{
  "id": "2312.11438",
  "version": "v2",
  "published": "2023-12-18T18:39:51.000Z",
  "updated": "2024-04-04T03:30:39.000Z",
  "title": "Nonlocal Approximation of Slow and Fast Diffusion",
  "authors": [
    "Katy Craig",
    "Matt Jacobs",
    "Olga Turanova"
  ],
  "categories": [
    "math.AP",
    "math.PR"
  ],
  "abstract": "Motivated by recent work on approximation of diffusion equations by deterministic interacting particle systems, we develop a nonlocal approximation for a range of linear and nonlinear diffusion equations and prove convergence of the method in the slow, linear, and fast diffusion regimes. A key ingredient of our approach is a novel technique for using the 2-Wasserstein and dual Sobolev gradient flow structures of the diffusion equations to recover the duality relation characterizing the pressure in the nonlocal-to-local limit. Due to the general class of internal energy densities that our method is able to handle, a byproduct of our result is a novel particle method for sampling a wide range of probability measures, which extends classical approaches based on the Fokker-Planck equation beyond the log-concave setting.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-04-04T03:30:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A15",
      "35Q70",
      "35Q35",
      "35Q62",
      "82C22"
    ],
    "keywords": [
      "nonlocal approximation",
      "dual sobolev gradient flow structures",
      "internal energy densities",
      "fast diffusion regimes",
      "nonlinear diffusion equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}