For discrete space Stochastic processes

Discrete time master equation

For discrete time, probability to be in state $n$ at time $t+\Delta t$ is:

$P(n, t+\Delta t) = \sum_{n'} W(n'\rightarrow n)P(n', t)$

where the $W(n'\rightarrow n)$ are the transition probabilities (which can be expressed as a transition matrix).

Continuous time master equation

For continuous time, we can subtract $P(n,t)$ from both sides of the discrete time equation, and divide by $\Delta t \rightarrow 0$. Then

$\frac{d P(n, t)}{dt} = \sum_{n'} w(n' \rightarrow n) P(n', t) - [\sum_{n'} w(n \rightarrow n')]P(n,t)$

$= \sum_{n' \neq n} w(n' \rightarrow n) P(n', t) - [\sum_{n' \neq n} w(n \rightarrow n')]P(n,t)$

where $w(n' \rightarrow n) = \lim_{\Delta t \rightarrow 0} \frac{W(n'\rightarrow n)}{\Delta t}$, and where for the bracketed part we used that probability is conserved (i.e. the particle has to go somewhere), $\sum_{n'} w(n \rightarrow n') = 1$, and in the second line we cancelled the $n' = n$ terms from both terms.

Solve using Fourier series, as if it is in a (discrete) lattice. For more general networks, Fouriers may not be appropriate.. You can then use eigenvector methods