I am working with a jumpdiffusion on the unit interval, with absorbing endpoints, and I was hoping someone has found a way to determine the invariant measure, similar to that of an Ito process.
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The general procedure is the following. Take the generator $L$ of the semigroup associated to your process. Then find its dual $L^*$. The latter governs the evolution of 1dim distributions (via forward Kolmogorov equation), and an invariant density $p$ satisfies $L^*p=0$, should be positive and integrate to 1. I suspect all this can be found in the Ethier&Kurtz book on Markov processes, but I do not have it at hand right now. 


Henry Tuckwell's 1976 paper, "On the FirstExit Time Problem for Temporally Homogeneous Markov Processes", gives an ODDE to solve to determine the invariant (ergodic) measure. 

