{
  "id": "2505.06183",
  "version": "v1",
  "published": "2025-05-09T16:49:56.000Z",
  "updated": "2025-05-09T16:49:56.000Z",
  "title": "Long time behaviour of Mean Field Games with fractional diffusion",
  "authors": [
    "Olav Ersland",
    "Espen Robstad Jakobsen",
    "Alessio Porretta"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper we study the long time behaviour of mean field games systems with fractional diffusion, modeling the case that the individual dynamics of the players is driven by independent jump processes and controlled through the drift term, while being confined by an external field in order to guarantee ergodicity. In the case of globally Lipschitz, locally uniformly convex Hamiltonian, and weakly coupled costs satisfying the Lasry-Lions monotonicity condition, we prove that there is a unique solution $(u_T,m_T)$ to the mean field game problem in $(0,T)$ and we show that, if $T$ is sufficiently large, $(u_T,m_T)$ satisfies the so-called turnpike property, namely it is exponentially close to the (unique) stationary ergodic state for any proportionally long intermediate time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-05-09T16:49:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "long time behaviour",
      "fractional diffusion",
      "mean field games systems",
      "mean field game problem",
      "proportionally long intermediate time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}