Let $u(s,t)$ be a continuous potential density of a symmetric L\'evy 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 \[ f(t)=\int_{ T} u(t,v)\,d\mu(v), \] 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 \[ v(s,t)=u(s,t)+f(t). \] Then when $\lim_{ t\to 0}u(t,t)=0$, \[ \limsup_{ t\downarrow 0}\frac{ X_{ \alpha}(t )}{ u(t,t)\log \log 1/t }\ge 1 ,\qquad \text{ a.s.} \] and \[ \limsup_{ t\downarrow 0}\frac{ X_{ \alpha}(t )}{ u(t,t)\log \log 1/t }\le 1+C_{ u,h} ,\qquad \text{ a.s.} \] 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.
Categories: math.PR