Gaussian-Markov process

A proposal based on Onsager s theory was made by Landau and Lifshitz [27] for the fluctuations that should be added to the Navier-Stokes hydrodynamic equations. Fluctuating stress tensor and heat flux temis were postulated in analogy with the Onsager theory. Flowever, since this is a case where the variables are of mixed time reversal character, tlie derivation was not fiilly rigorous. This situation was remedied by tlie derivation by Fox and Ulilenbeck [13, H, 18] based on general stationary Gaussian-Markov processes [12]. The precise fomi of the Landau proposal is confimied by this approach [14]. 

As our first example we shall define what is known as a gaussian Markov process. This process, as we shall see later, is a good model for thermal noise or vacuum-tube-generated noise that has been passed through an RC filter with time constant a 1. We begin by defining two functions / and Q as follows... 

This result can now be used to verify our earlier statement that the gaussian Markov process defined by Eq. (3-218) is a good model for RC filtered vacuum tube noise. We have already seen that vacuum tube noise is essentially gaussian (as long as n is large) and that its spectrum is essentially white .70 A reasonable model for RC filtered... 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

The last two results are rather similar to the quadratic forms given by Fox and Uhlenbeck for the transition probability for a stationary Gaussian-Markov process, their Eqs. (20) and (22) [82]. Although they did not identify the parity relationships of the matrices or obtain their time dependence explicitly, the Langevin equation that emerges from their analysis and the Doob formula, their Eq. (25), is essentially equivalent to the most likely terminal position in the intermediate regime obtained next. 

Gaussian approximation, heat flow, 60 Gaussian distribution, transition state trajectory, white noise, 206-207 Gaussian-Markov process, linear... 

Higher terms in the expansion for Z(p) can be straightforwardly calculated using the theory of Gaussian Markov processes. For example in the case of the Wiener process we obtain for the first term inside the summation of Eq. (35)... 

Let 7(t) be a stationary Gaussian Markov process. By shifting and rescaling we can ensure that Px(y) is equal to (3.10). The transition probability is Gaussian and has therefore the general form... 

Exercise. Let Y be an r-variable Gaussian Markov process. Apply a linear transformation so that Pi(y, t) exp[— %y2. Then derive for the autocorrelation matrix... 

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. ... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

The single relaxation time approximation can be applied to the angular momentum memory function in a Completely analogous way.68 Kj(t) can be interpreted as the time-correlation function of the random torque acting on the molecule. If this random torque has a Lorentzian spectrum or, more restrictively, is a Gaussian-Markov process, Kj(t) is exponential. 

A. Dhar and S. N. Majumdar, Residence time distribution for a class of gaussian Markov processes. Phys. Rev. E 59 6413-6418 (1999). 

The mean values of the c. t) are zero and each is assumed to be stationary Gaussian white noise. The linearity of these equations guarantees that the random process described by the is also a stationary Gaussian-Markov process [12]. Denoting the inverse of R j by L.j and using the definition... 

The model described in Brennan and Schwartz (1979) uses the short rate and the long-term interest rate to specify the term structure. The long-term rate is defined as the market yield on an irredeemable, or perpetual, bond, also known as an undated or consol bond. Both interest rates are assumed to follow a Gaussian-Markov process. A Gaussian process is one whose marginal distribution, where parameters are random variables, displays normal distribution behavior a Markov process is one whose future behavior is conditional on its present behavior only, and independent of its past. A later study, Longstaff and Schwartz (1992), found that Brennan-Schwartz modeled market bond yields accurately. 

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. 

---