Stochastic noise multiplicative

F(f) is usually named multiplicative stochastic noise, since it multiplies the function g x), which depends on the variable of interest x. Due to the multiplicative nature of the stochastic force F t), it is not immediately... 

I is an important parameter of this theory which expresses the ratio of the correlation time of the multiplicative stochastic noise to that of the velocity V. The second term on the right-hand side of Eq. (1.18) is a potential term independent of the intensity of the noise < )eq, the role of which becomes increasingly irrelevant as y - oo. We cannot, however, neglect the last two terms, which express the lowest-order contribution coming from the noise i. 

Figure 4. Steady state probability distribution

A further complication that has been much studied in the literature is that of multiplicative noise in which the random force in stochastic differential equations like Eq. (1) is modified by a modulating term, i.e.,... 

A major limitation of the dissipative mechanisms involving multiplicative noise —and by extension the iGLE and WiGLE models— is that they involve equilibrium changes only in the strength of the response with respect to the instantaneous friction kernel. They do not involve a change in the response time of the solvent at equilibrium limits. Presumably the response time also changes in some systems, and the inclusion of this variation is a necessary component of the minimal class of models for nonstationary stochastic dynamics. Plow this should be included, however, is an open problem which awaits an answer. 

Note that the subdivision refers to the form of the equation, not to the process described by it the term multiplicative noise is a misnomer. There are other categories, such as stochastic partial differential equations, eigenvalue problems 0, and random boundaries ), but they will not be treated here. 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

Let us now consider stochastic motion in an OB system. In general, noise in an OB system may result from fluctuations of the incident field, or from thermal and quantum fluctuations in the system itself. We shall consider the former. The fluctuations of the intensities of the input or reference signals give rise respectively to either multiplicative or additive noise driving the phase. Both types of fluctuations can be considered within the same approach [108], Here we discuss only the effects of zero-mean white Gaussian noise in the reference signal ... 

Recently, Gillespie (2001) introduced an approximate approach, termed the r-leap method, for solving stochastic models. The main idea is the same as in the WP-KMC method. One selects a time increment r that is larger than the microscopic KMC time increment, and multiple molecular bundles of fast events occur. However, one now samples how many times each reaction will be executed from a Poisson rather than a uniform random number distribution. Prototype examples indicate that the r-leap method provides comparable noise with the microscopic KMC when the leap condition is satisfied, i.e., the time increments are such that the populations do not change significantly between time steps. 

SA of SODEs describing chemically reacting systems was introduced early on, in the case of white noise added to an ODE (Dacol and Rabitz, 1984). In addition to expected values (time or ensemble average quantities), SA of variances or other correlation functions, or even the entire pdf, may also be of interest. In other words, in stochastic or multiscale systems one may also be interested in identifying model parameters that mostly affect the variance of different responses. In many experimental systems, the noise is due to multiple sources as a result, comparison with model-based SA for parameter estimation needs identification of the sources of experimental noise for meaningful conclusions. 

Dekker has studied multiplicative stochastic processes. In his work the stochastic Liouville equation was solved explicitly through first order in an expansion in terms of correlation times of the multiplicative Gaussian colored noise for a general multidimensional weakly non-Markovian process. He followed the suggestions of refs. 17 and 18 and applied, Novikov s theorem. In the general multidimensional case, however, he improved the earlier work by San Miguel and Sancho. ... 

Figures 12c and b show the dependence of on Q for some values of i as obtained by using the experimental method of Section III. We would like to stress again that a great deal of attention has been devoted to limiting the effects of spurious additive noise from the circuit and that if the well is not exactly symmetric the multiplicative noise can itself produce a spread of the variable x. In spite of our efforts, a weak additive stochastic force proves to be present in our electrical circuit.

Another very important feature of the stochastic equations considered here, when they are subjected to RMT analysis, is their resemblance to the general formalism arrived at in the thermodynamics of nonequilibrium processes this suggests an analogy between the effects of multiplicative noise and the continuous flux of energy which maintains the systems far from equilibrium. This is considered the main characteristic of self-organizing living systems and means that multiplicative stochastic models could take on a new and fundamentally important role. 

