Chapman-Kolmogorov Equation and Infinitesimal Generators

Note that some authors have used these operators interchangeably for the description of the mesoscopic transport process. It is clear that if L is self-adjoint, then it can be used as a transport operator and the function u(x, t) can represent the particle density. For example, the one-dimensional Brownian motion B t) has the infinitesimal generator L = 9 /9x which is self-adjoint. A symmetric a-stable Levy process on R has the generator L = 9 /9 x , which is self-adjoint too. In the next section we obtain L and L from the Chapman-Kolmogorov equation. 

Let us consider the Chapman-Kolmogorov equation for the transition density 

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. 

Let us replace s with small h in (3.259) and rewrite this equation for the density piy, t)  

Letting h 0, we obtain the Kolmogorov forward equation, the Master equation, 

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