# Stochastically modeled weakly reversible reaction networks with a single linkage class

## TL;DR

This paper proves that certain stochastically modeled weakly reversible reaction networks with a single linkage class are positive recurrent, under specific conditions, extending known deterministic properties to stochastic systems.

## Contribution

It establishes the positive recurrence of stochastically modeled weakly reversible reaction networks with a single linkage class under new assumptions, using a novel proof technique.

## Key findings

- Stochastically modeled systems are positive recurrent under specified conditions.
- A new proof technique involving the recurrence of the embedded Markov chain is developed.
- The results extend deterministic boundedness properties to stochastic reaction networks.

## Abstract

It has been known for nearly a decade that deterministically modeled reaction networks that are weakly reversible and consist of a single linkage class have trajectories that are bounded from both above and below by positive constants (so long as the initial condition has strictly positive components). It is conjectured that the stochastically modeled analogs of these systems are positive recurrent. We prove this conjecture in the affirmative under the following additional assumptions: (i) the system is binary and (ii) for each species, there is a complex (vertex in the associated reaction diagram) that is a multiple of that species. To show this result, a new proof technique is developed in which we study the recurrence properties of the $n-$step embedded discrete time Markov chain.