On Uniform Functions on Configuration Spaces of Large Scale Interacting Systems

TL;DR

This paper advances the theory of uniform functions on configuration spaces for large-scale stochastic systems, enabling the characterization of conserved quantities without stationary distributions and accommodating multiple state transitions.

Contribution

It generalizes the concept of uniform functions to be independent of base states and extends the interaction framework to multiple transitions, especially under exchangeability.

Findings

Uniform functions characterize conserved quantities without stationary distributions.

Exchangeable interactions allow expressing global conserved quantities as sums of local ones.

The theory is now independent of irreducible quantification assumptions.

Abstract

Stochastic large scale interacting systems can be studied via the observables, i.e. functions on the underlying configuration space. In our previous article, we introduced the concept of uniform functions, which are suitable class of functions on configuration spaces underlying stochastic systems on infinite graphs. An important consequence is the successful characterization of conserved quantities without introducing the notion of stationary distributions. In this article, we further develop the theory of uniform functions and construct the theory independent of any choice of a base state. Furthermore, we generalize the notion of interactions given in our previous article to accommodate the case where there are multiple possible state transitions on adjacent vertices. We then prove that if the interaction is exchangeable, then any uniform function which gives a global conserved quantity can be expressed as a sum of local conserved quantities of the interaction. Contrary to our previous article, we do not need to assume that the interaction is irreducibly quantified. This shows that our theory of uniform functions on configuration spaces over infinite graphs with transition structure given by an exchangeable interaction is a natural framework to study general stochastic large scale interacting systems. While some of the ideas in this article are based on our previous article, the article is logically independent and self-contained.