Asymmetric stochastic transport models with ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ symmetry

TL;DR

This paper introduces new asymmetric stochastic transport models based on the ${\\mathcal{U}}_q(\mathfrak{su}(1,1))$ algebra, revealing duality properties and deriving their diffusion limits and applications in current analysis.

Contribution

It presents novel asymmetric transport processes derived from quantum algebra, including self-dual and dual processes, with explicit diffusion limits and applications.

Findings

The asymmetric Inclusion Process is self-dual.

The diffusion limit is an asymmetric analogue of the Brownian Energy Process.

The asymmetric KMP Process has a symmetric dual process.

Abstract

By using the algebraic construction outlined in \cite{CGRS}, we introduce several Markov processes related to the ${\mathcal{U}}_q(\mathfrak{su}(1,1))$ quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the Brownian Energy Process and which turns out to have the symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.