Kernel Method for Stationary Tails: From Discrete to Continuous

In this paper, we will extend the kernel method employed for two-dimensional discrete random walks with reflecting boundaries. We provide a survey on how the kernel method, together with singularity analysis, can be applied to study asymptotic properties of stationary measures for continuous random walks. Specifically, we show that all key techniques in the kernel method for the discrete random walks can be extended for the continuous case. We use the semimartingale reflecting Brownian motion model as an example. We detail all key components in the analysis for a boundary measure, including analytic continuation, interlace between the two boundary measures, and singularity analysis. These properties allow us to completely characterize the tail behaviour of the boundary measure through a Tauberian-like theorem.