Transport Equations Forward vs Backward

Consider a collection of particles that move independently of each other in three-dimensional space R. We assume that the position of a particle X(r) is a time-homogeneous Markov process with transition density p(y, r x). 

The density of particles p(y,t) at point y at time t can be expressed in terms of the initial density of particles pq ) as 

As we have mentioned in the previous section, it is convenient to integrate with respect to the forward variable y. We define the transition operator 7 and the new function m(x, t) as 

It follows from (3.250) that the operator T, associated with the transition probability p(y, t x), can be written in terms of a conditional expectation over the particle position X(t) at time t, provided X(0) = x  

We always write the expectation E with the index x when we want to emphasize that the process X(t) starts at point x, i.e., X(0) = x  

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