Itô versus Stratonovich

A survey is given of the facts and fancies concerning the nonlinear Langevin or Itô equation. Actually, it is merely a pre-equation, which becomes an equation when an interpretation rule is added. The rules of Itô and Stratonovich differ, but both are mathematically consistent and therefore equally admissible conventions. The reason why they seem to lead to physical differences is that the Langevin approach used to arrive at the equation involves a tacit assumption. For systems with external noise this assumption can be justified, and it is then clear that the Stratonovich rule applies. Systems with internal noise, however, can only be properly described by a master equation and the Itô-Stratonovich controversy never enters. Afterward one is free to model the resulting fluctuations either with an Itô or a Stratonovich scheme, but that does not lead to any new information.     

Simulation of stochastic differential equations

A lot of discrete approximation schemes for stochastic differential equations with regard to mean-square sense were proposed. Numerical experiments for these schemes can be seen in some papers, but the efficiency of scheme with respect to its order has not been revealed. We will propose another type of error analysis. Also we will show results of simulation studies carried out for these schemes under our notion.     

The probability approach to numerical solution of nonlinear parabolic equations

A number of new layer methods for solving semilinear parabolic equations and reaction‐diffusion systems is derived by using probabilistic representations of their solutions. These methods exploit the ideas of weak sense numerical integration of stochastic differential equations. In spite of the probabilistic nature these methods are nevertheless deterministic. A convergence theorem is proved. Some numerical tests are presented. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 490–522, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10020     

Organizational Modes of Large-Scale Vortices in an Axisymmetric Turbulent Jet

Large vortices occurring in the axial plane of a self-similar axisymmetric turbulent jet are educed by spatially filtering PIV data. First, the instantaneous PIV frame is convolved with a Gaussian kernel to obtain a smooth (low-pass) field. Next, the low-pass field is Galilean transformed to expose the large vortices residing near the edges of the jet. Large vortices tend to organize themselves in preferred modes; evidence of ring and helical modes is revealed by Galilean transformation of the low-pass filtered field. Both modes seem to occur prominently in jets, with the helical mode being the more frequent. The overall diameter of both ring and helical modes is comparable with the local jet width. The low-pass field occasionally exhibits arrowhead shaped structures with large entrainment at their downstream tips. Stochastic estimates computed from the Galilean-transformed low-pass filtered field indicate that jet meander and a sweep-in of ambient fluid are sufficient to reconstruct large vortices. The frequency of occurrence of modes agrees with previously quoted results. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stochastic systems with cross-correlated Gaussian white noises

This paper theoretically investigates three stochastic systems with cross-correlation Gaussian white noises.Both steady state properties of the stochastic nonlinear systems and the nonequilibrium transitions induced by the cross-correlated noises are studied.The stationary solutions of the Fokker-Planck equation for three specific examples are analysed.It is shown explicitly that the cross-correlation of white noises can induce nonequilibrium transitions.     

Harmonic stochastic resonance-enhanced signal detecting in NW small-world neural network

The harmonic stochastic resonance-enhanced signal detecting in Newman-Watts small-world neural network is studied using the Hodgkin-Huxley dynamical equation with noise.If the connection probability p,coupling strength g syn and noise intensity D matches well,higher order resonance will be found and an optimal signal-to-noise ratio will be obtained.Then,the reasons are given to explain the mechanism of this appearance.     

Pathwise Convergence of a Rescaled Super-Brownian Catalyst Reactant Process

Consider the one-dimensional catalytic super-Brownian motion X (called the reactant) in the catalytic medium which is an autonomous classical super-Brownian motion. We characterize both in terms of a martingale problem and (in dimension one) as solution of a certain stochastic partial differential equation. The focus of this paper is for dimension one the analysis of the longtime behavior via a mass-time-space rescaling. When scaling time by a factor of K, space is scaled by K ^η and mass by K ^−η. We show that for every parameter value η ≥ 0 the rescaled processes converge as K→ ∞ in path space. While the catalyst’s limiting process exhibits a phase transition at η = 1, the reactant’s limit is always the same degenerate process.      

The Averaged Robbins – Monro Method for Linear Problems in a Banach Space

We consider a recursive method of Robbins–Monro type to solve the linear problem Ax=V in a Banach space. The bounded linear operator A and the vector V are assumed to be observable with some noise only. According to Polyak and Ruppert we use gains converging to zero slower than 1/n and take the average of the iterates as an estimator for the solution of the linear problem. Under weak conditions on the noise processes almost sure and distributional invariance principles are shown.     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

Stability of a quadratic Jensen type functional equation in the spaces of generalized functions

Making use of the fundamental solution of the heat equation we find the solution and prove the stability theorem of the quadratic Jensen type functional equation