Mikhail Menshikov, Dimitri Petritis
Markov chains in a wedge with excitable boundaries
(305K, postscript)

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.