{
  "id": "2502.09300",
  "version": "v1",
  "published": "2025-02-13T13:14:07.000Z",
  "updated": "2025-02-13T13:14:07.000Z",
  "title": "Optimal response for stochastic differential equations by local kernel perturbations",
  "authors": [
    "Gianmarco del Sarto",
    "Stefano Galatolo",
    "Sakshi Jain"
  ],
  "categories": [
    "math.DS",
    "math.OC"
  ],
  "abstract": "We consider a random dynamical system on $\\mathbb{R}^d$, whose dynamics is defined by a stochastic differential equation. The annealed transfer operator associated with such systems is a kernel operator. Given a set of feasible infinitesimal perturbations $P$ to this kernel, with support in a certain compact set, and a specified observable function $\\phi: \\mathbb{R}^d \\to \\mathbb{R}$, we study which infinitesimal perturbation in $P$ produces the greatest change in expectation of $\\phi$. We establish conditions under which the optimal perturbation uniquely exists and present a numerical method to approximate the optimal infinitesimal kernel perturbation. Finally, we numerically illustrate our findings with concrete examples.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-13T13:14:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37H05",
      "37H30",
      "47B34",
      "82C31",
      "93E03"
    ],
    "keywords": [
      "stochastic differential equation",
      "local kernel perturbations",
      "optimal response",
      "optimal infinitesimal kernel perturbation",
      "infinitesimal perturbation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}