Stochastic processes White and colored noises

Next we consider several noise sources. They are Gaussian sources if their support of values is due to a Gaussian distribution. Contrary the dichotomic telegraph process assumes two values, i.e. = A and 2 = A. We will always assume A = — A. 

The second classification of the noise classifies its temporal correlations. In case of white noise the noise is uncorrelated in time which corresponds 

The integral over the Gaussian white noise gives the Wiener process which stands for the trajectory of a Brownian particle. The integral during 

In case of a Brownian particle D is the spatial diffusion coefficient. 

The study of a Brownian particle suspended in a fluid lead also to the introduction of the exponentially correlated Ornstein-Uhlenbeck process [48], the only Markovian Gaussian non-white stochastic process [19, 22]. We present here the Langevin approach to this problem, hence we analyze the forces that act on a single Brownian particle. We suppose the particle having a mass m equal to unity, and we assume the force due to the hits with thermal activated molecules of the fluid to be a stochastic variable. Moreover, due to the viscosity of the fluid, a friction force proportional to the velocity of the particle has to be considered. All this yields the following equation 

---