The Scaling Limit of Markov Processes is a subject in probability theory that encompasses several remarkable results in the field. Donsker’s Invariance Principle is certainly the most celebrated among them, characterizing Brownian motion as the scaling limit of random walks. In this talk, I will show how a scaling limit approach can be employed to describe a form of asymptotic lumpability for Markov processes in a large state space. I will illustrate this idea using two examples.1) The first example connects a fundamental process in population genetics, known as the Kingman coalescent process, to the scaling limit of a system of coalescing random walks on a large transitive graph. 2) The second example involves the metastable behavior of a zero-range process, a phenomenon studied in statistical mechanics. We will describe this behavior using a Markov chain on the metastable states as a suitable scaling limit.

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