  We consider a recurrent Markov process which is an It\^o semi-martingale. The
L\'evy kernel describes the law of its jumps. Based on observations
X(0),X({\Delta}),...,X(n{\Delta}), we construct an estimator for the L\'evy
kernel's density. We prove its consistency (as n{\Delta}->\infty and
{\Delta}->0) and a central limit theorem. In the positive recurrent case, our
estimator is asymptotically normal; in the null recurrent case, it is
asymptotically mixed normal. Our estimator's rate of convergence equals the
non-parametric minimax rate of smooth density estimation. The asymptotic bias
and variance are analogous to those of the classical Nadaraya-Watson estimator
for conditional densities. Asymptotic confidence intervals are provided.
