  We consider several multiscale-in-time kinetic Monte Carlo models, in which
some variables evolve on a fast time scale, while the others evolve on a slow
time scale. In the first two models we consider, a particle evolves in a
one-dimensional potential energy landscape which has some small and some large
barriers, the latter dividing the state space into metastable regions. In the
limit of infinitely large barriers, we identify the effective dynamics between
these macro-states, and prove the convergence of the process towards a kinetic
Monte Carlo model. We next consider a third model, which consists of a system
of two particles. The state of each particle evolves on a fast time-scale while
conserving their respective energy. In addition, the particles can exchange
energy on a slow time scale. Considering the energy of the first particle, we
identify its effective dynamics in the limit of asymptotically small ratio
between the characteristic times of the fast and the slow dynamics. For all
models, our results are illustrated by representative numerical simulations.
