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 λ∧ T_0, 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 X_α={X_α(t),t∈ T } be an α-permanental process with kernel v(s,t)=u(s,t)+f(t). Then when lim_t→ 0u(t,t)=0, lim sup_t↓ 0X_α(t )/u(t,t)loglog 1/t ≥ 1 , a.s. and lim sup_t↓ 0X_α(t )/u(t,t)loglog 1/t ≤ 1+C_u,h , a.s. where C_u,μ≤ |μ| is a constant that depends on both u and μ, which is given explicitly, and is different in the different examples.