From entropic constraints to reinforced processes: a probabilistic origin of multiscale measures

Abstract: We investigate multiscale Gibbs measures from a variational and probabilistic viewpoint, focusing on the structural asymmetry among conditional entropies that characterizes their construction. We show how this asymmetry emerges both from variational principles with entropic constraints and from stochastic processes with reinforcement. We thus introduce the reinforced multinomial process and prove a large-deviation principle for its empirical histogram. The associated rate function reproduces precisely the entropy imbalance defining multiscale measures, thereby providing a genuine probabilistic mechanism for their emergence. The reinforced multinomial process thus offers a simple and rigorous stochastic foundation for multiscale Gibbs structures.