{
  "id": "2010.01533",
  "version": "v1",
  "published": "2020-10-04T10:29:58.000Z",
  "updated": "2020-10-04T10:29:58.000Z",
  "title": "A maximal $L_p$-regularity theory to initial value problems with time measurable nonlocal operators generated by additive processes",
  "authors": [
    "Jae-Hwan Choi",
    "Ildoo Kim"
  ],
  "comment": "44 pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Let $Z=(Z_t)_{t\\geq0}$ be an additive process with a bounded triplet $(0,0,\\Lambda_t)_{t\\geq0}$. Then the infinitesimal generators of $Z$ is given by time dependent nonlocal operators as follows: \\begin{align*} \\mathcal{A}_Z(t)u(t,x) &=\\lim_{h\\downarrow0}\\frac{\\mathbb{E}[u(t,x+Z_{t+h}-Z_t)-u(t,x)]}{h}=\\int_{\\mathbb{R}^d}(u(t,x+y)-u(t,x)-y\\cdot \\nabla u(t,x)1_{|y|\\leq1})\\Lambda_t(dy). \\end{align*} Suppose that L\\'evy measures $\\Lambda_t$ have a lower bound (Assumption 2.10) and satisfy a weak-scaling property (Assumption 2.11). We emphasize that there is no regularity condition on L\\'evy measures $\\Lambda_t$ and they do not have to be symmetric. In this paper, we establish the $L_p$-solvability to initial value problem (IVP) \\begin{equation} \\label{20.07.15.17.02} \\frac{\\partial u}{\\partial t}(t,x)=\\mathcal{A}_Z(t)u(t,x),\\quad u(0,\\cdot)=u_0,\\quad (t,x)\\in(0,T)\\times\\mathbb{R}^d, \\end{equation} where $u_0$ is contained in a scaled Besov space $B_{p,q}^{s;\\gamma-\\frac{2}{q}}(\\mathbb{R}^d)$ (see Definition 2.8) with a scaling function $s$, exponent $p \\in (1,\\infty)$, $q\\in[1,\\infty)$, and order $\\gamma \\in [0,\\infty)$. We show that IVP is uniquely solvable and the solution $u$ obtains full-regularity gain from the diffusion generated by a stochastic process $Z$. In other words, there exists a unique solution $u$ to IVP in $L_q((0,T);H_p^{\\mu;\\gamma}(\\mathbb{R}^d))$, where $H_p^{\\mu;\\gamma}(\\mathbb{R}^d)$ is a generalized Bessel potential space (see Definition 2.3). Moreover, the solution $u$ satisfies $$ \\|u\\|_{L_q((0,T);H_p^{\\mu;\\gamma}(\\mathbb{R}^d))}\\leq N(1+T^2)\\|u_0\\|_{B_{p,q}^{s;\\gamma-\\frac{2}{q}}(\\mathbb{R}^d)}, $$ where $N$ is independent of $u$, $u_0$, and $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-04T10:29:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S10",
      "60H30",
      "47G20",
      "42B25"
    ],
    "keywords": [
      "time measurable nonlocal operators",
      "initial value problem",
      "additive process",
      "regularity theory",
      "levy measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}