Local Asymptotics for Entrance Laws of Reflected Lévy Excursions

Source abstract

This paper establishes uniform local large-deviation asymptotics for the entrance law $\underline n_t(dx)$ of excursions of a Lévy process with negative drift and locally regularly varying Lévy measure $ν$, reflected at its running infimum. In the fixed-time regime, we prove that as $x\to\infty$, $$\underline n_t\big((x,x+δ]\big) \sim \underline n(ζ\wedge t)\, ν\big((x,x+δ]\big),$$ uniformly in $δ\in[δ_0,\infty]$ for every $δ_0>0$. Our principal result concerns the large-time regime and shows that, as $t\to\infty$, $$\underline n_t\big((x,x+δ]\big)\sim \int_x^{x+δ} V(z) \,ν(βt+dz),$$ uniformly in $x\geq0$ and $δ\in[δ_0,\infty]$, where $V$ denotes the renewal function. Using a fluctuation-theoretic decomposition of the killed semigroup, we further derive corresponding uniform local large-deviation asymptotics for the Lévy process killed upon its first passage below zero.