Local moduli of continuity for permanental processes that are zero at zero

Abstract

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.