{
  "id": "2412.15626",
  "version": "v1",
  "published": "2024-12-20T07:34:53.000Z",
  "updated": "2024-12-20T07:34:53.000Z",
  "title": "Stationary states for stable processes with partial resetting",
  "authors": [
    "Tomasz Grzywny",
    "Karol Szczypkowski",
    "Zbigniew Palmowski",
    "Bartosz Trojan"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We study a $d$-dimensional stochastic process $\\mathbf{X}$ which arises from a L\\'evy process $\\mathbf{Y}$ by partial resetting, that is the position of the process $\\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\\mathbf{Y}$ between these two consecutive moments, $c \\in (0, 1)$. We focus on $\\mathbf{Y}$ being a strictly $\\alpha$-stable process with $\\alpha\\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $\\rho_{\\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $\\rho_{\\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $\\alpha$-stable processes for $\\alpha \\in (0,2)$, and $\\alpha$-stable subordinator for $\\alpha\\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\\to +\\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $\\rho_{\\mathbf{Y}}(y)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-20T07:34:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G10",
      "60K40",
      "82C05",
      "82C31",
      "60J35",
      "35K08",
      "60J65",
      "60G51",
      "60G52"
    ],
    "keywords": [
      "stable process",
      "partial resetting",
      "transition density",
      "brownian motion",
      "asymptotic behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}