A combinatorial analysis of interacting diffusions

TL;DR

This paper analyzes a class of interacting diffusions with piecewise constant, scale-invariant drifts, linking their invariant measures to polyhedral geometry and combinatorial structures, including Coxeter groups and permutation-based distributions.

Contribution

It introduces a geometric framework connecting invariant measures of interacting diffusions to polyhedral structures and combinatorial fans, enabling explicit descriptions in special cases.

Findings

Invariant measures relate to polyhedral geometry and combinatorial fans.

Explicit invariant measures are derived for Coxeter group-invariant polytopes.

Permutation-based probability distributions emerge in the analysis of interaction models.

Abstract

We consider a particular class of n-dimensional homogeneous diffusions all of which have an identity diffusion matrix and a drift function that is piecewise constant and scale invariant. Abstract stochastic calculus immediately gives us general results about existence and uniqueness in law and invariant probability distributions when they exist. These invariant distributions are probability measures on the $n$-dimensional space and can be extremely resistant to a more detailed understanding. To have a better analysis, we construct a polyhedra such that the inward normal at its surface is given by the drift function and show that the finer structures of the invariant probability measure is intertwined with the geometry of the polyhedra. We show that several natural interacting Brownian particle models can thus be analyzed by studying the combinatorial fan generated by the drift function, particularly when these are simplicial. This is the case when the polyhedra is a polytope that is invariant under a Coxeter group action, which leads to an explicit description of the invariant measures in terms of iid Exponential random variables. Another class of examples is furnished by interactions indexed by weighted graphs all of which generate simplicial polytopes with $n !$ faces. We show that the proportion of volume contained in each component simplex corresponds to a probability distribution on the group of permutations, some of which have surprising connections with the classical urn models.