Dynamics of the Chemical Master Equation, a strip of chains of equations in d-dimensional space

TL;DR

This paper studies a multi-chain Chemical Master Equation model with inter-chain transitions, applying Hamilton-Jacobi methods to derive exact results for distribution dynamics in the infinite-particle limit.

Contribution

It introduces a multi-chain model with inter-chain transitions and provides exact analytical results for distribution dynamics using Hamilton-Jacobi equations.

Findings

Derived exact results for the maximum of the distribution.

Obtained variance dynamics in the infinite-particle limit.

Extended the Chemical Master Equation framework to multi-chain systems.

Abstract

We investigate the multi-chain version of the Chemical Master Equation, when there are transitions between different states inside the long chains, as well as transitions between (a few) different chains. In the discrete version, such a model can describe the connected diffusion processes with jumps between different types. We apply the Hamilton-Jacobi equation to solve some aspects of the model. We derive exact {(in the limit of infinite number of particles)} results for the dynamic of the maximum of the distribution and the variance of distribution.