I found that most references for the symmetric random walk on the integers are for the discrete time case, i.e. the ones that gives us explicit transition probabilities. Now, I am looking at a random walk $X:[0,\infty) \rightarrow \mathbb{Z}$ with rates $q(i,i-1)=q(i,i+1)=\frac{1}{2}$ and want to know the transition probabilities $p_t(0,n)=?$, but I would also like to know if there are more general results.

The more general question could be: Does anybody know if there is a way to state the transition probabilities explicitly for arbitrary transition rates?

Theoretically, we only need to solve the infinite system of ODE's:

$\frac{dp_t(x,y)}{dt} = \frac{1}{2}\left(p_t(x-1,y)+p_t(x+1,y)\right)-p_t(x,y)$ for all pairs of $x,y \in \mathbb{Z}$ and initial condition $p_t(x,y)=\delta_{x,y}$ but this is not that simple (if one does not know the answer).