Transition probability density

Channels with Continuous Input and Output. Let the channel input alphabet be the set of real numbers. Temporarily we will assume the input, a , to be bounded between two limits, A < x B. We will also assume that a channel transition probability density, Pr(y x), exists which is a continuous function of both x and y. 

Formula (2.2) contains only one-dimensional probability density W(xi, t ) and the conditional probability density. The conditional probability density of Markov process is also called the transition probability density because the present state comprehensively determines the probabilities of next transitions. Characteristic property of Markov process is that the initial one-dimensional probability density and the transition probability density completely determine Markov random process. Therefore, in the following we will often call different temporal characteristics of Markov processes the transition times, implying that these characteristics primarily describe change of the evolution of the Markov process from one state to another one. 

The transition probability density satisfies the following conditions ... 

The transition probability density becomes Dirac delta function for coinciding moments of time (physically this means small variation of the state during small period of time) ... 

If the initial probability density W(xq. to) is known and the transition probability density W(x, t xo, to) has been obtained, then one can easily get the one-dimensional probability density at arbitrary instant of time ... 

Equation (2.5) is a stochastic differential equation. Some required characteristics of stochastic process may be obtained even from this equation either by cumulant analysis technique [43] or by other methods, presented in detail in Ref. 15. But the most powerful methods of obtaining the required characteristics of stochastic processes are associated with the use of the Fokker-Planck equation for the transition probability density. 

The transition probability density of continuous Markov process satisfies to the following partial differential equations (WXo(x, t) = W(x, t xo, to)) ... 

The first way is to obtain the transition probability density by the solution of Eq. (2.6) with the delta-shaped initial distribution and after that averaging it over the initial distribution Wo(x) [see formula (2.4)]. 

Thus, the one-dimensional probability density of the Markov process fulfills the FPE and, for delta-shaped initial distribution, coincides with the transition probability density. 

Equation (2.5) can be interpreted physically as follows. E G is just the probability density p that a particle released at location (x , y, z ) at time / will be at location (jt, y, z) at time t, the transition probability density,... 

We have now derived two fundamentally different expressions for the mean concentration of an inert tracer in a turbulent flow Eqs. (2.7) and (2.19). To compute the mean concentration c from Eq. (2.7) requires only that p(x, y, z, t x, y, z, t ), the transition probability density, be specified, whereas the mean velocities and eddy diffiisivities must be prescribed to obtain c from Eq. (2.19). In the next section we see how forms of c are obtained from Eqs. (2.7) and (2.19). 

The approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

A third class of equations, which permit to study the effect of autocorrelation times of arbitrary length, has been encountered in IX.7. This class consists of equations (1.1) in which Y(t) is a Markov process. We write 77 for its transition probability density 77(y, t y0, t0) and... 

Using the path-integral expression for the transition probability density [64], one can write pi, in the form [60]... 

These results clearly show that the time development of the system is entirely determined by the transition probability densities f(xk,tkl xk x,tk x) and the initial probability density of states /(x, tx). 

In physical systems it can happen that the transition probability densities are homogeneous in time and/or in space. A stochastic process X(t) is stationary if X(t) and X(t + r) obey the same probability laws for every r this means that all joint probability densities verify time translation invariance... 

The essential difference between the two transition probability densities lies in the fact that for the gaussian distribution pw r, ) the different moments E[Xm], m = 1, 2,. . . , n, exist, while for the Cauchy distribution pc(j, x) they do not exist. The Levy distributions characterized by p(t, k) = exp -a k qT) with 0< <2U 127 128 play a prominent role in the theory of relaxation processes.129 133... 

It can easily be verified that the transition probability density verifies the forward Kolmogorov s equation (4.116), with M(t, y) = 0 and jS2(t, y) = D, which is the familiar diffusion equation homogeneous in space and time... 

Fig. 4.4. Transition probability density p(x0 t, x) at different times [Eq. (4.155)] of a Brownian particle in the high-friction regime and in a force field deriving from an harmonic potential [initial condition p0(0, j 0) = 5(ac - )] x0 = -0.2 A, D = 0.1 A2 ns"1, r = 0.1 ns.

From this, the version of the Fokker-Planck equation for the transition probability density with two variables r and v is seen to be... 

The probability density that if the particle is at x at t1, it will undergo a displacement to x at t. Denote this probability density Q(x. t x, t ) and call it the transition probability density for the particle. 

Let us consider, as shown in Figure 18.3, a single particle that is at position xo at time to and is at position x at some later time t in a turbulent field. The complete statistical properties of the particle s motion are embodied in the transition probability density Q(x,i x, r ). An analysis of this problem for stationary, homogeneous turbulence was presented by Taylor (1921) in one of the classic papers in the field of turbulence. If the... 

The transition probability density Q expresses physically the probability that a tracer particle that is at Jt, y, z/ at f will be at x,y,z at t. We showed that under conditions of stationary, homogenous turbulence Q has a Gaussian form. For example, in the case of a... 

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