  In this paper, we investigate a nonparametric approach to provide a recursive
estimator of the transition density of a non-stationary piecewise-deterministic
Markov process, from only one observation of the path within a long time. In
this framework, we do not observe a Markov chain with transition kernel of
interest. Fortunately, one may write the transition density of interest as the
ratio of the invariant distributions of two embedded chains of the process. Our
method consists in estimating these invariant measures. We state a result of
consistency and a central limit theorem under some general assumptions about
the main features of the process. A simulation study illustrates the well
asymptotic behavior of our estimator.
