Fields Colloquium

The distribution of company sizes in the U.S. stock market has kept roughly the same shape for a century, with a Pareto tail, even though firms enter and leave at a substantial rate. Rank-based models explain the stable shape through size-dependent growth rates and volatilities, but hold the set of firms fixed. We add entry and exit: firms are geometric Brownian particles that are born and die at rates depending on their rank. In the many-firm limit, the empirical distribution converges, uniformly in rank and time, to the solution of a reaction-diffusion equation whose reaction term is built from the birth and death rates. The long-run shape is a traveling wave in log size. Without entry and exit the equation is a conservation law, whose waves require an entropy-type condition on the drift. Calibrated to the data, the reaction term is bistable: the wave exists and attracts without such an entropy condition, and the condition fails at most ranks. We discuss the identity that ties the wave's shape to volatility, drift, and turnover, its use as an inverse problem for the drift, and, time permitting, portfolio growth along the wave. Joint work with Caroline Smyth (Columbia College Class of 2026).