Noise gaussian white

The mean values of the. (t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the a. is also a stationary Gaussian-... 

Since the f. are linearly related to the they are also stationary Gaussian white noises. This property is explicitly expressed by... 

Unconstrained ML for Gaussian white noise. For Gaussian stationary noise, the covariance matrix is diagonal and proportional to the identity matrix ... 

The regularized solution is easy to obtain in the case of Gaussian white noise if we choose a smoothness prior measured in the Fourier space. In this case, the MAP penalty writes ... 

In our case, i.e. gaussian white noise and smoothness prior, CV and GCV have the same expression ... 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

QUANTUM AND CLASSICAL DYNAMICS OF NONINTEGRABLE SYSTEMS 143 where l(t) is a Gaussian white-noise source with noise average... 

Alternatively one may postulate that all higher cumulants are zero. This specifies all stochastic properties of L(t) in terms of the single parameter F. The L(t) defined in this way is called Gaussian white noise. From the mathematical point of view it does not really exist as a stochastic function (no more than the delta function exists as a function) and in physics it never really occurs but serves as a model for any rapidly fluctuating force. 

Exercise. Show that the property of L(t) to be Gaussian white noise is expressed by the following identity of its characteristic functional ... 

Remark. The white noise limit is not sufficiently defined by just saying rc 0. We have to construct a sequence of processes which in this limit reduce to Gaussian white noise. For that purpose take a long time interval (0, T) and a Poisson distribution of time points Ta in it with density v. To each Ta attach a random number ca they are independent and identically distributed, with zero mean. Consider the process... 

This is the generating function of Gaussian white noise, see (3.2). 

In the standard overdamped version of the Kramers problem, the escape of a particle subject to a Gaussian white noise over a potential barrier is considered in the limit of low diffusivity—that is, where the barrier height AV is large in comparison to the diffusion constant K [14] (compare Fig.6). Then, the probability current over the potential barrier top near xmax is small, and the time change of the pdf is equally small. In this quasi-stationary situation, the probability current is approximately position independent. The temporal decay of the probability to find the particle within the potential well is then given by the exponential function [14, 22]... 

Here, m and rai are the mass and position vector of beads, respectively. is the friction tensor, which is assumed to be isotropic for simplicity in our simulation, that is, = Fl, where I is the unit dyad and r = 0.5t 1 (t = cr(m/ )° 5j (Grest, 1996). Further, f aj is the Brownian random force, which obeys the Gaussian white noise, and is generated according to the fluctuation—dissipation theorem ... 

The term is the frictional tensor. We assume it to be isotropic ( IT (I is unit tensor), is assumed to be Gaussian white noise that is generated according to the fluctuation-dissipation theorem [173,174]... 

S is the disease variable and gw accounts for Gaussian white noise. 

Gaussian white noise is implemented according to the Box-Mueller algorithm [101]. Ic is the coupling current in network simulations which accounts for electrotonic gap-junction connections between the neurons. For bidirectional coupling of two neurons the coupling current Ic are of the form [94] ... 

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