Stationary Markov process

If one chooses Pi(Vi, 0) = fi ) a non-stationary Markov process is defined, called the Wiener process or Wiener-Levy process. ) It is usually considered for f >0 alone and was originally invented for describing the stochastic behavior of the position of a Brownian particle (see VIII.3). The probability density for t > 0 is according to (2.2)... 

Even when a system is in a steady state other than equilibrium certain physical quantities may be stationary Markov processes. An example are the current fluctuations in the circuit of fig. 7 when a battery is added, which maintains a constant potential difference and therefore a non-zero average current. Another example is a Brownian particle in a homogeneous gravitational field its vertical velocity is a stationary process, but not its position. 

For stationary Markov processes the transition probability P1(1 does not depend on two times but only on the time interval for this case we introduce a special notation... 

This is an identity for all stationary Markov processes and may therefore be applied to physical systems in equilibrium without any additional derivation from the equations of motion. It should not, however, be confused with detailed balance, which differs from it by having + t in the right-hand member. Detailed balance is a physical property, which does not follow from the mere definition of Tt but requires a physical derivation, see V.6. In order to avoid erroneous use of the equations (3.2) and (3.3) we now stipulate that in the future the symbol Tt shall not be used for negative t. 

Exercise. The autocorrelation function of a stationary Markov process with zero mean is given by... 

The best known example of a stationary Markov process is the Ornstein-Uhlenbeck process510 defined by... 

Find the Pn for this non-Gaussian stationary Markov process. 

These processes are non-stationary because the condition singled out a certain time t0. Yet their transition probability depends on the time interval alone as it is the same as the transition probability of the underlying stationary process. Non-stationary Markov processes whose transition probability depends on the time difference alone are called homogeneous processes. 10 They usually occur as subensembles of stationary Markov processes in the way described here. However, the Wiener process defined in 2 is an example of a homogeneous process that cannot be embedded in a stationary Markov process. 

The extraction of a homogeneous process from a stationary Markov process is a familiar procedure in the theory of linear response. As an example take a sample of a paramagnetic material placed in a constant external magnetic field B. The magnetization Y in the direction of the field is a stationary stochastic process with a macroscopic average value and small fluctuations around it. For the moment we assume that it is a Markov process. The function Px (y) is given by the canonical distribution... 

Exercise. Let 7 be the stationary Markov process described by an M-equation with operator W. Let P2(yi, i y2>h) be its two-time distribution and G2(ki, ti k2,t2) the characteristic function thereof. Derive... 

Next consider a process Y(t) observed by the apparatus. We ask the a posteriori probability Pu(y, t) supposing that all values u(t ) from t = 0 to t = t have been monitored. We derive an equation for Pu(y, t) in the case that Y(t) is a stationary Markov process governed by the M-equation (1.5). 

If A < 0 the stationary solution (1.4) is Gaussian. In fact, in that case it is possible by shifting y and rescaling, to reduce (1.5) to (IV.3.20), so that one may conclude the stationary Markov process determined by the linear Fokker-Planck equation is the Ornstein-Uhlenbeck process. For Al 0 there is no stationary probability distribution. 

Two strategies that can be used to simplify the calculation of the mean walklength n) are now reviewed. In the first of these, it is shown that if the random walk is modeled by a stationary Markov process on a finite state... 

Markov process is completely characterized by its and T 2 or equivalently by only W2- A stationary Markov process has distributions satisfying the Smoluchowski equation (also called the Chapman-Kolmogorov equation)... 

Finally, it may be shown [7.13] that the conditional probability for a stationary Markov process is subject to the Smoluchowski-Chapman-Kolmogorov... 

If the time dependence of the Hamiltonian H(t) is derived from the interaction of paramagnetic centers with the environment in such a way that the Hamiltonian is defined by a complete set of random parameters Q, and if the time dependence of 12 is a stationary Markov process, then ... 

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