I have a continuous-state discrete-time markov chain (process) and I need to find the steady-state density function. I am familiar with discrete-state markov chains, but I am not sure how to handle the continuous-state case. In particular:
1.I am not sure what conditions are necessary and sufficient for the existence of a steady-state probability, or for its uniqueness.
2.I am not sure how to actually get to steady state densities. I can write down steady state conditions, same as I would for a discrete state markov chain, but then, how do I find them?
Here are some more details. State space is the interval $[0,N]$. Transition probabilities from x to y are:
$p(x,y\in(0,N))=\frac{1}{Q}, \;\;\;\;\; p(x,y=0)=\frac{D-x}{Q}, \;\;\;\;\; p(x,y=N)=\frac{Q-D-N+x}{Q}.$
Steady state conditions obtained by imposing stationarity of distribution (using $\pi$ for steady state density/probability mass) yield
$\pi(0)=\frac{Q}{2(Q-D)}\intop_{(0,N]}\pi(y)\frac{D-y}{Q}dy,$
$\pi(N)=\frac{Q}{2(Q-D)}\intop_{(0,N]}\pi(y)\frac{Q-D-N+y}{Q}dy,$
$\pi(x|x\in(0,N))=\frac{1}{Q}\left[\pi(0)+\pi(N)+\intop_{(0,N)}\pi(y)dy\right].$
Where do I go from here?
Any help to the above, including useful references to an introductory text on continuous-state markov chains or specific results therein, is highly appreciated.
Frank