Yaglom limit for stable processes in cones

TL;DR

This paper analyzes the tail behavior of the first exit time for isotropic stable processes in Lipschitz cones, establishing Yaglom limits and entrance laws, with spectral representations for specific cases.

Contribution

It provides the first detailed asymptotic analysis of exit times and Yaglom limits for stable processes in cones, including spectral representations for symmetric Cauchy processes.

Findings

Asymptotics of the tail distribution of exit times are derived.

Yaglom limits are established for killed stable processes in cones.

Spectral representations are obtained for specific stable processes.

Abstract

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha$-stable L\'evy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.