Kolmogorov backward equation

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 ... 

Kolmogorov backward equation. These equations are listed below. 

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... 

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],... 

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... 

In the language of applied maths, p(x, y, t) is the Green s function for the diffusion process. It is important to note that in the forward equation, differentiation is carried out with respect to y (the current position) and with respect to jc (the initial position) in the backward equation. In the simulation of chemical systems, the drift term (p.) arising in Kolmogorov s equation is normally due to the Coulombic interaction between charged species and can be expressed as... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

For the solution of real tasks, depending on the concrete setup of the problem, either the forward or the backward Kolmogorov equation may be used. If the one-dimensional probability density with known initial distribution deserves needs to be determined, then it is natural to use the forward Kolmogorov equation. Contrariwise, if it is necessary to calculate the distribution of the mean first passage time as a function of initial state xo, then one should use the backward Kolmogorov equation. Let us now focus at the time on Eq. (2.6) as much widely used than (2.7) and discuss boundary conditions and methods of solution of this equation. 

We will not present here how to derive the first Pontryagin s equation for the probability Q(t, x0) or P(f,x0). The interested reader can see it in Ref. 19 or in Refs. 15 and 18. We only mention that the first Pontryagin s equation may be obtained either via transformation of the backward Kolmogorov equation (2.7) or by simple decomposition of the probability P(t, xq) into Taylor expansion in the vicinity of xo at different moments t and t + t, some transformations and limiting transition to r — 0 [18]. 

These are the so-called forward and backward Kolmogorov equations for the Ornstein-Uhlenbeck process. Their paramount importance will appear in VIII.4 under the more familiar name of Fokker-Planck equation. 

The corresponding backward Fokker-Planck equation (Kolmogorov equation) for the conditional density / ( " / b t ) is given by (5)... 

Equation (92) is simpler than (60) but otherwise structurally identical to it. In particular, the Euler-Lagrange equation associated with minimizing (92) has the form of a backward Kolmogorov equation ... 

Therefore (95) instead of (60) can be taken as a starting point for what we did in Sect. 6.1. Observe that (95) is the backward Kolmogorov equation associated with the stochastic differential equation (compare (19))... 

This equation is the limiting equation for Q r) it is the backward Kolmogorov equation associated with the overdamped equation obtained from (21) in the limit as 7 oo (the overdamped dynamics is considered in Sect. 6.6). On the other hand, inserting (107) into (103) gives... 

The investigation of Brownian particle motion led to the development of the mathematical theory of random processes which has been widely adopted. A great contribution to the mathematical theory of Brownian motion was made by Wiener and Kolmogorov for example, Wiener proved that the trajectory of Brownian motion is almost everywhere continuous but nowhere differentiable, Kolmogorov introduced the concept of forward and backward Fokker-Planck equations. 

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 ... 

---