Mixing under monotone censoring

We initiate the study of mixing times of Markov chain under monotone censoring. Suppose we have some Markov Chain $M$ on a state space $Ω$ with stationary distribution $π$ and a monotone set $A \subset Ω$. We consider the chain $M'$ which is the same as the chain $M$ started at some $x \in A$ except that moves of $M$ of the form $x \to y$ where $x \in A$ and $y \notin A$ are {\em censored} and replaced by the move $x \to x$. If $M$ is ergodic and $A$ is connected, the new chain converges to $π$ conditional on $A$. In this paper we are interested in the mixing time of the chain $M'$ in terms of properties of $M$ and $A$. Our results are based on new connections with the field of property testing. A number of open problems are presented.

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