{
  "id": "2309.06970",
  "version": "v1",
  "published": "2023-09-13T14:04:29.000Z",
  "updated": "2023-09-13T14:04:29.000Z",
  "title": "Exponential ergodicity of continuous-time Markov chains on $\\mathbb Z^d$, with applications to stochastic reaction networks",
  "authors": [
    "David F. Anderson",
    "Daniele Cappelletti",
    "Wai-Tong Louis Fan",
    "Jinsu Kim"
  ],
  "comment": "44 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "This paper provides a new method that can be used to determine when an ergodic continuous-time Markov chain on $\\mathbb Z^d$ converges exponentially fast to its stationary distribution in $L^2$. Specifically, we provide general conditions that guarantee the positivity of the spectral gap. Importantly, our results do not require the assumption of time-reversibility of the Markov model. We then apply our new method to the well-studied class of stochastically modeled reaction networks. Notably, we show that each complex-balanced model that is also \"open\" has a positive spectral gap, and is therefore exponentially ergodic. We further illustrate how our results can be applied for models that are not necessarily complex-balanced. Moreover, we provide an example of a detailed-balanced (in the sense of reaction network theory), and hence complex-balanced, stochastic reaction network that is not exponentially ergodic. We believe this to be the first such example in the literature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-13T14:04:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J27",
      "60J28"
    ],
    "keywords": [
      "stochastic reaction network",
      "exponential ergodicity",
      "ergodic continuous-time markov chain",
      "spectral gap",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}