Equation (6.19) contains multiplicative noise terms G, (M)/i (t). This poses an interpretation problem as discussed by Risken [31]. Risken has shown, taking the Langevin equation for N stochastic variables =... 

For multiplicative noise the determination of these moments requires a more detailed consideration of the stochastic integral since white noise is too irregular for Riemann integrals to be applied. Application of Stieltjes integration yields a dependence of the moments on how the limit to white noise is taken. If t) is the limit of the Ornstein-Uhlenbeck -process with r —> 0 (Stratonovich sense) the coefficients read [50]... 

Since the complexity of the physiologic system identification problem rivals its importance, we begin by demarcating those areas where effective methods and tools currently exist. The selection among candidate models is made on the basis of the following key functional characteristics (1) static or dynamic (2) Hnear or nonhnear (3) stationary or nonstationary (4) deterministic or stochastic (5) single or multiple inputs and/or outputs (6) lumped or distributed. These classification criteria do not constitute an exhaustive list but cover most cases of current interest. Furthermore, it is critical to remember that contaminating noise (be it systemic or measurement-related) is always present in an actual study, and... 

In this section, we consider the combination of stochastic perturbation with a deterministic thermostat. Methods constructed in this way can be ergodic for the canonical distribution while also providing flexibility in way equilibrium is achieved. We distinguish in (6.16) between multiplicative noise, where B = B(z) varies with z and additive noise, where B is constant. In our treatment of this topic we will only consider additive noise. The presence of multiplicative noise may complicate discretization. As we shall see, the reliance on additive noise improves the performance of discretization schemes. 

Here M is a mobility parameter, y is the shear rate, which is zero if no shear is applied, p/is a stochastic term which is distributed according to a fluctuation-dissipation theorem [13], and kjK is the reaction rate, which can be either negative for reactants or positive for products. Notice that the reactive noise can be neglected here. Different order reactions or multiple reaction terms can be added without any difficulties, but as a proof of principle we focus here on the above type of reactions. In this subsection we study the effect of combined microphase separation, shear, and reaction to gain insight in the mechanisms that are important in pathway-controlled morphology formation and in particular in reactive blending. 

Haken, H., Wunderlin, A. (1982) Slaving principle for stochastic differential equations with additive and multiplicative noise and for discrete noisy maps. Z. Phys. 47, 179 Hastings, S. P. (1976) Periodic plane waves for the Oregonator. Stud. Appl. Math. 55, 293 Herschkowitz-Kaufman, M. (1975) Bifurcation analysis of nonlinear reaction-diffusion equations II. Steady state solutions and comparison with numerical simulations. Bull. Math. Biol. 37, 589 Howard, L. N., Kopell, N. (1977) Slowly varying waves and shock structures in reaction-diffusion equations. Stud. Appl. Math. 56, 95... 

Eq. (40) has to be interpreted as a stochastic differential equation with multiplicative noise, and its solution is a difficult task ... 

The CLE is a multivariate Ito stochastic differential equation with multiple, multiplicative noise. We define the CLE again and present methods to solve it in Chapter 18, where we discuss numerical simulations of stochastic reaction kinetics. 

In the fast-continuous region, species populations can be assumed to be continuous variables. Because the reactions are sufficiently fast in comparison to the rest of the system, it can be assumed that they have relaxed to a steady-state distribution. Furthermore, because of the frequency of reaction rates, and the population size, the population distributions can be assumed to have a Gaussian shape. The subset of fast reactions can then be approximated as a continuous time Markov process with chemical Langevin Equations (CLE). The CLE is an ltd stochastic differential equation with multiplicative noise, as discussed in Chapter 13. 

A CLE is an Ito stochastic differential equation (SDE) with multiplicative noise terms and represents one possible solution of the Eokker-Planck equation. From a multidimensional Fokker-Planck equation we end up with a system of CLEs ... 

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