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Let \(u(s,t)\) be a continuous potential density of a symmetric Lévy process or diffusion with state space T killed at \(T_{0}\) , the first hitting time of 0, or at \(\lambda \wedge T_{0}\) , where \(\lambda \) is an independent exponential time. Let \(\displaystyle f(t)=\int _{T} u(t,v)\,d\mu (v), \) where \(\mu \) is a finite positive measure on T. Let \(X_{\alpha }=\{X_{\alpha }(t),t\in T \}\) be an \(\alpha \) -permanental process with kernel \(\displaystyle v(s,t)=u(s,t)+f(t). \) Then when \(\lim _{t\to 0}u(t,t)=0\) , \(\displaystyle \limsup _{t\downarrow 0}\frac {X_{\alpha }(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \) and \(\displaystyle \limsup _{t\downarrow 0}\frac {X_{\alpha }(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,\mu } ,\qquad \text{a.s.} \) where \(C_{u,\mu }\le |\mu |\) is a constant that depends on both u and \(\mu \) , which is given explicitly, and is different in the different examples.

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Local Moduli of Continuity for Permanental Processes That Are Zero at Zero

  • 会议论文

  • 发表日期：

    2026年

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  • Michael B. Marcus,
  • Jay Rosen

摘要

Let \(u(s,t)\) be a continuous potential density of a symmetric Lévy process or diffusion with state space T killed at \(T_{0}\) , the first hitting time of 0, or at \(\lambda \wedge T_{0}\) , where \(\lambda \) is an independent exponential time. Let \(\displaystyle f(t)=\int _{T} u(t,v)\,d\mu (v), \) where \(\mu \) is a finite positive measure on T. Let \(X_{\alpha }=\{X_{\alpha }(t),t\in T \}\) be an \(\alpha \) -permanental process with kernel \(\displaystyle v(s,t)=u(s,t)+f(t). \) Then when \(\lim _{t\to 0}u(t,t)=0\) , \(\displaystyle \limsup _{t\downarrow 0}\frac {X_{\alpha }(t )}{u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{a.s.} \) and \(\displaystyle \limsup _{t\downarrow 0}\frac {X_{\alpha }(t )}{u(t,t)\log \log 1/t }\le 1+C_{u,\mu } ,\qquad \text{a.s.} \) where \(C_{u,\mu }\le |\mu |\) is a constant that depends on both u and \(\mu \) , which is given explicitly, and is different in the different examples.