{
  "abstract": "This paper establishes a local turnpike property for a two-population, second-order mean-field game system with a quadratic Hamiltonian, in which the running cost for each population may depend on both population densities. In the case of two populations, imposing a scalar stability condition separately on each species is not sufficient because it neglects cross-interaction effects. The appropriate requirement is instead a matrix-valued coercivity condition on vector perturbations $(μ_1,μ_2)$, which involves the symmetric part of the linearized interaction matrix together with weighted Poincaré inequalities associated with the two stationary measures. Consequently, in the multi-population framework, diffusion still acts independently on each population, but the interactions are controlled by the full coupling matrix. A naive scalar energy functional fails to capture mixed second-variation terms arising from costs that depend on both densities. We prove that matrix coercivity is precisely equivalent to the strict positivity of this second variation, and we provide a spectral characterization in terms of the principal spectral value of the coupled stationary linearized operator. For linear models with cross-interactions, this yields explicit algebraic conditions that separate stable parameter ranges from unstable ones. In the non-potential framework, the skew-symmetric part of the interaction matrix leaves the coercivity threshold unchanged, but it can induce transient oscillatory dynamics. Moreover, the scalar turnpike theory extends in a natural, though nontrivial, manner to multi-population systems, where stability is governed by the positivity of a coupled matrix-valued second variation that encodes both self-interactions and cross-interactions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.17103",
  "author_total": 3,
  "authors": [
    {
      "name": "Mohamed Ghattassi"
    },
    {
      "name": "Fanze Kong"
    },
    {
      "name": "Nader Masmoudi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3yg92g6eqed7h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.17103",
  "fulltext_url": null,
  "id": "3yg92g6eqed7h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.17103",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7221821416",
  "pdf_url": "https://arxiv.org/pdf/2610.17103",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3yg92g6eqed7h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.17103",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.17103"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Local Turnpike Stability for Two-Population Second-Order Mean-Field Game Systems",
  "title_text": "Local Turnpike Stability for Two-Population Second-Order Mean-Field Game Systems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/3yg92g6eqed7h",
  "venue": "arXiv",
  "versions": []
}