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

TL;DR

This paper explores the probabilistic origins of multiscale Gibbs measures by introducing reinforced stochastic processes and establishing a large-deviation principle that explains their structural asymmetry.

Contribution

It introduces the reinforced multinomial process and provides a probabilistic framework for understanding multiscale Gibbs measures through large deviations.

Findings

Reinforced multinomial process characterized and analyzed.

Large-deviation principle established for the process.

Entropy imbalance explained via probabilistic mechanism.

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.