## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 28, Number 3 (2018), 1821-1855.

### Large deviations theory for Markov jump models of chemical reaction networks

Andrea Agazzi, Amir Dembo, and Jean-Pierre Eckmann

#### Abstract

We prove a sample path Large Deviation Principle (LDP) for a class of jump processes whose rates are not uniformly Lipschitz continuous in phase space. Building on it, we further establish the corresponding Wentzell–Freidlin (W-F) (infinite time horizon) asymptotic theory. These results apply to jump Markov processes that model the dynamics of chemical reaction networks under mass action kinetics, on a microscopic scale. We provide natural sufficient topological conditions for the applicability of our LDP and W-F results. This then justifies the computation of nonequilibrium potential and exponential transition time estimates between different attractors in the large volume limit, for systems that are beyond the reach of standard chemical reaction network theory.

#### Article information

**Source**

Ann. Appl. Probab., Volume 28, Number 3 (2018), 1821-1855.

**Dates**

Received: January 2017

Revised: August 2017

First available in Project Euclid: 1 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1527840033

**Digital Object Identifier**

doi:10.1214/17-AAP1344

**Mathematical Reviews number (MathSciNet)**

MR3809478

**Zentralblatt MATH identifier**

06919739

**Subjects**

Primary: 60F10: Large deviations 80A30: Chemical kinetics [See also 76V05, 92C45, 92E20]

Secondary: 37B25: Lyapunov functions and stability; attractors, repellers 60J75: Jump processes

**Keywords**

Large deviation principle Wentzell–Freidlin theory jump Markov processes chemical reaction networks Lyapunov functions toric jets

#### Citation

Agazzi, Andrea; Dembo, Amir; Eckmann, Jean-Pierre. Large deviations theory for Markov jump models of chemical reaction networks. Ann. Appl. Probab. 28 (2018), no. 3, 1821--1855. doi:10.1214/17-AAP1344. https://projecteuclid.org/euclid.aoap/1527840033