Hamilton-Jacobi-Bellman equations for Chemical Reaction Networks

TL;DR

This paper explores the connection between Hamilton-Jacobi-Bellman equations, large deviations, and Lyapunov functions in Chemical Reaction Networks, highlighting the special case of complex-balanced networks and suggesting links to Kähler geometry.

Contribution

It clarifies how Lyapunov functions for CRNs relate to HJB equations and large deviations, especially for complex-balanced networks, and proposes new research directions involving Kähler geometry.

Findings

Lyapunov functions solve HJB equations for complex-balanced CRNs

Connection established between large deviations and deterministic stability

Potential link between CRN theory and Kähler geometry

Abstract

This is an expository note on large deviations, Hamilton-Jacobi-Bellman (HJB) equations, and the role of the Freidlin-Wentzell quasipotential in Chemical Reaction Networks (CRNs). The note was motivated by observations which identified Lyapunov functions for deterministic descriptions of CRNs by taking appropriate scaling limits of invariant distributions for the corresponding stochastic dynamics. We explain how this is a special case of a classical theory due to Freidlin and Wentzell. We also show that this Lyapunov function is a solution to the HJB partial differential equation if and only if the network is "complex-balanced". The target audience are researchers in the CRN community who are familiar with the Markov process description of CRNs, but do not have the time to invest in learning the technical machinery of large deviations. We conclude by exploring some possible relationships with K\"ahler geometry which suggest an interesting and unexplored research direction.