When does the stationary density of an homogeneous Markov process exist?
It is hard to be brief here, but I will try.
One answer is: when the corresponding stationary Fokker-Planck equation (aka forward Kolmogorov equation) has a nonnegative integrable solution. The density is then obtained by normalization of that solution. This is not a very good answer because FP equations are often not so easy to analyze.
In fact, it is hard to guess what is the question you really wanted to ask. For example, one may say that your question is twofold: 1) when is there an invariant probability distribution? 2) If it exists, is it absolutely continuous w.r.t. Lebesgue?
Existence is guaranteed if the drift prevents the trajectories from going to infinity so that they spend a lot of time in a compact set. Some of the relevant keywords are: Lyapunov-Foster functions, Harris recurrence. Among weakest known conditions guaranteeing existence of invariant distributions are those due to Veretennikov.
Given part 1), absolute continuity of the stationary distributions can essentially be deduced from absolute continuity of transition probabilities. This part is easy if your equations are elliptic and not so easy if the noise excites only some directions.
The analysis can be quite nontrivial depending on how bad your equation is, as a look at a recent paper http://arxiv.org/abs/0712.3884 might convince you.