Gaussian stochastic processes

A Gaussian process can be Markovian. As an example consider the Markovian process characterized by the transition probability 

This process is Gaussian by definition, since (7.54) is satisfied for any pair of times. The distribution (7.55) satisfies 

Therefore the Markovian property (7.48) is satisfied provided that 

If we further assume that ki is a function only of the time difference tk — ti, that is, Aa/ = — ti), it follows that its fonn must be 

Taken independently of the time ordering information, the distribution (7.54) is a multivariable, zi-dimensional, Gaussian distribution 

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The random force F t) can be viewed as corresponding to a stationary Gaussian stochastic process [the Gaussian character ensuing from the statistical properties of the bath operators involved in Eqs. (8) and (9)]. It is fully characterized by its average value and its correlation function. 

Equations (7.60)-(7.63) describe general properties of many-variable Gaussian distributions. For a Gaussian random process the set zy corresponds to a sample zy,Z, from this process. This observation can be used to convert Eq. (7.63) to a general identity for a Gaussian stochastic process z(t) and a general function of... 

Equation (7.64) is a general identity for a Gaussian stochastic process characterized by its average (/) and the time correlation functions Cz(Zi,Z2). In many applications the stochastic process under study is stationary. In such cases (z) = m does not depend on time while Cz(Zi, Z2) = Cz(Zi — t2) depends only on the time difference. 

In the classical limit (hcoj k T), the reaction coordinate X t) in each quantum state can be described as a Gaussian stochastic process [203]. It is Gaussian because of the assumed linear response. As follows from the discussion in Section II.A, if the collective solvent polarization follows the linear response, the ET system can be effectively represented by two sets of harmonic oscillators with the same frequencies but different equilibrium positions corresponding to the initial and final electronic states [26, 203]. The reaction coordinate, defined as the energy difference between the reactant and the product states, is a linear combination of the oscillator coordinates, that is, it is a linear combination of harmonic functions and is, therefore, Gaussian. The mean value is = — , for state 1 and = , for state 2, respectively. We can represent Xi(r) and X2 t) in terms of a single Gaussian stochastic process x(t) with zero mean as follows ... 

Figure 9.15. Typical trajectories of a Gaussian stochastic process x(t) with zero mean and Gaussian (a) or exponential (i>) correlation function. Circles are crossing points of x = 0. Trajectories were generated by regular sampling in the frequency domain, (c) corresponds to the Debye relaxation spectrum with a cutoff frequency. Reorganization energy of the discarded part of the spectrum is 7% of the total. The sampling pattern was the same as in (b).

Deodatis G, Micaletti RC (2001) Simulation of highly skewed non-Gaussian stochastic processes. Trans ASCE, J Eng Mech 127 1284-1295... 

The statistical properties of the random force f(0 are modeled with an extreme economy of assumptions f(t) is assumed to be a stationary and Gaussian stochastic process, with zero mean (f(0 = 0), uncorrelated with the initial value v(t = 0) of the velocity fluctuations, and delta-correlated with itself, f(0f(t ) = f25(t -1 ) (i.e it is a purely random, or white, noise). The stationarity condition is in reality equivalent to the fluctuation-dissipation relation between the random and the dissipative forces in Equation 1.1, which essentially fixes the value of y. In fact, from Equation 1.1 and the assumed properties of f(t), we can derive the expression y(f)v(t) = exp [v(0)v(0) -+ ylM °, where Xg = In equilibrium, the long-time asymptotic value y/M must coincide with the equilibrium average (vv) = (k TIM)t given by the equipartition theorem (with I being the 3 X 3 Cartesian unit tensor), and this fixes the value of y to y=... 

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