  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.
