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

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 $$f(t)={\int }_{T}u(t,v)d\mu (v),$$ 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 $$v(s,t)=u(s,t)+f(t)\mathrm{.}$$ Then when ${\lim }_{t\to 0}u(t,t)=0$, $$\underset{t\downarrow 0}{\mathrm{lim\hspace{0.17em}sup}}\frac{X_{\alpha }(t)}{u(t,t)\log \log 1\mathrm{/}t}\ge 1,\text{a.s.}$$ and $$\underset{t\downarrow 0}{\mathrm{lim\hspace{0.17em}sup}}\frac{X_{\alpha }(t)}{u(t,t)\log \log 1\mathrm{/}t}\le 1+C_{u,h},\text{a.s.}$$ where $C_{u,\mu }\le \mathrm{\mid }\mu \mathrm{\mid }$ is a constant that depends on both $u$ and $\mu$, which is given explicitly, and is different in the different examples.