Langevin equation noise properties

An interesting investigation on the influence of multiplicative non-white noise in an analog circuit simulating a Langevin equation of a Brownian particle in a double-well potential has been carried out by Sancho et al. This device allowed them to study the stationary properties as a function of the noise correlation time. Theory in a white-noise limit cannot provide a satisfactory explanation for experimental results such as a relative maximum of the probability distribution and the maximum position in the stationary distribution for noises of weak intensity. 

Langevin equation that the noise A(t) has the following statistical properties ... 

As seen above, a solution of the Langevin equation (Equation 6.50) (which is a nonlinear partial differential equation with random noise) consists of constructing the correlation functions of f (t) from the equation and then averaging the expressions with the help of the properties of the noise r(t). An alternative method of solution is to find the probability distribution function P(x, t) for realizing a situation in which the random variable f (t) has the particular value X at time t. P(x, t) is an equivalent description of the stochastic process f (0 and is given by the Fokker-Planck equation (Chandrasekhar 1943, Gardiner 1985, Risken 1989, Redner 2001, Mazo 2002)... 

Here the average was done both on 7 (t) and on the phase oscillations, and we have used again Eq. (14.128). Equation (14.132) is a Langevin-type equation characterized by a random noise Stf) whose statistical properties are given by (14.134). It is similar to others we had before, with one important difference The random force S(d) depends on the state K of the system. We can repeat the procedure we used in Section 8.4.4 to get a Fokker-Planck equation, but more caution has to be exercised in doing so. The result for the probability P(. K, t dK to find the action in the range K... K dK is... 

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