Harmonic analysis of the Langevin equation

If R(t) satisfies the Markovian property (8.20), it follows from the Wiener-Khintchine theorem (7.76) that its spectral density is constant 

Equation (8.28) implies that all frequencies are equally presented in this random force spectrum. A stochastic process of this type is called a white noise. 

From the spectral density of R t) we can find the spectral density of stochastic observables that are related to R via linear Langevin equations. For example, consider the Langevin equation (8.13) with V(x) = /2)mo (the so called Brownian harmonic oscillator) 

The power spectrum of any of these stationary processes is given by Eq. (7.74). Therefore, Eq. (8.33) implies a relation between these spectra 

In the absence of an external potential (free Brownian motion o = 0) Eq. (8.34b) becomes 

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