A Stochastic Model of Language Change Through Social Structure and Prediction-Driven Instability

Abstract

Children reliably learn their community's language; consequently human languages are relatively stable on short time scales. However, languages can change dramatically over the course of centuries, and once begun, such changes generally run monotonically to completion. We consider a stochastic model that reproduces this pattern of fluctuations via large deviations. We begin with a Markov chain that represents an age-structured population in which children learn which of two grammars their community prefers but are aware of age-correlated usage patterns and will use the dispreferred grammar more often if they infer that its use is spreading. The Markov chain is shown to converge in the limit of an infinite population to a stochastic differential equation that generalizes the Wright--Fisher SDE for population genetics. This proof is not routine because the dynamics are only defined in a Cartesian product of simplexes, and it must be verified that trajectories of the SDE cannot escape. Results are proved by ...