[Paper Review] Duality functions and stationary product measures

Frank Redig, Federico Sau|arXiv (Cornell University)|Feb 23, 2017

Stochastic processes and statistical mechanics12 references3 citations

TL;DR

This paper establishes a constructive duality framework linking stationary product measures to factorized (self-)duality functions in stochastic processes. By introducing a novel generating function approach, it systematically derives all (self-)duality functions for zero-range, symmetric inclusion/exclusion, and interacting diffusion processes like the Brownian energy process, including new dualities in continuous state spaces.

ABSTRACT

We investigate a general relation between stationary product measures and factorized (self-)duality functions. This yields a constructive approach to find (self-)duality functions from the stationary product measures. We introduce a new generating function approach which simplifies the search for (self-)duality functions. Using these methods, we find all (self-)duality functions for zero range processes, as well as for symmetric inclusion and exclusion processes. Moreover, we find new (self-)dualities for interacting diffusion processes such as the Brownian energy process, where both the process and its dual are in continuous variables.

Motivation & Objective

• To establish a general mathematical relationship between stationary product measures and (self-)duality functions in interacting particle systems.
• To address the challenge of systematically constructing (self-)duality functions, which are crucial for analyzing equilibrium and non-equilibrium dynamics.
• To develop a unified, constructive framework that simplifies the search for duality functions across diverse stochastic processes.
• To extend the theory to continuous-state processes, such as the Brownian energy process, where both the process and its dual evolve in continuous variables.

Proposed method

• Introduce a generating function approach that encodes duality functions in terms of generating functions derived from stationary product measures.
• Use the factorized structure of duality functions to reduce the problem to solving a system of equations based on the stationary measure's moments.
• Apply the method to zero-range processes by deriving duality functions from their product-form stationary measures.
• Extend the framework to symmetric inclusion and exclusion processes by leveraging their known stationary product measures and symmetry properties.
• Generalize the approach to continuous-state processes by adapting the generating function formalism to diffusion-type dynamics.
• Verify the resulting duality functions by checking the duality equation through direct substitution and moment matching.

Experimental results

Research questions

• RQ1 How can stationary product measures be systematically used to construct (self-)duality functions in interacting particle systems?
• RQ2 What is the role of generating functions in simplifying the derivation of duality functions across different classes of stochastic processes?
• RQ3 Can the proposed method recover all known (self-)duality functions for zero-range and symmetric inclusion/exclusion processes?
• RQ4 Are there new (self-)dualities that can be discovered for continuous-state interacting diffusion processes like the Brownian energy process?
• RQ5 How does the factorized structure of duality functions emerge from the underlying stationary product measures?

Key findings

• The generating function approach provides a systematic and constructive method to derive (self-)duality functions from stationary product measures, significantly simplifying the search process.
• All known (self-)duality functions for zero-range processes are recovered using this method, confirming its completeness for these models.
• The method successfully identifies new (self-)dualities for symmetric inclusion and exclusion processes, extending the known duality framework to these systems.
• For the Brownian energy process, the method yields new (self-)dualities where both the original process and its dual are in continuous state space, a novel contribution to the field.
• The framework reveals a deep structural connection between the factorized form of duality functions and the product-form nature of stationary measures.
• The approach is generalizable to a wide class of stochastic processes, including both discrete and continuous-state systems, demonstrating broad applicability.