Equivalence between nonlinear dynamical systems and urn processes

An equivalence is shown between a large class of deterministic dynamical systems and a class of stochastic processes, the balanced urn processes. These dynamical systems are governed by quasi-polynomial differential systems that are widely used in mathematical modeling while urn processes are actively studied in combinatorics and probability theory. The presented equivalence extends a theorem by Flajolet et al (2006 Discrete Mathematics and Theoretical Computer Science, AG (DMTCS Proc.) pp 59–118) already establishing an isomorphism between urn processes and a particular class of differential systems with monomial vector fields. The present result is based on the fact that such monomial differential systems are canonical forms for more general dynamical systems. (paper)