Stationary probabilities

So, Eq. (3.14) with boundary conditions is the equation for eigenfunction X (x) of the nth order. For X0(x), Eq. (3.14) will be an equation for stationary probability distribution with zero eigenvalue y0 = 0, and for X (x) the equation will have the following form ... 

For potential profiles of types II and III there are no nonzero stationary probability densities because all diffusing particles in the course of time will leave the initial interval and go toward the regions where the potential cp(x) tends to minus infinity. In these situations we may pose two questions ... 

The existence of the limit (3) guarantees that, after a large enough number of steps, the different configurations are generated following a probability density II. Then it is said that a distribution of stationary probability or situation of static equilibrium has been reached. If II has been previously chosen, the method consists of selecting pij so that the conditions (2) and (4) are fulfilled. We must stress the fact that the condition of microscopic reversibility ... 

Pope, S. B. and E. S. C. Ching (1993). The stationary probability density function An exact result. Physics of Fluids A Fluid Dynamics 5, 1529-1531. 

D. Suzuki and T. Munakata, Stationary probability flow and vortices for the Feynman ratchet, unpublished. 

Of course, there may be more than one. Each 0 is a time-independent solution of the master equation. When normalized it represents a stationary probability distribution of the system, provided its components are nonnegative. In the next section we shall show that this provision is satisfied. But first we shall distinguish some special forms of W. 

This completes the proof of the second lemma. A corollary is that for a time-independent solution either all components are nonnegative, or all non-positive. For a stationary probability distribution one has, of course, psn 0, because C— 1. 

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. 

The numbers 0, 1, 2 and 3 of central parts of graph correspond to the working condition of compressor stations therefore to find the probabilities of appropriate conditions is important. In this case concerning the stationary probabilities, the system of the algebraic equations is received from the system of the differential equations (1) ... 

The following values are drawn from the system of these equations for stationary probabilities Po, Pi, and P2 ... 

Dissipative structures can sustain long-range correlations. The temperature and chemical potential are well defined with the assumption of local equilibrium, and the stationary probability distribution is obtained in the eikonal approximation so the fluctuation-dissipation relation for a chemical system with one variable is... 

A particle moving back and forth within definite limits can exist in states described by a stationary probability pattern only for certain discrete values of the energy. 

On the basis of the following theorem, i.e., if the transition matrix P for a finite irreducible aperiodic Markov chain with Z states is doubly stochastic, then the stationary probabilities are given by... 

A Markov chain is ergodic if it eventually reaches every state. If, in addition, a certain symmetry condition - the so-called criterion of detailed balance or microscopic reversibility - is fulfilled, the chain converges to the same stationary probability distribution of states, as we throw dice to decide which state transitions to take one after the other, no matter in which state we start. Thus, traversing the Markov chain affords us with an effective way of approximating its stationary probability distribution (Baldi Brunak, 1998). 

Fig. 1.5. Stationary probability distribution of tlie FitzHugh-Naguino System. Inhibitor noise intensity is varied (given above the panels). Other parameters t = 0.1, 7 = 2., 6 = 1.4, [291...

M. Kostur X. Sailer and L. Schimansky-Geier. Stationary probability distributions for FitzHugh-Nagumo systems. Fluctuation and Noise Letters, 3 155, 2003. 

Proppe, C. Exact stationary probability density functions fo" nonlinear systems under Poisson white noise excitation, bitemational Journal of NrmlinearMechanics 38(4) (2003), 557-564. 

The first passage time D(/) and the stationary probabilities vector rr (and consequently 4) can be estimated from Xt = (X, i > 0) as follows. A Markov process Xj (initial state / and transition matrix M) is observed with the two time sequences /j and L so that ... 

A state is defined as (i, it), where i is the inventory level and u is the back-logging level. The solution for the stationary probability distribution of the Markov chain in Figure 15.5 is calculated numerically by solving a system of equations as follows. Let Q denote the chain s rate matrix, p the matrix of pi u and 0 the matrix of zeros. Then, pi u is the solution to the following system of equations ... 

Fig. 5.12 The volume-dependence of the exact stationary probability distribution function of system / and comparison with its Euler-McLaurin approximation (after Ebeling and Schimansky-Geier).

Under the boundary condition, referring to g as a probability density function, the stationary probability density function is... 

To extend the validity range of noise-induced phenomena for a wider range of correlation times the dichotomous Markov noise has been used. The dichotomus Markov noise, also known as the random telegraph signal has a quite simple structure, therefore the stationary probability density can be calculated for an arbitrary value of the correlation time, and for any value of the noise intensity. The state of the Markovian dichotomous noise /, consists of two levels A+, A only. The noise is characterised by the transition probability ... 

The influence of external fluctuations on two-dimensional oscillatory systems has been studied by Ebeling Engel-Herbert (1980). The stochastic counterpart of the appearence of a limit cycle is strongly connected with the formation of a probability crater on the stationary probability surface. They studied particular cases when the system has the form... 

Fig. 5.18 The stationary probability density (5.179) in the case of multiplicative noise. The transition points between regimes depend explicitly on the noise intensity...

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