This problem arose when solving a continuous Markov chain exercise from a book I'm studying. Given a set of positive $q\_i$, with $i \in \mathbb{Z} $, and non-negative $\lambda$ and $\mu$ that add to 1 the solution amounts to solving 


$p\_{0,i}^'(t) = \lambda q\_{i-1} p\_{0,i-1}(t) - q\_i p\_{0,i}(t) + \mu q\_{i+1} p\_{0,i+1}(t) \;\; \text{where} \; \; i \in \mathbb{Z}$


with initial conditions $p\_{0,i}(0) = \delta\_{0,i}$. How does one go about solving this problem ? 



Solution to difference differential equation with constant coefficients
Thanks to Michael Renardy for the analogy with the heat equation. Let's assume $q\_i = q$, i.e. independent on $i$. The difference-differential equation can then be solved by using generating function technique. Let $G(z) = \sum\_{i \in \mathbb{Z}} z^i p\_{0,i}(t)$. Then the equation and initial condition imply that $G(z) = \exp(-q t) \exp \left( q t (\lambda z + \mu/z) \right)$. Extracting series coefficients from here gives: $p\_{0,i}(t) = \exp(-q t) \lambda^{i/2} \mu^{-i/2} I\_{i}( 2 q t \sqrt{\lambda \mu})$, which is known as [Skellam](http://en.wikipedia.org/wiki/Skellam_distribution) distribution.


