{
  "id": "1705.01243",
  "version": "v1",
  "published": "2017-05-03T03:13:17.000Z",
  "updated": "2017-05-03T03:13:17.000Z",
  "title": "An $L_p$-theory for diffusion equations related to stochastic processes with non-stationary independent increment",
  "authors": [
    "Ildoo Kim",
    "Kyeong-Hun Kim",
    "Panki Kim"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $X=(X_t)_{t \\ge 0}$ be a stochastic process which has an (not necessarily stationary) independent increment on a probability space $(\\Omega, \\mathbb{P})$. In this paper, we study the following Cauchy problem related to the stochastic process $X$: \\label{main eqn} \\frac{\\partial u}{\\partial t}(t,x) = \\cA(t)u(t,x) +f(t,x), \\quad u(0,\\cdot)=0, \\quad (t,x) \\in (0,T) \\times \\mathbf{R}^d, \\end{align} where $f \\in L_p( (0,T) ; L_p(\\mathbf{R}^d))=L_p( (0,T) ; L_p)$ and \\begin{align*} \\cA(t)u(t,x) = \\lim_{h \\downarrow 0}\\frac{\\mathbb{E}\\left[u(t,x+X_{t+h}-X_t)-u(t,x)\\right]}{h}. We provide a sufficient condition on $X$ to guarantee the unique solvability of equation (\\ref{ab main}) in $L_p\\left( [0,T] ; H^\\phi_{p}\\right)$, where $H^\\phi_{p}$ is a $\\phi$-potential space on $\\mathbf{R}^d$ . Furthemore we show that for this solution, \\| u\\|_{L_p\\left( [0,T] ; H^\\phi_{p}\\right)} \\leq N \\|f\\|_{L_p\\left( [0,T] ; L_p\\right)}, where $N$ is independent of $u$ and $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-03T03:13:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stochastic process",
      "non-stationary independent increment",
      "diffusion equations",
      "cauchy problem",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}