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Suppose $(X_n)_{n\in \mathbb N_0}$ is a discret-time Markov process on state space $(0,1]$ with transition kernel $\kappa(x,\cdot)$ possessing an absolute continuous (Lebesgue-)density for all $x\in (0,1]$ with support $(0,1]$. The transition density $p(x,y) $ may be written as continuous function of $(x,y)\in(0,1]^2$. Also, $\kappa(x,\cdot)$ converges weakly to $(\delta_0 + \delta_1)/2$ as $x\downarrow 0$.

Now, I would like to analyse, whether $(X_n)$ has an invariant measure, if it's unique and the law of $X_n$ converging to it for any initial distribution.

Are there references (text books etc.) for that context?

Thanks in advance :)

maliesen
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  • I think, one may interpret $(X_n)$ as Markov process on the compact space [0,1] and something like Theorem 5.4.1 in https://stanford.edu/class/msande321/Handouts/05%20Feller%20Chains.pdf looks promising. – maliesen May 16 '18 at 15:52

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