- 99-406 Mikhail Menshikov, Dimitri Petritis
- Markov chains in a wedge with excitable boundaries
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.