Doob s theorem

Exercise. Find three examples of processes, each of which has two but not three of the properties stipulated in Doob s theorem. 

Exercise. The trivial exception to Doob s theorem mentioned above is the completely random process defined by... 

Doob s theorem states that a Gaussian process is Markovian if and only if its time correlation function is exponential. It thus follows that V is a Gaussian-Markov Process. From this it follows that the probability distribution, P(V, t), in velocity space satisfies the Fokker-Planck equation,... 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

Equation (30) can be solved if Fy is sampled at each time step in some specified way. If Fy is assumed to be a Gaussian Markov process it follows from Doob s theorem that Kj(t) is an exponential function of time. Then only two parameters need be specified before Fy can be sampled from the Gaussian two-time probability distribution and these are the mean square value (Fy) and the correlation time of Fy, say xy. Equation (30) then forms a set of coupled stochastic differential equations that can be solved by methods similar to those already mentioned. 

Such a noise is easy to generate electronically, iii) Since we are interested in macroscopic systems, we will observe the system usually only on macroscopic time scales. It is then reasonable to assume that Z is a Markov process. Furthermore, it has been argued that a non-Markovian noise will not introduce any essentially new physics into the problem [5]. Properties i) - iii) uniquely specify the noise process. In the case of ii)a) we find, in light of DOOB s theorem [6], that Z is given by a stationary Ornstein-Uhlenbeck process, i.e. it obeys.the following Lan-gevin equation ... 

Berne and Pecora s demonstration is a special case of Doob s theorem(5), which shows for random processes that are also Markoff processes, such as the processes generated by the Langevin equation, that P(Ax,t) must be a Gaussian having uncorrelated sequential steps, that Eq. 9.5 is correct, and that g q, t) must be a single exponential in t. A contrapositive to Doob s theorem shows that if g q, t) is not a simple exponential then 8x is not described by a Gaussian Markoff process, and Eq. 9.5 is not applicable. An explicit calculation has been given that correctly... 

At elevated concentrations, light-scattering spectra of solutions of colloidal spheres become multimodal. Correspondingly, as required by Doob s theorem, particle motions cease to be described by a Gaussian random Markoff process. Segre, et al. report t), as obtained using a two-color, two-detector homo-... 

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