Forward equation

To solve a system of equations, our first step is to put zeros into the second and the third rows of the first column, and into the third row of the second column. For our exercise we will bring forward equations 2-1 through 2-3 as (equation set 3-1) ... 

These differential equations depend on the entire probability density function / (x, t) for x(t). The evolution with time of the probability density function can, in principle, be solved with Kolmogorov s forward equation (Jazwinski, 1970), although this equation has been solved only in a few simple cases (Bancha-Reid, 1960). The implementation of practical algorithms for the computation of the estimate and its error covariance requires methods that do not depend on knowing p(x, t). 

Letting A/- 0 in this relation, one obtains the forward equation (4.42). The backward equation (4.41) can be deduced by a similar... 

Kolmogorov s Forward Equation The transition probability p(s, x t, y) satisfies the forward equation ... 

The n-variable version of Kolmogorov s forward equation (or Fokker-Planck equation) can be written as... 

Klier et al. (5<5) presented a kinetic model based on previous observations of the physical and chemical characteristics of the binary Cu/ZnO catalysts and put forward equations that quantitatively account for the C02/C0 dependence of the synthesis rates. Their model was required to accommodate the following observations ... 

The Kolmogorov equation. In the preceding model, the major assumption made was that the one-step transition probabilities p and q remain constant. In the following it is assumed that these probabilities are state depending, namely, designated as pk and qk. The following forward equation for the present model reads ... 

The forward equation The Fokker-Planck equation is now derived as the differential form of the Chapman-Kolmogorov equation For any function f (x)... 

Recently, Swain et al. (7) have put forward equation 2, where Alog k is the incremented log k relative to the lowest value in the series, A and B are measures of solvent electrophilicity and nucleophilicity, and a, b, and c are determined by details of the solvolysis under consideration. Although this equation appears to have an advantage over equation 1 in that A and B values can be obtained without the study of solvolysis reactions, use of such values frequently leads to unreasonable predictions (8). 

MP here is metalloporphyrin and PH2 is free base porphyrin. In general the forward equation is demetallation and the reverse is metallation. The dissociation of metalloporphyrins is governed by chemical stability. 

These are called Kolmogorov s backward and forward equations, respectively. The Chapman-Kolmogorov equations P(s + t) = P(s)P(t) may be deduced from (30) as follows ... 

This equation states that the particle density at time n + 1 is the sum of the densities at intermediate points x - z at time n multiplied by the probability of transition from X - z to X. This is a mesoscopic description. Although it only deals with the mean density of particles p(x,n), it involves a detailed description of the movement of particles on the microscopic level. Equation (3.13) is the same as the Kolmogorov forward equation (3.12). The solution to (3.13) can be rewritten as aconvolution... 

The mesoscopic density is governed by the Kolmogorov forward equation, the Master equation,... 

Of course (3.234) and (3.240) are identical in form, but only the forward equation (3.240) has the physical meaning of a transport equation for particles. We will discuss the difference between forward and backward equations in the next section. It turns out that, it is more convenient to deal with the backward equation (3.234). Let us give an example. The Brownian motion B t) starting at x can be rewritten in terms of the standard Wiener process W(t) as... 

Our goal is to derive the Kolmogorov forward and backward equations and to discuss the main difference between them. The forward equation deals with the events during the small time interval (t, t+h] and gives us the answer for how those events define the probability density p(y, t+h x) at time while the backward equation is concerned with events just after the time t = 0. 

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