Multiplicative noise, Fokker-Planck equation

Having illustrated the problem associated with multiplicative noise we will now illustrate how the procedure is applied to obtain the drift and diffusion coefficients for the two-dimensional Fokker-Planck equation in phase space for a free Brownian particle and for the Brownian motion in a one-dimensional potential. This equation is often called the Kramers equation or Klein-Kramers equation [31]. 

The most important difference between the nonlinear Fokker-Planck equation (11.11) and van Kampen s expansion (11.7) is in the diffusion term. It is constant in van Kampen s expansion - describing additive noise - and linear in (f> and 0 in the nonlinear Fokker-Planck equation thus describing multiplicative noise. As expected intuitively, the results in figure 11.2 show a better agreement between the nonlinear Fokker-Planck equation and... 

At a constant Ca concentration, the main difference between van Kampen s expansion and the nonlinear Fokker-Planck equation is in the character of fluctuations. They correspond to additive noise for the D expansion and to multiplicative noise in the latter approach. Although the noise is intrinsically multiplicative, van Kampen s expansion provides a reasonable approximation, which improves with increasing base level and growing IP3 concentration. It opens up the opportunity for further studies since the Q expansion is the only method that yields analytic expressions for the probability density and all higher moments. That distinguishes it from the master equation and the nonlinear Fokker-Planck equation, for which only the first moment is directly accessible. 

The fluctuation is state-dependent, i.e. the noise is multiplicative. Equation (5.158) can be associated with the Fokker-Planck equation ... 

The effect of additive as well as multiplicative noise by the centre-manifold approach was studied by Knobloch Wiesenfeld (1983). The main point of their procedure is that in the vicinity of bifurcation points a reduced Fokker-Planck equation is sufficient to describe the dynamics (after a short relaxation time). 

If the deterministic system is subject to multiplicative noise the stationary solution of the reduced Fokker-Planck equation is... 

The probability density of the response state vector of a nonlinear system under the excitation of Gaussian white noises is governed by a parabolic partial differential equation, called the Fokker-Planck equation. Exact solutions to such equations are difficult especially when both parametric (multiplicative) and external (additive) random excitations are present. In this paper, methods of solution for response vectors at the stationary state are discussed under two schemes based on the concept of detailed balance and the concept of generalized stationary potential, respectively. It is shown that the second scheme is more general and includes the first scheme as a special case. Examples are given to illustrate their applications. 

When parametric (or multiplicative) white noise excitations are also present, namely, the excitations also appear in the coefficients of the unknowns in the equations of motion, solution to a reduced Fokker-Planck equation becomes... 

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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