{
  "id": "2406.15974",
  "version": "v1",
  "published": "2024-06-23T00:36:22.000Z",
  "updated": "2024-06-23T00:36:22.000Z",
  "title": "Explosion by Killing and Maximum Principle in Symmetric Markov Processes",
  "authors": [
    "Masayoshi Takeda"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Keller and Lenz \\cite{KL} define a concept of {\\it stochastic completeness at infinity} (SCI) for a regular symmetric Dirichlet form $(\\cE,\\cF)$. We show that (SCI) can be characterized probabilistically by using the predictable part $\\zeta^p$ of the life time $\\zeta$ of the symmetric Markov process $X=({\\bf P}_x,X_t)$ generated by $(\\cE,\\cF)$, that is, (SCI) is equivalent to $\\bfP_x(\\zeta=\\zeta^p<\\infty)=0$. We define a concept, {\\it explosion by killing} (EK), by $\\bfP_x(\\zeta=\\zeta^i<\\infty)=1$. Here $\\zeta^i$ is the totally inaccessible part of $\\zeta$. We see that (EK) is equivalent to (SCI) and $\\bfP_x(\\zeta=\\infty)=1$. Let $X^{\\rm res}$ be the {\\it resurrected process} generated by the {\\it resurrected form}, a regular Dirichlet form constructed by removing the killing part from $(\\cE, \\cF)$. Extending work of Masamune and Schmidt (\\cite{MS}), we show that (EK) is also equivalent to the ordinary conservation property of time changed process of $X^{\\rm res}$ by $A^k_t$, where the $A^k_t$ is the positive continuous additive functional in the Revuz correspondence to the killing measure $k$ in the Beurling-Deny formula (Theorem \\ref{ma-sh}). We consider the maximum principle for Schr\\\"odinger-type operator $\\cL^\\mu=\\cL-\\mu$. Here $\\cL$ is the self-adjoint operator associated with $(\\cE,\\cF)$ %with non-local part and $\\mu$ is a Green-tight Kato measure. Let $\\lambda(\\mu)$ be the principal eigenvalue of the trace of $(\\cE,\\cF)$ relative to $\\mu$. We prove that if (EK) holds, then $\\lambda(\\mu)>1$ implies a Liouville property that every bounded solution to $\\cL^\\mu u=0$ is zero quasi-everywhere and that the {\\it refined maximum principle} in the sense of Berestycki-Nirenberg-Varadhan \\cite{BNV} holds for $\\cL^\\mu$ if and only if $\\lambda(\\mu)>1$ (Theorem \\ref{RMP}).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-23T00:36:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric markov process",
      "maximum principle",
      "regular symmetric dirichlet form",
      "equivalent",
      "green-tight kato measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}