Multiplicative noise

K. Lindenberg, K. E. Shuler, V. Seshadri, and B. J. West, Langevin equations with multiplicative noise theory and applications to physical processes, in Probabilistic Analysis and Related Topics, Vol. 3, A. T. Bharucha-Reid (ed.), Academic Press, San Diego, 1983, pp. 81-125. 

SPC techniques are hardly affected by additive noise and multiplicative noise is absent. However, subtractive noise due to the collection efficiency and transmission of optics and the quantum efficiency of the detector do play a role. In addition, at high count rates, the efficiency goes down due to pileup effects. 

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. 

This equation is commonly characterized by the term additive noise , but a transformation of the variable y changes the additive noise into the multiplicative noise of (4.5), compare XVI. 1. 

III. Nonlinear equations, i.e., equations that are nonlinear in the unknown function u. Here the distinction between additive and multiplicative is moot. In section 4 they will be transformed into linear equations with multiplicative noise. 

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. 

As a result of AEP, the initial system of the set of Eqs. (81) is reduced to the equation describing the diffusional motion of a Brownian particle which undergoes the action of an additive and a multiplicative noise (with intensities D and Q, respectively) in the presence of a renormalized bounding potential, Eq. (90). The Markovian l t corresponds to X 00. If we take such a limit at a ed value of y, = d, and the case studied by Htoggi is recovered. Of course, having neglected the condition X c y we have reduced the problem to a trivial diffusional (lowest-order) approximation. 

Figure 10. k as a function of the intensity of the multiplicative noise, k is defined as (0) with (jc(t))/(ac(0)). The two arrows on the left denote the point where the discrete branch of the eigenvalue spectrum disappears (see Schenzle and Brand ). The two arrows on the right denote the phase transition threshold. [Taken from S. Faetti et al., Z. Ffiys., B47, 353 (1982).]... 

EXPERIMENTAL INVESTIGATION ON THE EFFECT OF MULTIPLICATIVE NOISE BY MEANS OF ELECTRIC CIRCUITS... 

A. The Case of a Purely Multiplicative Noise Equilibrium Distribution.460... 

Note, however, that Eq. (4.21) loses its validity in the case of strong multiplicative noise, as it is based on a linearization assumption. The remarks above have to be related to a purely qualitative level of interpretation. 

In ref. 39 a computer calculation of the rate of escape was done based on the CFP of Chapter III. The agreement between this calculation and Eq. (4.2S) is good, as shown in Fig. 11. Note that the combination of weak inertia with multiplicative noise results in a finite rate of escape which is rigorously forbidden by the AEP when no additive noise is present. 

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. 

We form the average of Eqs. (5.20)-(5.22) noting that (L(r)) will vanish throughout because, in the inertia corrected Langevin equation, M is statistically independent of the white noise field h(r). This is not, however, true of the noninertial Langevin equation where the multiplicative noise term L(f) contributes a noise induced drift term to the average (see Section VI). The averages so formed are... 

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 =... 

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