{
  "id": "2408.15697",
  "version": "v1",
  "published": "2024-08-28T10:52:14.000Z",
  "updated": "2024-08-28T10:52:14.000Z",
  "title": "Analysis of Stochastic Chemical Reaction Networks with a Hierarchy of Timescales",
  "authors": [
    "Lucie Laurence",
    "Philippe Robert"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We investigate a class of stochastic chemical reaction networks with $n{\\ge}1$ chemical species $S_1$, \\ldots, $S_n$, and whose complexes are only of the form $k_iS_i$, $i{=}1$,\\ldots, $n$, where $(k_i)$ are integers. The time evolution of these CRNs is driven by the kinetics of the law of mass action. A scaling analysis is done when the rates of external arrivals of chemical species are proportional to a large scaling parameter $N$. A natural hierarchy of fast processes, a subset of the coordinates of $(X_i(t))$, is determined by the values of the mapping $i{\\mapsto}k_i$. We show that the scaled vector of coordinates $i$ such that $k_i{=}1$ and the scaled occupation measure of the other coordinates are converging in distribution to a deterministic limit as $N$ gets large. The proof of this result is obtained by establishing a functional equation for the limiting points of the occupation measure, by an induction on the hierarchy of timescales and with relative entropy functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-28T10:52:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic chemical reaction networks",
      "timescales",
      "chemical species",
      "coordinates",
      "relative entropy functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}