If we have a Markovian Jump Process $x(t)$, we let $\varepsilon > 0$ be small, we scale jump sizes by $\varepsilon^{1/2}$ and the jump intensity by $\varepsilon^{-1/2}$. I.e. we make the process jump much more frequent and decrease the jump size.
One can then find the stationary distribution for $\varepsilon \rightarrow 0$ (call it $x_d$). This is then the diffusion approximation for $x(t)$. I think this is correct?
My question now is: can we obtain the stationary distribution for $x(t)$ from $x_d$? I don't understand how to use $x_d$ to analyze $x(t)$..