Diffusion-Driven Surface Autocatalysis in Exterior Domains

Abstract We study the collective dynamics of independent particles that diffuse outside a spherical surface, on which they are replicated with a prescribed catalytic rate. In spatial dimensions three and higher, the transient nature of diffusion creates a competition between autocatalytic branching and escape to infinity, leading to three distinct regimes: subcritical (finite limiting population), critical (algebraic growth), and supercritical (exponential growth). The rotational symmetry of the domain and an explicit form of the single-particle diffusion propagator allow us to fully investigate the time evolution of the population size, including the mean population size, variance, higher-order moments, as well as the whole distribution. In particular, we obtain an explicit form of the distribution at long times and describe a slow, power-law approach to this steady-state limit. We further extend the steady-state analysis to general exterior domains using the Dirichlet-to-Neumann operator, which determines the critical catalytic rate and reveals how boundary geometry affects the population-size distribution.