The K-process on a tree as a scaling limit of the GREM-like trap model

We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the parameters of the trap model, we establish its infinite volume limit, given by what we call a $K$-process in an infinite $k$-level tree. From this we deduce that the $K$-process also is the scaling limit of the GREM-like trap model on extreme time scales under a fine tuning assumption on the volumes.