  We present two data-driven procedures to estimate the transition density of
an homogeneous Markov chain. The first yields to a piecewise constant estimator
on a suitable random partition. By using an Hellinger-type loss, we establish
non-asymptotic risk bounds for our estimator when the square root of the
transition density belongs to possibly inhomogeneous Besov spaces with possibly
small regularity index. Some simulations are also provided. The second
procedure is of theoretical interest and leads to a general model selection
theorem from which we derive rates of convergence over a very wide range of
possibly inhomogeneous and anisotropic Besov spaces. We also investigate the
rates that can be achieved under structural assumptions on the transition
density.
