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I have a process $\{X_t\}_t$, where $x_t$ in $[0,1]$, and a Lyapunov function $V(x)=x$ such that the drift $$E[X_{t+1}|x_t] - x_t < -k(1-x_t)$$

for $x_t \in (1-\epsilon,1]$, where $k>0$. Thus the drift is negative on $(1-\epsilon,1]$ though not uniformly upper bounded. Is there any equivalent Foster-Lyapunov theorem that ensures stability of the process on $[0,1-\epsilon]$?

I have more structure in the process such that if $x_t = \alpha$, then $x_{t+1}\in \left((1-\gamma) \alpha, (\gamma+(1-\gamma)\alpha\right)$, where $\gamma<1$. Thus the process can not directly jump to 1, but can only do that in the limit (1 is the absorbing point of the process).

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  • $\begingroup$ You must also assume something about the process in $[0,1-\epsilon]$, or else it could jump to 1 from any point in that interval. Also can you specify exactly which notion of stability you are trying to prove, and if the process can jump to 1. $\endgroup$ Nov 15, 2019 at 19:32
  • $\begingroup$ I clarified the question to incorporate your point. So the process can not jump to 1 in any finite steps (even though 1 is an absorbing point). I am trying to prove existence of stationary distribution of this MArkov process on the set $[0,1-\epsilon]$. I have already proven U-irreducibility, where U is the uniform measure on the set $[0,1-\epsilon]$. $\endgroup$ Nov 15, 2019 at 20:03
  • $\begingroup$ correction-- trying to prove existence of stationary distribution of this Markov process on the set [0,1). $\endgroup$ Nov 15, 2019 at 20:57

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