**MSC:**- 60J10 Markov chains with discrete parameter
- 31C05 Harmonic, subharmonic, superharmonic functions
- 60J50 Boundary theory

We define the canonical Markov chain for ${\mathcal P}$ and denote its Markov operator by $P$.

We show that its Martin boundary ${\mathcal M}$ is homeomorphic to ${\mathcal P}$.

The associated Dirichlet problem $(P-I)f=0$ and $f=g$ on ${\mathcal P}$ has a unique solution such that $f(\xi)={\mathcal P}_{\xi}$ for $\xi \in {\mathcal P}$.

We obtain an integral representation for kernel functions on ${\mathcal P}$ (Poisson integral type).