## Abstract

We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as 1 / ϵ, and we prove the convergence in the fast-reaction limit ϵ→ 0. We establish a Γ -convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the Γ -convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

Original language | English |
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Pages (from-to) | 865-906 |

Number of pages | 42 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 35 |

Issue number | 1 |

Early online date | 22 Jul 2021 |

DOIs | |

Publication status | Published - Mar 2023 |

### Bibliographical note

Funding Information:This research has been funded by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project C08. We thank Robert Patterson for the useful discussions.

### Funding

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, Project C08. We thank Robert Patterson for the useful discussions.

Funders | Funder number |
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Deutsche Forschungsgemeinschaft | CRC 1114 |

## Keywords

- Fast reaction limit
- Linear network
- Quasi-steady state approximation
- Rate functional
- Γ -Convergence