of calcium fluxes to astrocyte spontaneous calcium oscillations in deterministic and stochastic models

Spontaneous Ca²⁺ oscillation is a special kind of astrocyte signal that is crucial in several brain functions. However, the underlying mechanism of its initiation is still unclear. Experimental investigations reveal quite different results of the contribution of Ca²⁺ influxes from the extracellular space. We developed a new biophysical model, which includes two kinds of extracellular Ca²⁺ influx: through voltage gated Ca²⁺ channels and through capacitative Ca²⁺ entry. Besides, two stochastic models are constructed to discuss the influence of channel noise in voltage gated influx and stochastic IP₃ receptor dynamics on the contribution of Ca²⁺ fluxes. Simulation results reveal that the contribution of a given flux can be considerably strengthened or weakened by special cellular states, such as the change of membrane potential, inter-flux feedback; the special cellular condition (e.g. endoplasmic reticulum depletion, other flux inhibition). In stochastic environment, the contribution of fluxes may be strongly altered and the respond of different fluxes varies qualitatively. Our results may help to understand the different results obtained in experiments.     

Statistical analysis of stochastic resonance with ergodic diffusion noise

A subthreshold signal is transmitted through a channel and may be detected when some noise--with known structure and proportional to some level--is added to the data. There is an optimal noise level, called stochastic resonance that corresponds to the highest Fisher information in the problem of estimation of the signal. As noise we consider an ergodic diffusion process and the asymptotic is considered as time goes to infinity. We propose consistent estimators of the subthreshold signal and we solve further a problem of hypotheses testing. We also discuss evidence of stochastic resonance for both estimation and hypotheses testing problems via examples.     

Robustness in stochastic programs with risk constraints

This paper is a contribution to the robustness analysis for stochastic programs whose set of feasible solutions depends on the probability distribution?P. For various reasons, probability distribution P may not be precisely specified and we study robustness of results with respect to perturbations of?P. The main tool is the contamination technique. For the optimal value, local contamination bounds are derived and applied to robustness analysis of the optimal value of a portfolio performance under risk-shaping CVaR constraints. A?new robust portfolio efficiency test with respect to the second order stochastic dominance criterion is suggested and the contamination methodology is exploited to analyze its resistance with respect to additional scenarios.     

An Averaging Principle for Multivalued Stochastic Differential Equations

The averaging principle for multivalued stochastic differential equations (MSDEs) driven by Brownian motion with Brownian noise is investigated. An averaged MSDEs for the original MSDEs is proposed, and their solutions are quantitatively compared. Under suitable assumptions, it is shown that the solution of the MSDEs converges to that of the original MSDEs in the sense of mean square and also in probability. Two examples are presented to illustrate the averaging principle.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

Approximation properties for solutions to non‐Lipschitz stochastic differential equations with Lévy noise

In this paper, we consider the non‐Lipschitz stochastic differential equations and stochastic functional differential equations with delays driven by Lévy noise, and the approximation theorems for the solutions to these two kinds of equations will be proposed respectively. Non‐Lipschitz condition is much weaker condition than the Lipschitz one. The simplified equations will be defined to make its solutions converge to that of the corresponding original equations both in the sense of mean square and probability, which constitute the approximation theorems. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit conditions for asymptotic stability of stochastic Liénard-type equations with Markovian switching

This paper deals with the boundedness in probability and convergence for solutions of stochastic Liénard-type equations. Some explicit conditions to ensure that all solution is bounded in probability and convergent to some fixed sets are obtained. We also show that white noise can affect the boundedness of corresponding deterministic Liénard equations.