Random Walk with Discrete States in Continuous-Time

2 Random Walk with Discrete States in Continuous-Time 

So far we have considered the homogeneous case for which the waiting time density is independent of the position of the particles or their state. Let us formulate the general equations describing a random walk with discrete states in continuous time for which the waiting time PDF depends on the current state. (CTRWs with space-dependent waiting time PDFs have been studied in [75].) We introduce the mean density of particles Pmit) in state m and the density of particles j (t) arriving in state m exactly at time t. The balance equations can be written as 

Here P (t) = p iT)dr is the survival probability in the state m, is the transition probability from state i to tn, and PiQ is the initial density of particles in state i. Using the Laplace transform, we obtain from (3.54) and (3.55) 

It should be noted that (3.58) cannot be written in the standard form (3.41), which makes it difficult to give its probabilistic interpretation. 

This equation can be useful for studying multi-component systems where the chemical reactions do not obey classical kinetics. 

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