  The paper deals with the asymptotic properties of a random jump process in a
high contrast periodic medium in $\mathbb R^d$, $d\geq 1$. We show that if the
coordinates of the random jump process in $\mathbb R^d$ are equipped with an
extra variable that characterizes the position of the process inside the
period, then the limit dynamics of this two-component process is Markov. We
describe the limit process and prove the convergence in law in the path space.
We show that the components of the limit process are coupled and derive the
evolution equation with a memory term for the spatial component of the process.
We also discuss the construction of the limit process in the $L^2$ space and
study the spectrum of the generator of the limit semigroup.
