{
  "id": "2405.14429",
  "version": "v1",
  "published": "2024-05-23T10:57:56.000Z",
  "updated": "2024-05-23T10:57:56.000Z",
  "title": "Invariance of Gaussian RKHSs under Koopman operators of stochastic differential equations with constant matrix coefficients",
  "authors": [
    "Friedrich Philipp",
    "Manuel Schaller",
    "Karl Worthmann",
    "Sebastian Peitz",
    "Feliks Nüske"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR",
    "math.DS"
  ],
  "abstract": "We consider the Koopman operator semigroup $(K^t)_{t\\ge 0}$ associated with stochastic differential equations of the form $dX_t = AX_t\\,dt + B\\,dW_t$ with constant matrices $A$ and $B$ and Brownian motion $W_t$. We prove that the reproducing kernel Hilbert space $\\bH_C$ generated by a Gaussian kernel with a positive definite covariance matrix $C$ is invariant under each Koopman operator $K^t$ if the matrices $A$, $B$, and $C$ satisfy the following Lyapunov-like matrix inequality: $AC^2 + C^2A^\\top\\le 2BB^\\top$. In this course, we prove a characterization concerning the inclusion $\\bH_{C_1}\\subset\\bH_{C_2}$ of Gaussian RKHSs for two positive definite matrices $C_1$ and $C_2$. The question of whether the sufficient Lyapunov-condition is also necessary is left as an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-23T10:57:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic differential equations",
      "constant matrix coefficients",
      "gaussian rkhss",
      "invariance",
      "reproducing kernel hilbert space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}