Backward equation

As a companion to the M-equation it is sometimes advantageous to consider the adjoint or backward equation... 

Exercise. For Markov processes that are not stationary or homogeneous one also has a forward, or master equation and a backward equation,... 

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

From the Kolmogorov equations (4.41) and (4.42), one obtains the difference-differential equations for the birth-death process. The backward equation is given by... 

From Eq. (4.115) one can easily derive the corresponding backward equation... 

The marching-ahead equation becomes a marching-backward equation. The method is called reverse shooting. The procedure is to guess aj = aout and then to set aj / = aj. The index j in Equation (9.30) begins at J — 2 and is decremented by 1 until / = 0 is reached. The reaction rate continues to be evaluated at the central, /th point. The test condition is whether ain is correct when calculated using the inlet boundary condition... 

Obviously, if Plx, Z xo) is known we can compute the mean first passage time from Eq. (8.154). We can also find an equation for this function, by operating with backward evolution operator t (xo) on both sides ofEq. (8.154). Recall that when the operator L is time independent the backward equation (8.122) takes the form (cf. Eq. (8.123b)) 9P(x, Z xo)/9Z = T f (xo)T (x, Z xo) where D is the adjoint... 

In this section we remind the reader of the Kolmogorov forward and backward equations, infinitesimal generators, stochastic differential equations, and functional integrals and then consider how the basic transport equations are related to underlying Markov stochastic processes [141, 142],... 

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. 

To derive the backward equation, we consider the events just after the time t = 0 during the short time interval (0, ft]. The Chapman-Kolmogorov equation is... 

This equation is written for two variables, the time t and the initial position x. The final position y plays the role of a parameter. The function m(jc, t) defined in (3.250) obeys the Kolmogorov backward equation ... 

Referring to (3.250) and (3.257), we conclude that the solution to the finite difference backward equation... 

The backward equation, Liouville s equation, for the particle density takes the form... 

If the transition rates a = = const, then the forward and backward equations are identical. In particular, the mesoscopic equation for the density p is... 

As we discussed earlier, the backward equation (3.307) does not describe the average transport process of particles that follow the process X(t). The transport equation for the density p(y, t) with the convection-diffusion flux J = v y)p — D y)dp/dy can be written as... 

The anisotropic convection-diffusion equation (3.311) can be written in the form of a backward equation, dpjdt = Lp, if... 

Note that the pair (13.101) and (13.102) are backward equations in just the sense discussed for the infinite medium Eq. (13.37). The general procedure would be to choose a normalized space form for the... 

Mathematically, the system consists of parabolic PDEs, which were solved numerically by discretization of the spatial derivatives with finite differences and by solving the ODEs thus created with respect to time (Appendix 2). Typically, 3-5-point difference formulae were used in the spatial discretization. The first derivatives of the concentrations originating from a plug flow (Equations 9.1 through 9.3) were approximated with BD formulae, whereas the first and second derivatives originating from axial dispersion in the bulk phases and diffusion inside the catalyst particles were approximated by central difference formulae. Some simple backward (Equation 9.14) and central difference (Equation 9.15) formulae are shown here as examples ... 

The directional transport times T (x) can be calculated from the corresponding backward equation (40),... 

Kolmogorov backward equation. These equations are listed below. 

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