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

Let $u(s,t)$ be a continuous potential density of a symmetric L\'evy process or diffusion with state space T killed at T₀, the first hitting time of $0$, or at λ T₀, where λ is an independent exponential time. Let \[ f(t)=∫T u(t,v)\,dμ(v), \] where μ is a finite positive measure on T. Let Xα=α(t),t∈ T \ be an α-permanental process with kernel \[ v(s,t)=u(s,t)+f(t). \] Then when limt→ 0u(t,t)=0, \[ t↓ 0{Xα(t )}{u(t,t)log log 1/t }≥ 1 , a.s. \] and \[ t↓ 0{Xα(t )}{u(t,t)log log 1/t }≤ 1+Cu,h , a.s. \] where Cu,μ≤ |μ| is a constant that depends on both u and μ, which is given explicitly, and is different in the different examples.