99-406 Mikhail Menshikov, Dimitri Petritis
Markov chains in a wedge with excitable boundaries (305K, postscript) Oct 25, 99
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Abstract. We consider two models of Markov chains with unbounded jumps. In the first model the chain evolves in a quadrant with boundaries having internal structure; when the chain is in the interior of the quadrant, it moves as a standard Markov chain without drift. When it touches the boundary, it can spend some random time in internal --- invisible --- degrees of freedom of the boundary before it emerges again in the quadrant. The second model deals with a Markov chain --- again without drift --- evolving in two adjacent quadrants with excitable boundaries and interface with some invisible degrees of freedom. We give, for both models, conditions for transience, recurrence, ergodicity, existence and non existence of moments of passage times that are expressed in terms of simple geometrical properties of the wedge, the covariance matrix of the chain and its average drifts on the boundaries, by using martingale estimates coming from Lyapunov functions.

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