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

2024/02/11 by Marcus, Michael B., Rosen, Jay
#60E07 #60G15 #60G17 #60G99 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2402.07074

Abstract

Let u(s,t) be a continuous potential density of a symmetric Lévy process or diffusion with state space T killed at T₀, the first hitting time of 0, or at λ\wedge 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 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→ 0}u(t,t)=0, \limsup_{t\downarrow 0}(X_α(t ))/(u(t,t)log log 1/t )≥ 1 , a.s. and \limsup_{t\downarrow 0}(X_α(t ))/(u(t,t)log log 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.

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